general vector and dyadic analysis 0780334132

Generalized Vectorand Dyadic Analysis

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Generalized Vectorand Dyadic Analysis

Applied Mathematics in Field TheorySecond Edition

-----..,--Chen-To Tai

Professor EmeritusRadiation Laboratory

Department ofElectrical Engineeringand Computer ScienceUniversity ofMichigan

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Library of Congress Cataloging-in-Publication Data

Tai, Chen-To (date)Generalized vector and dyadic analysis: applied mathematics in

field theory / Chen-To Tai -2nd ed.p. ern.

Includes bibliographical references and index.ISBN 0-7803-3413-2 (cloth)1. Vector analysis. I. Title.

QA433.T3 1997515'.63-dc21 96-29863

CIP

Contents

Preface to the Second Edition xi

Preface to the First Edition xiii

Acknowledgments for the First Edition xv

1 Vector and Dyadic Algebra 11-1 Representations ofVector Functions1-2 Products and Identities 41-3 Orthogonal Transformation of Vector

Functions 81-4 Transform of Vector Products 141-5 Definition of Dyadics and Tensors 161-6 Classification of Dyadics 171-7 Products Between Vectors and Dyadics 19

2 Coordinate Systems 232-1 General Curvilinear System (GCS)2-2 Orthogonal Curvilinear System (OCS)

2328

vii

viii

2-3 Derivatives of Unit Vectors in OCS 332-4 Dupin Coordinate System 352-5 Radii of Curvature 37

3 Line Integrals, Surface Integrals, and VolumeIntegrals 433-1 Differential Length, Area, and Volume 433-2 Classification of Line Integrals 443-3 Classification of Surface Integrals 483-4 Classification of Volume Integrals 56

4 Vector Analysis in Space 584-1 Symbolic Vector And Symbolic Vector

Expressions 584-2 Differential Formulas ofthe Symbolic Expression

in the Orthogonal Curvilinear Coordinate Systemfor Gradient, Divergence, and Curl 61

4-3 Invariance of the Differential Operators 654-4 Differential Formulas of the Symbolic

Expression in the General CurvilinearSystem 69

4-5 Alternative Definitions of Gradientand Curl 75

4-6 The Method of Gradient 784-7 Symbolic Expressions with Two Functions

and the Partial Symbolic Vectors 814-8 Symbolic Expressions with Double Symbolic

Vectors 864-9 Generalized Gauss Theorem in Space 91

4-10 Scalar and Vector Green's Theorems 934-11 Solenoidal Vector, Irrotational Vector,

and Potential Functions 95

5 Vector Analysis on Surface 995-1 Surface SymbolicVector and Symbolic Expression

for a Surface 995-2 Surface Gradient, Surface Divergence, and Surface

Curl 1015-2-1 Surface Gradient 101

Contents

Con~n~ ~

5-2-2 Surface Divergence 1025-2-3 Surface Curl 103

5-3 Relationship Between the Volume and SurfaceSymbolic Expressions 104

5-4 Relationship Between Weatherbum's SurfaceFunctions and the Functions Definedin the Method of Symbolic Vector 104

5-5 Generalized Gauss Theorem for a Surface 1065-6 Surface Symbolic Expressions with a Single

Symbolic Vector and Two Functions 1115-7 Surface Symbolic Expressions with Two Surface

Symbolic Vectors and a Single Function 113

6 Vector Analysis of Transport Theorems6-1 Helmholtz Transport Theorem 1166-2 Maxwell Theorem and Reynolds Transport

Theorem 119

116

7 Dyadic Analysis 1217-1 Divergence and Curl of Dyadic Functions

and Gradient of Vector Functions 1217-2 Dyadic Integral Theorems 124

8 A Historical StUdy of Vector Analysis8-1 Introduction 1278-2 Notations and Operators 129

8-2-1 Past and Present Notationsin Vector Analysis 129

8-2-2 QuatemionAnalysis 1318-2-3 Operators 132

8-3 The Pioneer Works of J. Willard Gibbs(1839-1903) 1358-3-1 Two Pamphlets Printed in 1881

and 1884 1358-3-2 Divergence and Curl Operators

and Their New Notations 1388-4 Book by Edwin Bidwell Wilson Founded

Upon the Lectures of J. Willard Gibbs 141

127

x Contents

8-4-1 Gibbs's Lecture Notes 1418-4-2 Wilson's Book 1418-4-3 The Spread of the Formal

ScalarProduct (FSP) and FormalVector Product (FVP) 146

8-5 Vin the Hands of Oliver Heaviside(1850-1925) 149

8-6 Shilov's Formulation of Vector Analysis 1518-7 Formulations in Orthogonal Curvilinear

Systems 1528-7-1 Two Examples from the Book

by Moon and Spencer 1528-7-2 A Search for the Divergence

Operator in OrthogonalCurvilinear CoordinateSystems 154

8-8 The Use of V to Derive Vector Identities 1558-9 A Recasting of the Past Failures by the Method

of Symbolic Vector 1578-9-1 In Retrospect 159

Appendix A

Appendix B

Appendix C

Appendix D

Appendix E

Appendix F

References

Index 189

Transformation Between UnitVectors 161

Vector and Dyadic Identities 165

Integral Theorems 169

Relationships Between IntegralTheorems 170

Vector Analysis in the SpecialTheory of Relativity 174

Comparison of the Nomenclaturesand Notations of the QuantitiesUsed in This Book and in the Bookby Stratton 181

185

Prefaceto the

Second Edition

After the publication of the first edition of this book (IEEE Press, 1992), severalprofessional friends commented that I should have used the new notations forthe divergence and the curl of a vector function, namely VF and V'F, instead ofpreserving Gibbs's notations V· F and V x F, commonly used in many books. Inthis edition, I have added a chapter on the history of vector analysis to pointout more emphatically the contradiction and the confusion resulting from themisinterpretation of Gibbs's notations. It seems beyond doubt that the adoptionof the new notations is preferable from the logical point of view.

In 1994, I had the opportunity to teach a course at the University of Michiganin which I used the method ofsymbolic vector and the new notations to teach vectoranalysis. The reaction from the students was very favorable. Similar views havebeen communicated to me by colleagues from other institutions. My motivationfor revising the book is principally due to these encouragements.

In addition to the overhaul of the notations, the present edition considerablyexpands the coverage. The method of symbolic vector is now formulated notonly in the curvilinear orthogonal system, but also in the general nonorthogonalcurvilinear system. The reciprocal base systems, originally introduced by Gibbs,have been used very effectively in the formulation. New vector theorems andvector and dyadic identities have been derived to make the list as complete aspossible. The relationship between dyadic analysis and tensor analysis has alsobeen explained. The transformation of electromagnetic field vectors based on the

xi

xii Preface to the Second Edition

special theory of relativity is explained by both the conventional method, usingdifferential calculus, and the more sophisticated method due to Sommerfeld, withthe aid of four-dimensional vector analysis.

I am most grateful to Professor Phillip Alexander of the UDiversity ofWindsorand to my colleague, Mr. Richard Carnes of the University of Michigan, fortechnical assistance and manuscript editing. For all their help and encouragement,I want to thank Dr. John H. Bryant, Professor Fawwaz T. Ulaby, Professor JohnH. Kraus, Mr. Jui-Ching Cheng, Mrs. Carol Truszkowski, Ms. Patricia Wolfe, Dr.Roger DeRoo, and Dr. Jian Gong.

lowe my gratitude to Prof. Donald G. Dudley, Editor of the IEEE Press/OUPSeries on Electromagnetic Wave Theory, and to Mr. Dudley Kay, IEEE Press, andhis staff, particularly, Ms. Denise Phillip, for valuable suggestions in the productionof this book by the IEEE Press.

CHEN-ToTAlAnn Arbor, MI

Publisher's Acknowledgement

The IEEE Press and the Editor of the IEEE Press/OUP Series on Electromagnetic Wave Theory, Donald Dudley, would like to thank Associate Editor,Professor Ehud Heyman, for coordinating the reviews for this book. We wouldalso like to thank the anonymous reviewers for their helpful and incisive reviews.

Preface to the First Edition

Mathematics is a language.

The whole is simpler than its parts.

Anyone having these desires will make these researches.

-J. Willard Gibbs

This monograph is mainly based on the author's recent work on vector analysis anddyadic analysis. The book is divided into two main topics: Chapters 1-6 covervector analysis, while Chapter 7 is exclusively devoted to dyadic analysis. Onthe subject of vector analysis, a new symbolic method with the aid of a symbolicvector is the main feature of the presentation. By means of this method, theprincipal topics in vector analysis can be developed in a systematic way. All vectoridentities can be derived by an algebraic manipulation of expressions with twopartial symbolic vectors without actually performing any differentiation. Integraltheorems are formulated under one roof with the aid of a generalized Gausstheorem. Vector analysis on a surface is treated in a similar manner. Some basicdifferential functions on a surface are defined; they are different from the surfacefunctions previously defined by Weatherbum, although the two sets are intimatelyrelated. Their relations are discussed in great detail. The advantage of adoptingthe surface functions advocated in this work is the simplicity of formulating thesurface integral theorems based on these newly defined functions.

The scope of topics covered in this book on vector analysis is comparable tothose found in the books by Wilson [21], Gans [4], and Phillips [11]. However,the topics on curvilinear orthogonal systems have been treated in great detail.One important feature of this work is the unified treatment of many theoremsand formulas of similar nature, which includes the invariance principle of thedifferential operators for the gradient, the divergence, and the curl, and the relationsbetween various integral theorems and transport theorems. Some quite useful

xiii

xiv Preface to the First Edition

topics are found in this book, which include the derivation of several identitiesinvolving the derivatives of unit vectors, and the relations between the unit vectorsof various coordinate systems based on a method of gradient.

Tensor analysis is outside the scope of this book. There are many excellentbooks treating this subject. Since dyadic analysis is now used quite frequently inengineering sciences, a chapter on this subject, which is closely related to tensoranalysis in a three-dimensional Euclidean space, may be timely.

As a whole, it is hoped that this book may be useful to instructors and studentsin engineering and physical sciences who wish to teach and to learn vector analysisin a systematic manner based on a new method with a clearpicture ofthe constituentstructure of this mature science not critically studied in the past few decades.

Acknowledgmentsfor the First Edition

Without the encouragement which I received from my wife and family, and theloving innocent interference from my grandchildren, this work would never havebeen completed. I would like to express my gratitude to President Dr. Qian WeiChang for his kindness in inviting me as a Visiting Professor at The ShanghaiUniversity of Technology in the Fall of 1988 when this work was started. Most ofthe writing was done when I was a Visiting Professor at the Chung Cheng Instituteof Technology, Taiwan, in the Spring of 1990. I am indebted to President Dr. ChenChwan-Haw, Prof. Bor Sheau-Shong, and Prof. Kuei Ching-Ping for the invitation.

The assistance of Prof. Nenghang Fang of The Nanjing Institute of ElectronicTechnology, China, currently a Visiting Scholar at The University of Michigan,has been most valuable. His discussion with me about the Russian work on vectoranalysis was instrumental in stimulating my interest to formulate the symbolicvector method introduced in this book. Without his participation in the early stageof this work, the endeavor could not have begun. He has kindly checked all theformulas and made numerous suggestions. I am grateful to many colleagues foruseful information and valuable comments. They include: Prof. J. Van Bladelof The University of Gent, Prof. Jed Z. Buchwald of The University of Toronto,Prof. W. Jack Cunningham of Yale University, Prof. Walter R. Debler and Prof.James F. Driscoll ofThe University of Michigan, Prof. John D. Kraus and Prof. H.C. Ko of The Ohio State University, and Prof. C. Truesdell of The Johns HopkinsUniversity. My dear old friend Prof. David K. Cheng ofSyracuse University kindlyedited the manuscript and suggested the title of the book. The teachings of Prof.

xv

xvi Acknowledgments for the First Edition

Chih-Kung Jen of The Johns Hopkins Applied Physics Laboratory, formerly ofTsing Hua University, and Prof. Ronold W. P. King of Harvard University remainthe guiding lights in my search for knowledge. Without the help of Ms. BonnieKidd, Dr. Jian-Ming Jin, and Dr. Leland Pierce, the preparation of this manuscriptwould not have been so professional and successful.

I wish to thank Prof. Fawwaz T. Ulaby, Director of the Radiation Laboratory,for providing me with technical assistance. The speedy production of this bookis due to the efficient management of Mr. Dudley Kay, Executive Editor, and thevaluable technical supervision of Ms. Anne Reifsnyder, Associate Editor, of theIEEE Press. Some major changes have been made in the original manuscript asa result of many valuable suggestions from the reviewers. I am most grateful tothese reviewers.

CHEN-ToTAlAnn Arbor, MI

Chapter 1

Vector and Dyadic Algebra

1-1 Representations of Vector Functions

A vector function has both magnitude and direction. The vector functions thatwe encounter in many physical problems are, in general, functions of space andtime. In the first five chapters, we discuss only their characteristics as functions ofspatial variables. Functions of space and time are covered in Chapter 6, dealingwith a moving surface or a moving contour.

A vector function is denoted by F. Geometrically, it is represented by a linewith an arrow in a three-dimensional space. The length of the line corresponds toits magnitude, and the direction of the line represents the direction of the vectorfunction. The convenience of using vectors to represent physical quantities isillustrated by a simple example shown in Fig. 1-1, which describes the motionof a mass particle in a frictionless air (vacuum) against a constant gravitationalforce. The particle is thrown into the space with an initial velocity vo, making anangle 90 with respect to the horizon. During its flight, the velocity function of theparticle changes both its magnitude and direction, as shown by VI, V2, and so on, atsubsequent locations. The gravitational force that acts on the particle is assumedto be constant, and it is represented by F in the figure. A constant vector functionmeans that both the magnitude and the direction of the function are constant, beingindependent of the spatial variables, x and z in this case.

The rule of the addition of two vectors a and b is shown geometrically byFig. 1-2a, b, or c. Algebraically, it is written in the same form as the addition of

2 Vector and Dyadic Algebra Chap. I

g = gravitational constant

Figure 1-1 Trajectory of a mass particle ina gravitational field showing the velocity vand the constant force vector F at differentlocations.

b

(a) (b)

b

(c)

Figure 1·2 Addition of vectors, a + b = c.

two numbers of two scalar functions, that is,

c =a+ b.

The subtraction of vector b from vector a is written in the form

d = a-b.

(1.1)

(1.2)

Now, -b is a vector that has the same magnitude as b, but of opposite direction; then (1.2) can be considered as the addition of a and (-b). Geometrically,the meaning of (1.2) is shown in Fig. 1-3. The sum and the difference of twovectors obey the associative rule, that is,

a + b = b + a (1.3)

and

a- b = -b+a. (1.4)

They can be generalized to any number of vectors.The rule of the addition of vectors suggests that any vectorcan be considered as

being made of basic components associated with a proper coordinate system. The

Sec. 1-1 Representations of Vector Functions

-b b

3

Figure 1-3 Subtraction of vectors, a - b = d.

most convenient system to use is the Cartesian system or the rectangular coordinatesystem, or more specifically, a right-handed rectangular system in which, whenx is turning to y, a right-handed screw advances to the z direction. The spatialvariables in this system are commonly denoted by x, y, z. A vector that has amagnitude equal to unity and pointed in the positive x direction is called a unitvector in the x direction and is denoted by x. Similarly, we have y, z. In such asystem, a vector function F that, in general, is a function of position, can be writtenin the form

F = Fxx + FyY + Fzz. (1.5)

The three scalar functions Fx , Fy, Fz are called the components of F in thedirection ofx, Y, and z, respectively, while Fxx, FyY, and Fzz are called the vectorcomponents of F. The geometrical representation of F is shown in Fig. 1-4. It isseen that Fx , F y , and F z can be either positive or negative. In Fig. 1-4, F; and F zare positive, but F y is negative.

z

<,<,

F,z F'

F'yyY

..;----~y

x

Figure 1-4 Components of a vector in a Cartesian system.

In addition to the representation by (1.5), it is sometimes desirable to expressF in terms of its magnitude, denoted by IFI, and its directional cosines, that is,

F= IFI(cosaX+cosIlY+cos1z). (1.6)

4 Vector and Dyadic Algebra Chap. 1

a, 13, and y are the angles F makes, respectively, with x, y, and Z, as shownin Fig. 1-4. It is obvious from the geometry of that figure that

IFI = (F; + F;+ F;)1/2 (1.7)and

(1.8)r;

cosy = IFI.r; s;

cos a = IFI' cos P= IFI'Furthermore, we have the relation

cos2 a + cos2 ~ + cos2 Y = 1. (1.9)In view of (1.9), only two of the directional cosine angles are independent. Fromthe previous discussion, we observe that, in general, we need three parameters tospecify a vector function. The three parameters could be Fx , Fy , and F; or IFI andtwo of the directional cosine angles. Representations such as (1.5) and (1.6) canbe extended to other orthogonal coordinate systems, which will be discussed in alater chapter.

1-2 Products and Identities

The scalar product of two vectors a and b is denoted by a · b, and it is defined bya- b = [al lb] cos 9, (1.10)

where 9 is the angle between a and b, as shown in Fig. 1-5. Because of the notationused for such a product, sometimes it is called the dot product. By applying (1.10)to three orthogonal unit vectors UI,U2, U3, one finds

"" {I, i = j }u;·Uj= 0, ii=j' i,j=1,2,3. (1.11)

The value of a · b can also be expressed in terms of the components of a and bin any orthogonal system. Let the system under consideration be the rectangularsystem, and let c = a - b; then

Icl2 = la- bl 2 = lal 2 + Ibl2 - 21allbl cos 9.Hence

a. b = lallbl cos B= lal2 + Ibl

2-Ia - bl

2

2

_ a; + a~ + a; + b; + b~ + b; - (ax - bx)2 - (ay - by)2 - (az - bz)2- 2

(1.12)

b

....--...----·8 Figure 1·5 Scalar product of two vectors,a . b = lallblcos 9.

Sec. 1-2 Products and Identities 5

(1.13)

(1.14)

By equating (1.10) and (1.12), one finds

1cos e = lallbl (oxbx + oyby + ozbz)

= cos aa cos CXb + cos Pa cos J3b + cos 'Ya cos 'Yb,

a relationship well known in analytical geometry. Equation (1.12) can be used toprove the validity of the distributive law for the scalar products, namely,

(8 + b) . c =8· C + b- c.

According to (1.12), we have

(8 + b) · c = (ax + bx) Cx + (ay + by) cy + ta, + bz ) Cz

=(axcx + aycy + azcz) + (bxcx + bycy + bzcz)

= 8· C + b- c.

Once we have proved the distributive law for the scalar product, (1.12) can beverified by taking the sum of the scalar products of the individual terms of a and b.

The vector product of two vector functions 8 and b, denoted by 8 X b, isdefined by

8 X b = [al lb] sin aue , (1.15)

where 9 denotes the angle between 8 and b, measured from a to b; Uc denotes aunit vector perpendicular to both 8 and b and is pointed to the advancing directionof a right-hand screw when we tum from 8 to b. Figure 1-6 shows the relativeposition ofue with respect to a and b. Because of the notation used for the vectorproduct, it is sometimes called the cross product, in contrast to the dot product orthe scalar product. For three orthogonal unit vectors in a right-hand system, wehave Ul x "2 = "3, U2 X U3 = UI, and U3 x UI = U2. It is obvious that Ui x Ui = 0,i = 1, 2, 3. From the definition of the vector product in (1.15), one finds

b x 8 = -a x b. (1.16)

The value of ax b as described by (1.15) can also be expressed in terms of thecomponents of 8 and b in a rectangular coordinate system. If we let 8 x b = v =

axb

Figure 1-6 Vector product of two vectors,a x b = lallbl sin QUe; Ue .L a, ue .L b.

b

.-........... -------II~ 8


VxX + vyY + vzz, which is perpendicular to both a and b, then

a· v = axvx + QyVy + azvz = 0,

b· v = bxvx + byvy + bzvz = O.

Solving for v.J», and vy/vz , from (1.17) and (1.18) we obtain

Vx Qybz - azby vy Qzbx - axbz-= ,-=..Vz Qxby - aybx Vz Qxby - Qybx

Thus,

(1.17)

(1.18)

Vx vy Vz----=-----aybz - a.b, a.b, - a.b, - a.b; - aybx .

Let the common ratio of these quantities be denoted by c, which can be determinedby considering the case with a = x, b = y; then v = a x b = z; hence from thelast ratio, we find c = 1 because Vz = 1 and ax = by = 1, while ay = bx = O.The three components of v, therefore, are given by

Vx = aybz - azby }VY, = azbx - a.b, , (1.19)

Vz = Qxby - aybxwhich can be assembled in a determinant form as

x .Y zv = Qx ay az (1.20)

bx by bz

We can use (1.20) to prove the distributive law of vector products, that is,

(a + b) x c = a x c + b x c. (1.21)

To prove (1.21), we find that the x component of (a + b) x c according to (1.20)is equal to

(ay + by) Cz - (az + bz) cy = (ayc z - QzCy) + (bycz - bzcy) . (1.22)

The last two terms in (1.22) denote, respectively, the x component ofax c andb x c. The equality of the y and z components of (1.21) can be proved in a similarmanner.

In addition to the scalar product and the vector product introduced before,there are two identities involving the triple products that are very useful in vectoranalysis. They are

a- (b x c) = b· (c x a) = c· (a x b), (1.23)a x (b x c) = (a . c) b - (a . b) c. (1.24)

Identities described by (1.23) can be proved by writing a- (b x c) in a determinantform:

al

a· (b x c) = hI

Sec. 1-2 Products and Identities 7

According to the theory of determinants,

at az a3 b, b2 b3 CI C2 C3

bv b2 b3 = CI C2 C3 = at az a3

CI C2 C3 al a2 a3 b t b2 b3

The last two determinants represent, respectively, b · (c x a) and c . (a x b);hence we have the validity of (1.23). To prove (1.24), we observe that the vector8 x (b x c) lies in the plane containing band c, so we can treat 8 x (b x c) asbeing made of two components ab and J3c, as shown in Fig. 1-7, that is,

a x (b x c) = ab + J3c. (1.25)

Because

8 · [8 x (b x c)] = 0,

hence

a (8 . b) + J3 (8 · c) = o.

bxc

a

b

ax(bx c) = ab + f3c

c

Figure 1-7 Orientation of various vectors in a x (b xc).

Equation (1.25), therefore, can be written in the form

a x (b x c) = a [b - a· b cJ = a ' [(8 . c) b - (a . b) c], (1.26)a·c

where a' is a constant to be determined. By considering the case a = y, b = x,c = y, we have

a x (b x c) = X,(a·c)b=x,

(a . b) c = o.

Hence rt' = 1. All other choices of a, b, and c yield the same answer. The validityof (1.23) and (1.24) is independent of the choice of the coordinate system in whichthese vectors are represented.

1·3 Orthogonal Transformation of Vector Functions

A vector function represented by (1.5) in a specified rectangular system canlikewise be represented in another rectangular system. To discuss the relationbetween these two representations, we must first show the geometry of the twocoordinate systems. The relative orientation of the axes of these two systems canbe formed by three successive rotations originally due to Leonhard Euler (17071783).

Let the coordinates of the original system be denoted by (x, y, z). We firstrotate the (x, y) axes by an angle <t>l to form the (Xl, Yl) axes, keeping ZI = Z asshown in Fig. 1-8; then the coordinates of a point (x, y, z) change to (Xl, Yl, Zl)with

Xl = X cos <PI + y sin <PI,

Yl = -x sin <PI + yCOS'Pl,

Zl =z.

(1.27)

(1.28)

(1.29)

Now we tum the (YI, Zl) axes by an angle <P2 to form the (Y2, Z2) axes with x- = Xl;

then,

Yl = Yt cos~ + Zl sin~2,

Z2 = - Yl sin <1>2 + Zl cos <P2,

X2 = Xl.

(1.30)

(1.31)

(1.32)

Finally, we rotate the (Z2, X2) axes by an angle «1>3 to form the (Z3, X3) axes withY3 = Y2; then,

Z3 = Z2 cos cl>3 + X2 sin <P3,

X3 = -Z2 sin <t>3 + X2 cos <P3,

)'3 = Yl.

(1.33)

(1.34)

(1.35)

By expressing (X3, )'3, Z3) in terms of (x, y, z) and changing the letters (x, y, z) and(X3, Y.h Z3), respectively, to the unprimed and primed indexed letters (Xl, X2, X3)

and (x~, x~, x;), we obtain

3

x; = LQ;jXj,j=l

;=1,2,3, (1.36)

Sec. 1-3 Orthogonal Transformation of Vector Functions 9

YI Y

XI =xcoscl>l +ysinet>l

YJ = Y cos <1>1 -x sin ~I

------ ........... ----"-.........---~ z2

------.....~---'-_......--~ Yl z2 = Zl cos <1>2 - Yl sin ~2

Figure 1-8 Sequences of rotations of the axes of a rectangular coordinate system.

whereall = cos ~1 cos ~3 - sin <1>1 sin <1>2 sin <1>3,

a)2 = sin e, COSCP3 + cos <1>. sin<l>2sin~3,

a13 = -cos~sin<l>3,

a21 = - sin cp) cos CP2,

a22 = cos ~1 cos ~2,

a23 = sincl»2,a31 = cos CPl sin <1>3 + sin <l>t sin~ cosep3,a32 = sin <1>1 sin <1>3 - cos <1» sin <1>2 cos <1>3,a33 = cos <1>2 cos <1>3.

(1.37)


The coefficients aij correspond to the directional cosines between the x; and x j

axes, that is,

aij = cos ~ij, (1.38)

where ~ij denotes the angle between the two axes.If we solve (x, y, z) in terms of (X3, )'3, Z3) or xj (j = 1, 2, 3) in terms of x;

(i = 1,2,3) from (1.27) to (1.32), we obtain

3

Xj = EaijX;,i=1

j = 1,2,3, (1.39)

where aij denotes the same coefficients defined in (1.37). It should be observedthat the summation indices in (1.36) and (1.39) are executed differently in thesetwo equations. For example,

(1.40)

but

(1.41)

and aij i= a ji when j ¥= i. Henceforth, whenever a summation sign is used, it isunderstood that the running index goes from 1 to 3 unless specified otherwise. Amore efficient notation is to delete the summation sign in (1.36) and (1.39). Whenthe summation index appears in two terms, we write (1.36) in the form

(1.42)

The single index (i) means i = 1, 2, 3 and the double index (j) represents asummation of the terms from j = 1 to j = 3. Such a notation, originally due toEinstein, can be applied to more than three variables. In this book we will use thesummation sign in order to convey the meaning more vividly, particularly whenseveral summation indices are involved in an equation. The summation index willbe placed under the sign for one summation or several summations separated by acomma. The linear relations between the coordinates x; and xj, as stated by (1.36)and (1.39), apply equally well to two sets of unit vectors x; and xj and also to thecomponents of a vector A, denoted respectively by A~ and A j in the two systems.This is evident from the processes by which the primed system is formed fromthe unprimed system. To recapitulate these relations, it is convenient to constructa 3 x 3 square matrix, as shown in Table 1-1. We identify i, the first subscriptofaij, as the ordinal number of the rows, and j, the second subscript, as the ordinalnumber of the columns. The quantities involved in the transformation are listed atthe side and the top.

The matrix can be used either horizontally or vertically, for example,

A; = a31 Al + a32 A2 + a33A3,

Sec. 1-3

and

Orthogonal Transformation of Vector Functions

Table I-I: The Matrix ofTransformation [aij] for Quantities Definedin Two Orthogonal RectangularSystems

i\j Xj or Xj or Aj

x~ or alJ al2 al3IAI or a21 a22 a23Xi

A~ a31 a32 Q33I

11

(1.45)

(1.46)

A2 = a12A~ + a22A; + a32A;,

which conform with (1.36) and (1.39) after x; and x} are replaced by A; and A I:For convenience, we will designate the aij's as the directional coefficients and itsmatrix by raj}]. There are several important properties of the matrix that must beshown. In the first place, the determinant of [aij], denoted by lai}l, is equal tounity for a right-hand system under consideration. In such a system, when oneturns XI to X2 using a right-hand screw, it advances to the X3 direction. To prove

laijl = 1, (1.43)

we consider a cubic made of x; with i = 1, 2, 3. Its volume is equal to unity, that is,

xi . (x~ x xi) = 1. (1.44)

The expression on the left side of (1.44) is given by

Eai/x/ . (Ea}mxm x Laknxn) = la;}I,I m n

where (1, m , n) and (i, j, k) = (1, 2, 3) in cyclic order. The identity between (1.44)and (1.45) yields (1.43). A second identity relates the directional coefficients aijwith the cofactors or the signed minors of [aij]. If we solve (Xl, X2, X3) in termsof (xi, x~, x~) from (1.36) based on the theory of linear equations, we find

1" 'x} = --~ Aijxi ,fa;} I ;

where A;} denotes the cofactor or the signed minor of [a;}] obtained by eliminatingthe ith row and jth column. By comparing (1.46) with (1.39) and because laijI = 1,we obtain

Aij = Qij.

An alternative derivation is to start with the relation

x} = LAijx;,;

(1.47)

(1.48)

12 Vector and Dyadic Algebra Chap. I

(1.51)

which is the same as (1.46) with laijl = 1, and to replace Xj, x; with Xj and Xi;then the scalar product of (1.48) with x; yields

aij = Aij. (1.49)

As an example, let i = 1, j = 2; then

a12 = -(a21Q33 - a23Q31). (1.50)

The validity of (1.49) can also be verified by using the expressions of aij definedin terms of the Eulerian angles listed in (1.37); that is,

-a21 a33 + a23 a31 = sin 'I cos '1>2 cos <P2 cos <P3

+ sincl>2(coscl>l sincl>3 + sin e, sin<P2 cos'l>3)

= cos 'I sin <P2 sin «1>3 + sin <PI cos <P3

=a12·

Equation (1.47) is a very useful identity in discussing the transformation of vectorproducts.

Because the axes of the two coordinate systems x; and xj or the unit vectorsx; and xj are themselves orthogonal, then

AI AI ~ {I, i = i.Xi • X j = Uij = 0,

i =F i.and similarly,

(1.52)

where ai j denotes the Kronecker a function defined in (1.51). In terms of theunprimed unit vectors, (1.51) becomes

LaimXm · LajnXn = Oij; (1.53)m n

because

(1.53) reduces to

Laimajm = aij,m

and similarly, by expressing (1.52) in terms of x~ and x~, we obtain

Lamiamj = aij.m

(1.54)

(1.55)

Either (1.54) or (1.55) contains six identities. Looking at the rotational relationsbetween the unprimed and the primed coordinates, we observe that the threeEulerian angles or parameters generate nine coefficients. Only three of themare therefore independent, provided they are not the triads of

La~ = 1, i = 1,2,3, (1.56)j

Sec. 1-3

or

Orthogonal Transformation of Vector Functions 13

j= 1,2,3. (1.57)

The remaining six coefficients are therefore dependent coefficients that are relatedby (1.54) or (1.55) or a mixture of six relations from both of them. For example, ifall, al2, and a23 have been specified, then we can determine al3 from the equation

a~l +a;2 +a~3 = 1,and subsequently the coefficient a33 from

2 221a t3 +a23 +a33 = .The remaining four coefficients a21, a22, G3J, a32 can be found from the equations

allf!.2l + a12~22 + a13 a23 = 0,and

allf!.31 + a12f!32 + a13 a33 =o.We underline the unknowns by placing a bar underneath these coefficients. Theymust also satisfy

and

a23f!.22 + a33f!32 + a13 a12 = o.For convenience, we summarize here the important formulas that have beenderived:

x; = Laijxj, i=I,2,3, (1.58)j

Xj = Laijx;, i> 1,2,3, (1.59)i

AI L A i=I,2,3, (1.60)Xi = aijxj,j

A L AI J=1,2,3, (1.61)Xj = Gijxi ,

i

A; = LaijA j, i=I,2,3, (1.62)j

A j = LaijA;, J=1,2,3, (1.63)

laijl = 1, (1.64)

aij = Aij, (1.65)

Laimajm = s.; (1.66)m

Lamiamj = Oijo (1.67)m


In expressions (1.58) through (1.67), all the summation signs are understood to beexecuted from 1 to 3. The conglomerate puts these relations into a group.

Equations (1.62) and (1.63) are two important equations or requirements forthe transformation of the components ofa vector in two rectangular systems rotatedwith respect to each other. These relations also show that a vector function has aninvariant form, namely,

and

A= LA;x; = LAjij; j

(1.68)

(1.69)A·A= LA; = L Aj2.j

Equation (1.69) shows that the magnitude of a vector is an invariant scalar quantity,independent ofthe defining coordinate system. The speed ofa car running 50 milesper hour is independent of its direction. However, its direction does depend onthe reference system that is being used, namely, either (Xl, X2, X3) or (X~, X~, x~).

The vector functions that transform according to (1.62) and (1.63) are called polarvectors, to be distinguished from another class of vectors that will be covered inthe next section.

1-4 Transform of Vector Products

A vector product formed by two polar vectors A and B in the unprimed systemand their corresponding expressions A' and B' in the primed system is

or

C=AxB

C'=A' xB'.

(1.70)

(1.71)

According to the definition of a vector product, (1.19), its expression in a righthanded rectangular system is

c~ = A~Bj - AjB;with i, j, k = 1,2,3 in cyclic order. Now,

A; = La;mAm,m

Hence

Ck= A~Bj - AjB; =L L(a;majn - ajma;n)AmBnm n

=L La;mOjn(AmBn - AnBm).m n

(1.72)

(1.73)

(1.74)

(1.75)

Sec. 1-4 Transform of Vector Products IS

(1.77)

It appears that the components AmBn - AnBm in (1.75) do not transform like thecomponents of a polar vector as in (1.62). However, if we inspect the terms in(1.75), for example, with k = 1, i = 2, j = 3, then

C~ = (a22a33 - a23a32)(A2B3 - A3B2)

+(a23a31-a2Ia33)(A3BI-AIB3) (1.76)

+ (a21a32 - Q22a31)(A IB2 - A2 BI).

The terms involving the directional coefficients are recognized as the cofactorsAll, A12, and A I3 of [aij] and, according to (1.65), they are equal to all, a12,

and a13; hence

c~ = AIIC I + AI2C2 + AI3C3

= allCI +a12C2 +at3C3

= L:aijCj .j

Equation (1.77) obeys the same rule as the transformation of two polar vectors.Thus, in a three-dimensional Euclidean space, the vector product does transformlike a polar vector even though its origin stems from the vector product of two polarvectors. From the physical point of view, the vector product is used to describea quantity. associated with rotation, such as the angular velocity of a rotatingbody, the moment of force, the vorticity in hydrodynamics, and the magnetic fieldin electrodynamics. For this reason such a vector was called a skew vector byJ. Willard Gibbs (1837-1903), one of the founders of vector analysis. Nowadays,it is commonly called an axial vector. From now on, the word vector will be usedto comprise both the polar and the axial vectors in a three-dimensional or Euclideanspace. In a four-dimensional manifold as in the theory of relativity, the situationis different. In that case, we have to distinguish the polar vector, or the fourvector, from the axial vector, or the six-vector. This topic will be briefty discussedafter the subjects of dyadic and tensor analysis are introduced. Even though thetransformation rule of a polar vector applies to an axial vector, we must rememberthat we have defined an axial vector according to a right-hand rule. In a left-handcoordinate system, obtained by an inversion of the axes of a right-hand system,the components of a polar vector change their signs; then we must use a left-handrotating rule to define a vector product to preserve the same rule of transformationbetween a polar vector and an axial vector. We would like to mention that in aleft-hand system the determinant of the corresponding directional coefficients isequal to -1.

Before we close this section, we want to point out that as a result of theidentical rule of transformation of the polar vectors and the axial vectors, thecharacteristics of the two triple products A . (B x C) and A x (D x C) can beascertained. The scalar triple product is, indeed, an invariant scalar because

A . (D x C) = A' . (D' x C'), (1.78)


For the vector triple product, it behaves like a vector because by decomposing itinto two terms using the vector identity (1.24),

A x (B x C) = (A . C)B - (A · B)C, (1.79)

we see that A · C and A ·B are invariant scalars, and B and C are vectors; the sum atthe right side of (1.79) is therefore again a vector. This synthesis may appear to betrivial but it does offer a better understanding of the nature of these quantities. Thereader should' practice constructing these identities when the vectors are definedin a left-hand coordinate system.

1-5 Definition of Dyadlcs and Tensors

A vector function F in a three-dimensional space defined in a rectangular systemis represented by

If we consider three independent vector functions denoted by

F j = L fijx;, j = 1,2,3,;

(1.80)

(1.81)

then a dyadic function can be formed that will be denoted by F and defined by

F= LFjxj. (1.82)j

The unit vector xj is juxtaposed at the posterior position of F j. By substitutingthe expression of F, into (1.82), we obtain

F= LLF;jx;xj = LF;jx;xjo; j i,j

(1.83)

(1.84)

Equation (1.83) is the explicit expression of a dyadic function defined in arectangular coordinate system. Sometimes the name Cartesian dyadic is used. Adyadic function, or simply a dyadic, therefore, consists ofnine dyadic components;each component is made of a scalar component fij and a dyad in the form of apair of unit vectors XiX j placed in that order.

Because a dyadic is formed by three vector functions and three unit vectors,the transform of a dyadic from its representation in one rectangular system (theunprimed) to another rectangular system (the primed) can most conveniently beexecuted by applying (1.61) to (1.83); thus,

F = L F;j L am;x~ L anjX~t.] m n

Sec. 1-6 Classification of Dyadics 17

If we denote

F~n = L Qm;anj Fij,i.]

then

= L fijXiXj.i.]

(1.85)

(1.86)

Equation (1.85) describes the rule of transform of the scalar components_Fij of a

dyadic in two rectangular systems. By starting with the expression of F' in theprimed system, we find

Fij = EamianjF~n.m.n

(1.87)

When the 3 x 3 scalar components fij are arranged in matrix form, denoted by[F;j], it is designated as a tensor or, more precisely, as a tensor of rank 2 or a tensorof valance 2. The exact form of [F;j] is

[Fij] = [~~: ~~ ~~:]. (1.88)F31 F32 F33

In tensor analysis, a vector is treated as a tensor of rank 1 and a scalar as a tensorof rank O. In this book, tensor analysis is not one of our main topics. The subjecthas been covered by many excellent books such as Brand [1] and Borisenko andTarapov [2]. However, many applications of tensor analysis can be treated equallywell by dyadic analysis. In the previous section, we have already correlated a tensorof rank 2 in a Euclidean space with a dyadic. Tensor analysis is most useful in thetheory of relativity, but one can formulate problems in the special theory of relativity using conventional vector analysis, as illustrated in Appendix E of this book.

1-6 Classification of Dyadics

When the scalar components of a dyadic are symmetrical, such that

(1.89)

it is called a symmetric dyadic and the corresponding tensor, a symmetric tensor.When the components are antisymmetric, such that

(1.90)

such a dyadic is called an antisymmetric dyadic and the corresponding tensor, theantisymmetric tensor. For an antisymmetric dyadic, (1.90) implies Dii = O. An


anti symmetric tensor of this dyadic, therefore, has effectively only three distinctcomponents, namely, D12 , D13 , and D23 • The other three components are -DI2 ,

- D13, and - D23, which are not considered to be distinctly different.Let us now introduce three terms with a single index, such that

D1 = D23,

or

D; = Djk

with i, j, k = 1, 2, 3 in cyclic order; then the tensor of this antisymmetric dyadichas the form

(1.91)

The transform of these components to a primed system, according to (1.85), hasthe form

D;j = L a;ma jn Dmn,m,n

(1.92)

where we have interchanged the roles of the indices in (1.85). In terms of thesingle indexed components for D;j'

D~ = La;mQjnDmn,m,n

(1.93)

(1.94)

where (i, j, k) and (1,m, n) = (1,2,3) in cyclic order. For example, the explicitexpression of Di is

D~ = (a22a33 - a23a32)~3

+ (a23a31 - a21a33)D31

+ (a21a32 - a22a31)D12 .

The coefficients attached to Dmn are recognized as three cofactors of [aij]; theyare, respectively, All, A12, and A 13, which are equal to all, a12, and a13. Bychanging D23, D31, and D12 to D I , D2, and D3, we obtain

Di = AIIDI + A l 2 D2 + A 13 D3

=allDI + a12 D2 + a13D3

= LaijDj.j

(1.95)

Equation (1.95) describes, precisely, the transform of a polar vector in the twocoordinate system. By tracing back the derivations, we see that an antisymmetric

Sec. 1-7 Products Between Vectors and Dyadics 19

tensor is essentially an axial vector (defined in a right-hand system) and its components transform like a polar vector. The connection between an antisymmetrictensor, an axial vector, and a polar vector is well illustrated in this exercise. Wenow continue on to define some more quantities used in dyadic algebra.

When a symmetric dyadic is made of three dyads in the form

(1.96)

it is called an idem/actor. Its significance and applications will be revealed shortly.When the positions ofF j and xj are interchanged in (1.82), we form another dyadic

that is called the transpose of F, and it is denoted by [F]T, that is,

[FJT =LXjFj = L Fijxjx; = L Fj;x;xjoj j,; i,j

(1.97)

(1.99)

The corresponding tensor will be denoted by

[fij]T = [Fj i ] = [~:~ ~:~ ~~]. (1.98)

F J3 F23 F33

We therefore transpose the columns in [F;j] to form the rows in [F';j]T. It is obvious

that the transpose of [fij]T goes back to [h.

1·7 Products Between Vectors and Dyadlcs

There are two scalar products between a vector A and a dyadic D. The anteriorscalar product is defined by

B = A . b = A . L D jXj

j

= A· L o..s,s,i.]

= LAiDijXj,i,j

which is a vector; hence we use the notation B. Following the rules of the transformof a vector and a dyadic, we find that the same vector becomes

Hence

-,B = L A~ D~ni~ = A' . b .

m.n

B=B'.

(1.100)

(1.101)


That means the product, like the scalar product A · B, is independent of thecoordinate system in which it is defined, o! it is form-invariant.

The posterior vector between A and b is defined by

C = D·A = L Dijx;x j • A = L o., AjX;.i.] i.]

(1.102)

The scalar components of (1.102) can be cast as the product between thesquare matrix or tensor [Dij] and a column matrix [A j], that is,

A typical term of (1.103) reads

C1 = D11A 1 + D12 A2 + D13 A3 .

(1.103)

(1.104)

Linear relations like (1.103) occur often in solid mechanics, crystal optics, andelectromagnetic theory. Equation (1.102) is a more complete representation ofthese relations because the unit vectors are also included in the equation. Wethen speak of a scalar product between a vector and a dyadic instead of a productbetween a tensor and a column matrix.

By transforming A j, Dij, and Xi into the primed functions, we find

c = C' = LA~D~nx~.m,n

(1.105)

The anterior scalar product between A and [D]T, denoted by T for the time being,is given by

T = A · [D]T = A · L Dj;X;Xj;,j

=LA;Djixj = LAjDijXi't.t i.]

(1.106)

(1.107)

which is equal to C given by (1.102); hence we have a very useful identity:

= T =A· [D] = D·A.

Similarly, one finds

= T =[D] ·A = A· D.

For a symmetric dyadic, denoted by b;= T =[Dsl = o;

Hence

A · Ds = b, · A.

(1.108)

(1.109)

(1.110)

(1.111)

Sec. 1-7 Products Between Vectors and Dyadics 21

When Ds is the idemfactor defined by (1.96), we have

A·I = I·A =A. (1.112)

The tensor of I can be called a unit tensor, with the three diagonal terms equal tounity and the rest are null. _

There are two vector products between A and D. These products are bothdyadics. The anterior vector product is defined by

B=AxD= A x L DijXiXj

i.]

= L DijAk(Xk x x;)Xj,i,j,k

(1.113)

where i, j, k = 1, 2, 3 in cyclic order. The posterior vector product is defined by

C= b x A = L D;jAkX;(Xj x Xk). (1.114)i,j,k

One important triple product involving three vectors is given by (1.23):

A · (B x C) = B · (C x A) = C · (A x B). (1.115)

In dyadic analysis, we need a similar product, with one of the vectors changed toa dyadic. We can obtain such an identity by first changing (1.115) into the form

A· (B x C) = -B· (A x C) = (A x B)· C.

Now we let

C=C·F,

where F is an arbitrary vector function and C is a dyadic. Then

A· (B x C). F = -B· (A x C). F = (A x B)· C· F.

Because this identity is valid for any arbitrary F, we obtain

A · (B x C) = -B· (A x C) = (A x B) · C.

(1.116)

(1.117)

(1.118)

An alternative method of deriving (1.118) is to consider three sets of identitieslike (1.116) with three distinct C], with j = 1,2,3. Then, by juxtaposing a unitvector xj at the posterior position of each of these sets and summing the resultantequations, we again obtain (1.118) with

C= LCjXj.j

Other dyadic identities can be derived in a similar manner. Many of them will begiven in Chapter 7, which deals with dyadic analysis.


Finally, we want to introduce another class of dyadics in the form of

then,

S=MN=LM;Xi LNjxj

; j

=LMiNjxixj;i.]

(1.119)

(1.120)

(1.121)

(1.122)

Because there are three components of M and three components of~, all togethersix functions, and they have generated nine dyadic components of L, three of therelations in (1.120) must be dependent. For example, we can write

~3 = M2N3 = (M2N2)N3/ N2

= (M2N2)(MtN3)/(MtN2)

=~2S13/812.

When the vectors in (1.120) are defined in two different coordinate systemsunrelated to each other, we have a mixed dyadic. Let

M = M(Xl, X2, X3)

and

N" = N"(x~, x;, x;),

where we use a double primed system to avoid a conflict of notation with(x~, x~, x~), which have been used to denote a system rotated with respect to(Xl, X2, X3). Here xj with j = 1,2,3 are independent of Xi with i = 1,2,3. Themixed dyadic then has the form

T=MN"=L u.s, L N'Jx'Ji j

~IIN"" "II=L..J lY.li jXjXj •i,j

We can form an anterior scalar product of t with a vector function A definedin the X; system, but the posterior scalar product between A and i is undefinedor meaningless. Mixed dyadics can be defined in any two unrelated coordinatesystems not necessarily rectangular, such as two spherical systems. Thesedyadics are frequently used in electromagnetic theory [3]. Many commonly usedcoordinate systems are introduced in the following chapter.

Chapter 2

Coordinate Systems

2-1 General Curvilinear System (GCS)

In the general curvilinear coordinate system (GCS), the coordinate variables willbe denoted by roi with i = 1, 2, 3. The total differential of a position vector isdefined by

"" 8Rp idRp = ~ 8roi dro ·I

(2.1)

The geometrical interpretation of (2.1) is shown in Fig. 2-1. The vector coefficient8Rp/8ro

i is a measure of the change of Rp due to a change of Wi only; it will bedenoted by

8RpPi = 8roi . (2.2)

The vectors Pi with i = 1, 2, 3 are designated as the primary vectors. They are,in general, not orthogonal to each other. If the system is orthogonal, we willspecifically use the term orthogonal curvilinear system. Orthogonal linear systemis the same as the rectangular system. In terms of the primary vectors,

(2.3)

23

24 Coordinate Systems Chap. 2

oFigure 2-1 Position vector and its total differential.

The primary vectors are not necessarily of unit length, nor of the dimension oflength. The three differential vectors Pi droi define a differential volume given by

dV = PI . (P2 X P3) dro l dro 2 dro 3 = A dro 1 dro 2 dro3, (2.4)

where

A = Pi· (Pi x Pk) (2.5)

with (i, t. k) = (1, 2, 3) in cyclic order.We now introduce three reciprocal vectors defined by

. 1r' = A (Pi x Pk) (2.6)

with (i, j, k) = (1, 2, 3) in cyclic order. They are called reciprocal vectors because

. {I, i = j,P · r' - (2 7)

i - 0, i =F j. ·

The primary vectors can be expressed in terms of the reciprocal vectors in the form

Pi = Ari x r!', (2.8)

which can be verified by means of (2.6). The reciprocal systems of vectorswere originally introduced by Gibbs [4] without giving a nomenclature to thetwo systems of vectors. In Stratton's book [5], he designates our primary vectorsas the unitary vectors and our reciprocal vectors as the reciprocal unitary vectors.His notations for our Pi and r' are, respectively, a, and a'. Superscript and subscriptindices are used, following a tradition in tensor analysis. Incidentally, it should bementioned that in Stratton's book, the relation described by (2.8) was inadvertentlywritten, in our notation,

r i x rkPi = A

A vector function F can be expressed either in terms of the primary vectorsor the reciprocal vectors. We let

F = EJiri = EgiPi. (2.9)i

Sec. 2-1 General Curvilinear System (GCS) 25

On account of the relations described by (2.7), one finds

/;=Pi· F,

s' = r j. F.

(2.10)

(2.11 )

fi with i = 1,2,3 are designated as the primary components of F, and s' withj = 1,2,3 as the reciprocal components of F. In tensor analysis, /; are calledthe covariant components and s'. the contravariant components. By substituting(2.10) and (2.11) into (2.9), we can write (2.9) in the form

F= LF.p;r; = LF.rjpji j

or

F = Lripi· F = LPjrj. F.

j

In the language of dyadic analysis, we denote

L~Pi = LPjrj = /,

j

where I is an idernfactor, such that

(2.12)

(2.13)

(2.14)

This is an alternative representation of I defined by (1.96).The primary vectors Pi defined by (2.2), being used to represent dRp in (2.3),

can be found if the scalar relations between the curvilinear coordinate variables ro i ,

i = 1, 2, 3, and the rectangular variables x j, j = 1, 2, 3, are known or given. Interms of the rectangular unit vectors xj and the differentials dxI» the differentialposition vector can be written as

dRp = L,xjdxj.j

By equating it to (2.3), we have

LP;droi = LXjdxj.j

Thus,

(2.15)

(2.16)

i=I,2,3. (2.17)

This is the explicit expression of P; in terms of xj and the derivative of x j withrespect to 0);. Later, we will illustrate the application of (2.17) to determine Pi formany commonly used orthogonal curvilinear systems.


To determine the parameter A defined by (2.5), it is convenient to introducethe coefficients au defined by

Pi . Pj = Cl;j, i, j = I, 2, 3. (2.18)

It is obvious that Clij = Clji, thus we have only six distinct coefficients. By meansof (2.17), we find

(2.19)

(2.20)

When i = j,

(Xii =~(:;r.These coefficients can now be used to determine the parameter A. By definition,

(2.21)

The vectorfunctionpj xP3 in(2.21) can be expressed in terms ofr' with i = 1, 2, 3.According to (2.12) and (2.14),

P2 x P3 = L[(P2 x P3) · ri]p;. (2.22)

The reciprocal vectors ri in (2.22) can be changed to

. 1r' = A Pj X Pi

with (i, I. k) = (1,2,3) in cyclic order. Thus, (2.22) becomes

P2 x P3 = ~ [(P2 x P3) · (P2 x P3)PI

+ (P2 x P3) · (P3 x PI)P2

+ (P2 x P3) · (PI x P2)P3].

The scalar products in (2.24) can be simplified using

(a x b) · (c x d) = (a · c)(b · d) - (a · d)(b · c).

Thus,

(P2 x P3) · (P2 x P3) = (P2 · P2)(P3 · P3) - (P2 · P3)(P3 . P2)

= a22(X33 - (X23(X,32·

Similarly,

(P2 x P3) . (P3 x PI) = (X,23(X31 - a21a33,

(P2 x P3) · (PI x P2) = (X21 (132 - (X22(X31·

(2.23)

(2.24)

Sec. 2-1 General Curvilinear System (GCS) 27

Hence by taking the scalar product of PI with (2.24), we obtain

A = PI . (P2 x P3)

= ~ [(111 «(122(133 - (123(132)

+ (X12(a23(X31 - a21 (33)

+ (113«(121(132 - (122(131)]

or1/2

(X13 all al2

(X23 = (X12 a22

(X33 (X13 a23

(2.25)

(2.26)

We take the positive square root of the determinant as the proper expression for A.Before we leave this section, a theorem involving the sum of the derivative

of the vectors Ari should be presented. It is known in geometry that the totalvectorial area of a closed surface vanishes, that is,

!Is ds = O.

If we consider a small volume 6. V bounded by six coordinate surfaces located atCii ± ~CJ); /2 with i = 1, 2, 3 in GCS, then

6. V = A 6.(01 /).(02 6m 3,

AS; = A"; ~CJ)j ~CJ)k

with (i, j, k) = (1, 2, 3) in cyclic order. The differential form of (2.26) can beobtained by taking the limit of the following identity:

lim _1_ U ds = 0~v--.o 6V 11s

or

31 "[ . .]. klim - L...J (Ar')cd+~cd/2 - (Ar')roi-6roi /2 f:J.ro J 6.00 = 0,

~v-+o 6. V ;=1

which yields

La .

-.(Ar') =0.. am',(2.27)

Equation (2.27) is a very useful theorem; it will be designated as the closed surfacetheorem.

28

2-2 Orthogonal Curvilinear System (OCS)

Coordinate Systems Chap. 2

(2.29)

(2.31)

The GCS degenerates to the orthogonal curvilinear system (OCS) when the primaryvectors are mutually perpendicular to each other. In this case, we let

P; = ::~ = h;u;, i = 1, 2, 3, (2.28)

where Uj denote the unit vectors along the coordinates CJ)i and hi, the metriccoefficients in the OCS. For a specific system, such as the sph~rical ~oordinate

system with coordinate variables R, a, cp, we use the notations R, a, and ep to denotethe unit vectors in that system. Ui are used when the orthogonal curvilinear systemis arbitrary or unspecified. For the rectangular system, which is a special case ofOCS, hi = 1, and mi = Xi, or more specifically, X, y, z. The reciprocal vectors inOCS now become

r j = Pk X P; = hkh;u j = u j ,

A n h j

where n = hih jhk = hth2h3. The metric coefficients hi can be found if we knowthe relations between roi and xi: In the rectangular system,

oRp ~- = Xj. (2.30)OXj

By the chain rule of differentiation, we find

h ." . - oRp _ L 8Rp aXj _ L 8xj e .,u, - . - 0 a· - ':l. Xl·oro' j X j ro' Jam'

In (2.31), the summation with respect to j goes from j = 1 to j = 3; that labelingwill be omitted henceforth. Thus,

(2.32)

and the positive square root of (2.32) yields

(2.33)

Equation (2.33) can be used to determine hi when the relations between x j, thedependent variables, and (J)i, the independent variables, are known. Another expression ofhi is sometimes useful when the roles ofx j and mi are interchanged withx j as independent variables and CJ)i as dependent variables. By definition,

dR p = LXjdxj = LhiUi dmi; (2.34)

j i

Sec. 2-2 Orthogonal Curvilinear System (OCS)

hence

i 1,"",,,, "ddt» = J; L....,Ui ·Xj Xj;l j

thus,

Let

Then

1" '" aroiCij = - Ui • X j = - .

hj OXj

Equation (2.37) therefore becomes

Uj _ " oro; x'" .•- L.-.J J'h; j OXj

hence

-; = L (oroi )2.

hi j OXj

We take the positive square root of (2.40) to be the expression for 1/ hi:

[. 2] 1/2:i = ~(:=~) ·

29

(2.35)

(2.36)

(2.37)

(2.38)

(2.39)

(2.40)

(2.41)

In contrast to (2.33), roi's are now the dependent variables and x i» the independentvariables. Unlike functions with one independent variable,

oc.oi(x, y, z) =1= l/OX(c.oi, c.o. j

, ol),ax oro'

while

dy(x) _ l/dX(Y)rz:': dYe

The list that follows shows the metric coefficients of some commonly used OCS,and the relations between (VI, V2, V3) and (x, y, z), based on which these metriccoefficients are derived by means of (2.33).

30 CoordinateSystems Chap. 2

RectangularCoordinate variables: (x, y, z)

Metric coefficients: (1,1,1).

CylindricalCoordinate variables: (r,~, z)

Metric coefficients: (1, r, 1)

Relations: x = r cos <p, y = r sin <p, z = z.

SphericalCoordinate variables: (R, 9, ~)

Metric coefficients: (1, R, R sin 9)

Relations: x = R sin 9 cos ep, y = R sin 9 sin <p, z = R cos 8.

Elliptical CylinderCoordinate variables: (11,~, z)Metric coefficients:

[ (~2 _ 112 ) 1/2 ( ~2 _ 112 ) 1 /2 ]C 1 _ '1'\2 • C 1;2 _ 1 • 1

Relations: x = CI1~, y = c [(1 _1")2) (~2 - 1)]1/2, Z = z.

Parabolic CylinderCoordinate variables: (11,~, z)Metric coefficients:

[ ('1'\2 + 1;2)1/2 , (112+ 1;2) 1/2 , 1]Relations: x = ! (112 - ~2), Y = 1")~, z = z.

Prolate SpheroidalCoordinate variables: (11,~, ep)Metric coefficients:

Relations:

x = c [( 1 - 112) (~2 - 1)]1/2 cos ep,

y = c [(1 _1")2) (~2 - 1)]1/2 sin o,

z = C1l~.

Sec. 2-2 Orthogonal Curvilinear System (OCS) 31

Oblate SpheroidalCoordinate variables: (~, 11, <t»

Metric coefficients:

[ ( ~2 _112)1/2 (~2 _112)1/2 ]C ~2 _ 1 ,c 1 _ 112 • c~T1

Relations:

x = c~11 cos cp,

y = c~l1 sin «1>,

z = c[(~2 -1) (1_,,2)]1/2.

Bipolar CylindersCoordinate variables: (11,~, z)Metric coefficients:

[COSh~~ COS 11 • cosh~~ COS 11 • IJRelations:

a sinh];x= ,

cosh l; - cos 11

a sin 11y = coshj; - cos 11 .'z = z.

In this list, for the case of the elliptical cylinder, the governing equations for theelliptical cylinder and the conformal hyperbolic cylinder are

(2.42)

(2.43)

00 > ~ ~ 1,

1~11~-1,

x 2 rc2~2 + c2 (~2 _ 1) = 1,

x2 rc2112 - c2 (1 -112) = 1,

where c denotes half of the focal distance between the foci of the ellipse. For theprolate spheroid, the governing equations are

Z2 r2

c2~2 + c2 (~2 -1) = 1,

z2 r 2

-- -1c2112 c2(1-11 2 ) - ,

27t :::: «I> ~ 0, 00 > ~ ~ 1,

27t :::: <t> 2: 0, 1 ~ 11 2: -1,

(2.44)

(2.45)

(2.46)27t ~ ~ 0, 00 > ~ 2: 1,

where r 2 = x 2 + j1. Equation (2.44) represents an ellipse of revolution revolvingaround the z axis, which is the major axis. The conformal hyperboloid is represented by (2.45). For the oblate spheroid, the governing equations are

r2 z2

c2~2 + c2 (~2 _ 1) = I,

(2.47)2n ~ ep 2: 0, 1 ~ 11 ~ -1.r2 z2-- -1c2112 c 2 (1 - 112) - ,

Equation (2.46) represents an oblate spheroid generated by revolving an ellipsearound the z axis, which is the minor axis in this case, and (2.47) is the equationfor the conformal hyperboloid.

For the bipolar cylinders, the governing equations are

(x - a coth ~)2 + y2 = a2 csch2 ~,

(y - a cot 11)2 + x 2 = a2 csc2 11,

(2.48)

(2.49)

with 00 > ~ > -00 and 27t > 11 > 0, and where a denotes half of the distancebetween two pivoting points from which these circles are generated. In fact, (2.48)and (2.49) can be derived by considering a conformal transformation between thecomplex variables x + j Y and 11 + j~ in the form

(x + jy) - a = e j (T1+jl;) , (2.50)(x + jy) +a

which is called a bilinear transformation in the theory of complex variables.In the complex (x, y) plane, the numbers c) = (x - a) + jy and C2 =

(x + a) + jyare shown graphically in Fig. 2-2, where we assume a to be real.These numbers can also be written in the form

CI = [(x - a)2 + r]I/2 e j a . ,

C2 = [(x +a)2 + r]I/2 e j a 2 ,

-I YQ) = tan --,

x-a-1 y

Q2 = tan --.x+a

(2.51)

Equation (2.50) is therefore equivalent to

c [(X - a)2 + .2J

1/2...!. = r ej (a . - a 2) = e-~ejll;c2 (x +a)2 + y2

(2.52)

hence

[(x - a)2 + rJI/2 __;

-e ,(x + a)2 + y2

(2.53)

-I Y -1 Ytan -- - tan -- = 11.x-a x+a

(2.54)

Sec. 2-3 Derivatives of Unit Vectors in OCS

jy

33

------...,.--------t~~~iIlp_--------__to---x

Figure 2-2 Locus of the complex number x + j y resulting from a bilineartransformation.

It is not difficult to deduce (2.48) from (2.53), and (2.49) from (2.54). From thisdiscussion, we see that the locus of constant ~, which is a circle, corresponds toa constant ratio of the magnitude of CI and C2, and the locus of constant 11 is alsoa circle conformal to the circle of constant j; The fact that they are conformal isbecause (2.50) is a conformal transformation in the theory of complex numbers.

2-3 Derivatives of Unit Vectors in OCS

Equation (2.28) states

i=I,2,3.aRp _ h. u·aroi - I I,

Let us change the variables Ci'; to Vi for OCS; then,aRp " aR p "--=hjuj, --=hkUk.OVj OVk

Hence

(2.55)

o(hjUj) a(hkUk)----- = (2.56)aVk OVj

because both are equal to o2R p/ovj OVt. We assume that all of the first and secondderivatives of Rp do exist. Equation (2.56) can be written in the form

h ou j oh j " h OUk 8hk A

j-+ -Uj = k- + - us, (2.57)a~ a~ a~ 8~

34

Because

hence


Uj • CJUj =O.aVk

au} /aUk is therefore perpendicular to Iii: In the (vI» VIe) plane, it is parallel to Iik;

similarly, aUk/aVj is parallel to Uj. Thus, we can write

aUj A aUk AA- = a. Uk, - = pUj.aVk av}

Equation (2.57) can now be put in the form

(ahk ) A (ah j A ) A- - ah . Uk = - - ph/c u·.aVj J aVk J

Because Uk and u} are independent and orthogonal to each other, this equation canbe satisfied only if

hence

(2.58)

and

aUk 1 Bhj A

- = --Uj. (2.59)av} hk aVk

Equations (2.58) and (2.59) hold true for j and k = 1,2,3 with j :1= k.The derivative aUj/aVj can be found by considering the relationship among

the three orthogonal unit vectors Uj,Ii}, and Uk in a right-hand system:

{

I , 2, 3 orUj = Uj X Uk, i, j, k = 2, 3, 1 or (2.60)

3,1,2.

The coordinate variables of the system are denoted by (VI, V2, V3). The partialderivative of (2.60) with respect to V; yields

au; A aUk aUj A- = Uj x - + - X Uk.Bu, av; OVt

In view of (2.58) or (2.59), this equation is equivalent to

aUj A 1 ah j A 1 Bh, A A- = U j x - -U; + - -U; X U/cav; h k aVk h j aVj

(2.61)

(1 ah; A I ah; A )=- - -Uk+- -Uj .

hk aVk h j av}

Sec. 2-4 Dupin Coordinate System 3S

(2.62)

Equations (2.58) and (2.59) are very important formulas that will be used frequentlyin subsequent sections. It can be easily verified that as a result of (2.61),

L~ (QUi) =0,i 8v; hi

where Q = hlh2h3. Equation (2.62) can be derived readily from the generalclosed surface theorem stated in (2.27) by letting A = nand ri = uif hi. Thederivation of this theorem for the OCS appears to be more complicated than forthe GCS. However, the relations between the derivatives of the unit vectors giveus a deeper understanding of the vector relations in OCS.

Another identity that can be proved with the aid of (2.58) and (2.62) is

"Uj x ~ (Ui) = 0, j = 1,2,3. (2.63)~ hi 8v; hi

Equations (2.61) and (2.63) will be used in the derivation of many importantformulas, The interpretation of these two identities from the point of view ofvector theorems will be discussed in Chapter 4.

2-4 Dupin Coordinate System

The Dupin coordinate system is an indispensable tool to treat vector analysison a surface. In the general Dupin system, the coordinate variables will bedenoted by (VI, V2, V3) and the corresponding unit vectors by (ill, "2, "3) withmetric coefficients (h), h 2, 1). The variables (VI, V2) are used to describe thecoordinate lines on the surface, while V3 denotes the normal distance measuredlinearly from the surface; hence h3 = 1. For a right-hand system, the direction of"3 is determined by "I X"2. ") and "2 are assumed to be orthogonal, and bothare tangential to the surface. Figure 2-3 shows the disposition of these quantities.The total differential of the position vector measured from a point on the surfaceto a neighboring point in the space is then written as

dRp = h) dVI UI + h2 dV2 "2 + dV3 U3. (2.64)

When VI and V2 have not yet been specified, we designate the system as the generalDupin system. The surface ofa circular cylinder and that of a sphere are two of thesimplest surfaces belonging to the Dupin system. The one-to-one correspondenceof the variables, the unit vectors, and the metric coefficients is listed in Table 2-1.

Let us now consider a spheroidal surface described by

x 2 + r z2---+-=1

b2 a 2

or

(2.65)

36 Coordinate Systems

Lines of constant v2

Lines of constant VI

Figure 2-3 Dupin coordinate system, dRp = hI dv; Ul + h: dV2 U2 + dV3 U3-

Table 2-1: Two Dupin coordinate systems

Chap, 2

System

Cylindrical surfaceSpherical surface

('1>, Z, r)(9, 4l, R)

(~, z, r)(9,~, R)

(r, 1, 1)(R, R sin S, 1)

where (<1>, z, r) are the cylindrical variables. If we choose (<1>, z) as (VI, V2), then

where

(2.66)

Sec. 2-5

Hence

Radii of Curvature 37

hi = b [1 - (~)2r/2,

h2 = [1 + (~:YJ/2 = sec a ,

dr- = tan a = slope of the tangent to the ellipse, (2.65), making an angled Z a with the z axis.

The corresponding unit vectors are

UI =~,U2 = sin a r+ cos a z,U3 = cos 0. r - sin a z.

(2.67)

(2.68)

(2.69)

The choice of (VI, V2) in a Dupin system is not unique. In the previous example,we can use (<1>, r) as (VI, V2); then,

dS2 = 1 + (~;) 2 dr = esc a dr = h z dV2. (2.70)

Hence h2 = esc a, but h i. U1, and U2 remain the same. In the case of a sphericalsurface, we can use (z, x) as (VI, V2). In certain problems dealing with integrations,such a choice sometimes is desirable, particularly from the point of view ofnumerical calculations.

2-5 Radii of Curvature

For a surface described in the general Dupin coordinate system, there are two radiiof curvature of the surface associated with two contours in the (VI, V3) plane andthe (V2, V3) plane. These are two normal planes containing (Ut, U3) and (U2' U3)at P (VI, V2, V3) (see Fig. 2-3). These radii of curvature are closely related to themetric coefficients hI, h2 and the rate of change of these coefficients with respectto V3. Figure 2-4 shows a section of the contour C in the neighborhood of P,resulting from the intersection of the (U2, U3) plane with the surface.

Referring to Fig. 2-4, we denote

PQ = h2 dV2,

oP = R2 (second principal radius of the curvature),

PS = QT = dV3.

38

r

o


PQ=h2dv2

OP=R2

PS =QT= Q'T = dV3

ST=(hz+ ~~ dV3)dV2

Then,

......--...-.----....-----------~zFigure 2-4 Radius of curvature of a surface in the plane containing U2 and U) (r-zplane for the example illustrated in the text).

(ah2 )ST = b: + - dV3 dV2,aV3

, 8h 2Q Q = PQ- ST = -- dV3dv2.aV3

The triangles 0 P Q and T Q' Q are similar; hencePQ Q'QOP = QT '

which yields

or

(2.71)

(2.72)

(2.73)

(2.74)

1 1 ah2

R2 = - h2 aV3 •

Equation (2.73) relates the second principal radius ofcurvature in the (V2, V3) planeat P to the metric coefficient h2 and its derivative with respect to V3 at that point.To find the expression of R2 in terms of the shape of the curve, we have to knowthe governing equation for C. Let this equation be given in the form

r = fi(z),

where z represents V2; then,

dS2 = J(dr)2 + (dz)2 = 1 + (~y dz = hx dvz.

h2 = 1 + (~:)2

= J1 + tan2 a = sec a,

where a denotes the angle of inclination of the tangent at P made with the z axis.Now,

hence

1 1 da da-=--=cos<x-.R2 h2 dz dz

Because

dr ,tan<x=-=r,

dz

a differentiation of this equation with respect to z yields

1 da dr' "---=-=r.cos2 a dz dz

We have

1 3" r"- = (cos a)r = 3/2 .R2 (1 + r,2)

(2.75)

(2.76)

The derivation of this formula is found in many books on calculus. It isrepeated here to show its relationship with the derivative of the relevant metriccoefficient. Equation (2.75) shows that for a concave surface, r" > 0, so R2 ispositive, and for a convex surface, R2 is negative. Similarly, one finds that in the(U., U3) plane,

1 -1 8h l-=--,R 1 hi 8V3

where R I denotes the first principal radius ofcurvature. A formula similar to (2.75)can be derived if the governing equation of the curve in the (VI, V3) plane is known.The reciprocals of the two radii of curvature are called Gaussian curvatures of thesurface in the two orthogonal planes.

As an illustration of the application of these formulas, we consider theequation of a paraboloidal surface defined by

r2 = 4/z, (2.77)

where f denotes the focal length of the paraboloid and (r, cp, z) denote thecylindrical variables. The coordinates in the Dupin system for the surface areidentified as (VI, V2, V3) = (cp, z, V3) with

hI =r,


For the surface under consideration,

r' = dr = rz, r" = _!J7 z-3/2.dz V~ 2

Upon substituting r' and r" into (2.75), we find

R2 = -2/(1 + 7)3/2. (2.78)

At z = 0, R2 = -2f, and at z = f, R2 = -4.J2 I, and so on. To find RI, itis simpler to use (2.76) instead of finding the equation of the cross section in the(U2' U3) plane. Now,

-1 Bh 1 -1 ar sin P- = - - = - - = -- (2.79)R I hI aV3 r aV3 r

where ~ denotes the angle between the normal to the surface U3 and the z axis, and

AI. A 1tanp = -- , slnp = 1/2 •

r' (1 + rt2 )

Thus,

R1 = -r (1 + r 12 ) 1/ 2 = -21 (1 + ]:) 1/2 .

It can be proved that -Rl is the distance measured from the point P(r, z) onthe parabola along the inward normal to its intersect with the z axis. At z = 0,R, = -2f, and at z = I, R1 = -2.J2/ = !R2 • The relationship between R}and R2, in general, is R~ = 4flR2.

As an exercise, the reader may be interested to verify that for a spheroidalsurface defined by

and

2 [ 2 (b2 )J3/2R2 = - ~ 1 + ~ - - 1

b a 2 a 2(2.80)

(2.81)[ 2(b2 )J1/2R1 = -b 1 + =2 a2 - 1 ·

The relationship between R1 and R2 is R~ = (b4 / a 2)R2.The two Gaussian curvatures that are defined by (2.73) and (2.76) are related

to the rate ofchange ofan elementary area of the curved surface. For an orthogonalDupin coordinate system under consideration,

(2.82)

Sec. 2-5

hence


(2.85)

(2.83)

(2.84)

. 8~S3 a(h 1h2)lim -- --- = -- ---

.1.S3-'O ~S3 aU3 h 1h 2 aU3

1 8h t 1 8h2 ( 1 1 )= h; 8V3 + h2 OV3 = - Rl + R2 .

The sum of the two principal Gaussian curvatures is therefore equal to the decreaseof the normal derivative of an elementary area per unit area. We will denote thesum of the two principal curvatures by J:

1 1J=-+R] R2 '

and it will be designated as the surface curvature. It is convenient to define a meanradius of curvature for a surface by Rm , such that

1 1 1-+-=-;R} R2 s;

then,

1J= -. (2.86)

Rm

There is a simple graphical method to determine Rm for a given pair of R 1 and R2based on (2.85). We erect two vertical lines with lengths equal to R1 and R2 ona graded paper and draw two lines from the tips of these two vertical lines tothe bases, as shown in Fig. 2-5. The intersecting point of the two inclined linesyields Rm . The validity of this method is based on the relation

s; n; d2 d}-+-=--+--=1.R 1 R2 d, +d2 d, +d2

The same method applies to the case when R1 > 0 and R2 < 0 (a saddle surface),as shown in Fig. 2-6. In this case,

s; s; - D2 D1 + D2-+- = --+ = 1.R} R2 D1 D1

This simple graphical method applies to problems involving the sum of tworeciprocals such as two resistances or two reactances in parallel and two opticallenses with different focal lengths aligned in cascade. All these quantities couldbe of the same sign or opposite signs. The method is a visible aid to an algebraicidentity.

Figure 2-5 A graphical method to determine Rm for R I , R2 > o.

42 Coordinate Systems

Figure 2-6 s, > 0, R2 < o.

Chap. 2

Chapter 3

Line Integrals, SurfaceIntegrals, and Volume

Integrals

3-1 Differential Length, Area, and Volume

In this section, we shall give a brief review of the differential quantities to be usedin vector analysis and, particularly, their notation. A differential length, in general,will be denoted by di. It is the same as the total differential of the position vectordRp • In OCS,

ae = L h; du, e.. (3.1)

(3.2)

For a cell with its center located at (VI, V2, V3) and bounded by six surfaces locatedat Vi ± du, /2 with i = 1, 2, 3, the vector differential area of the three surfaceslocated at V; + du, /2 is then given by

dS; = hjdvjuj X hkdvkUk Iv;+dv;/2

=hjhkdvjdvkU; IVi+dv;/ 2 ,

where (i, j, k) follows the cyclic order of (1,2,3), (2,3,1), and (3, 1,2). Thevector differential areas of the other three surfaces are

(3.3)

All of these vector areas are pointed away, or outward, from their surfaces. Weshould emphasize that the metric coefficients and the corresponding unit vectors

43

44 Line Integrals, Surface Integrals, and Volume Integrals Chap. 3

(3.4)

are evaluated at the sites of these surfaces Vi ± d Vi /2, not at the center of the cell.The differential volume of the cell is given by

dV = hi dv, Uj . (h j dv j Uj X hk dVk Uk)

=b, h j hk dv, dv j dVk

=hI h2 h3 do, dV2 dV3.

3-2 Classification of Line Integrals

If we continuously change the position vector of a point in space in a certainspecified manner, the locus of the point will trace a curve in space (Fig. 3-1). Leta typical point on the curve be denoted by Ptx , y, z) in the Cartesian coordinatesystem. If (x, y, z) are functions of a single parameter t, then as t varies, x(t),y(t), and z(t) will vary accordingly. We call such a description the parametricrepresentation of a curve. We assume that there is a one-to-one correspondencebetween t and (x, y, z). The vector differential length of the curve can now bewritten as

" A "(dX A dy A dz ")dl=dxx+dyy+dzz= dt x + dt Y + dtZ dt. (3.5)

(3.6)

It should be pointed out here that we use dx, dy, dz, and dt to denote the totaldifferential of these variables, but dx idt , dyf dt , and d zf dt are the derivatives ofx, y, z with respect to t. As an example, let

21tx =acos-t

T. 21t

y = a sin Ttb

z= -tT

where a, b. T are constants and t is the parameter. As t varies, the locus of Pdescribes a right-hand spiral, advancing in the positive z direction as t is increased.When t changes from 0 to T, the spiral starts at (a, 0, 0) and ends at (a, 0, b);therefore, b denotes the height of the spiral after making one complete tum, and

o Figure 3-1 Curve in a three-dimensionalspace.

Sec. 3-2 Classification of Line Integrals 45

a denotes the radius of the circular projection of the spiral on the x-y plane. Tocalculate the length of the spiral, one starts with

(di)2 = .u ·ae = [ (~~Y+ (~~Y+ (~;YJ (dt)2;

hence

(3.7)

The integral of (3.7) from t = 0 to T yields

l L liTL= dl=- [(2na)2+b 2]1/2 dto T 0

= [(21tQ)2 + b2] 1/2 = (c 2 + b2) 1/2 ,

(3.8)

where c denotes the circumference of the projected circle. The pitch angle of thespiral is defined by

a.=tan-I(~). (3.9)

Equation (3.8) represents the simplest form of a line integral. In general, we definethe line integral of Type I as

II =1f(x, y, z) dt; (3.10)

where c denotes the contour of the curve wherein the integration is executed.As an example, let c be a parabolic curve described by

1 = 2x in the plane z = 0, (3.11)

and the contour extends from x = 0, y = 0 to x = ~, y = .J3', and the functionin (3.10) is supposed to be f'(x, y) = xy.

If we choose y as the parameter, then

dl = dx x+ dy Y= y dy x + dy y,dl = (y2 + 1)1/2 dy,

If(x,y) = "21 ;

thus,

( (../3 1 28II = 1c f'tx, y)di = 10 "2 1 (1 + 1)1/2dy = 15 ·

We purposely choose f(x, y) and c in such a manner that the integral can beevaluated in a closed form in order to clearly illustrate the steps.


The second type of line integral is defined by

12 = 1f(x, y, z) ae, (3.12)

where f is a vector function.If we write rin its component form in the Cartesian coordinate system,

f(x, y, z) = h(x, y, z) x + !y(x, y, z) y + !z(x, y, z) z, (3.13)

and because X, y, zare constant vectors, we can change (3.12) into the form

12 = X1fxdl + Y1fydl + Zf Jzdl. (3.14)

The three integrals contained in (3.14) are of Type I, which can be evaluatedaccording to the method described previously. Line integrals of Types ill, IV, andV are defined by

13 =1f(x, y, z)dl,

/4 = 1f(x, y, z) · ae.

Is =1f(x, y, z) x se.

Integrals of Type In can be resolved into three integrals, that is,

(3.15)

(3.16)

(3.17)

. 21tY = asm t;z.

13=x l f dx+ y! fdy+zlfdz. (3.18)

The three scalar integrals in (3.18) can be evaluated by choosing a proper parameterfor each integral. In fact, one can, for example, use x as a parameter for the firstintegral and express both y and z in terms of x. In the case of the spiral contour,if we let z be the parameter, then

21tx =acos-z,

bIntegrals of Type IV can be converted to

/4 = 1(fx dX + fydy+ Jzdz). (3.19)

Again, each term in (3.19) can be evaluated by the parametric method. Integralsof Type V can be written in the form

Is =x1(h dZ- hdy)+ Y f(h dx - ft dy)+z1(ft dy- hdx). (3.20)

All six terms in (3.20) can be evaluated in the same way. Thus, if the functions!and r and the differential lengths d i and d l are expressed in a Cartesian system,

Sec. 3-2 Classification of Line Integrals 47

we have a systematic method to evaluate all different types of line integrals. Inmany cases, the curve under consideration may correspond to the intersection oftwo surfaces represented by

z = Fit», y),

z = F2(x , y).

In that case, we can eliminate z between (3.21) and (3.22) so that

F1(x, y) = F2(X, y)

and then solve for x in terms of y to yield

x = F3(y),

z = Fl[F3(Y), Yl.

(3.21)

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

It is obvious that y can be used as the parameter for the curve. In many problems,it is sometimes rather difficult to find the explicit form of F3 unless F1 and F2 arerelatively simple functions.

If the integrals and the contour c are described in an orthogonal system otherthan a Cartesian system, then

[

3 ] 1/2

ae = ~(h; dv;)2

and

3

dl = Lh; dv, u;.i=1

(3.27)

A scalar function f is then assumed to be a function of (VI, V2, V3), and a vectorfunction f would be a function of both the V; 's and u; 's, Integrals of Types I andIV can be evaluated by expressing the V; 's and h;'s in terms of a single parameter,as was done previously. The integrands of integrals of Types II, III, and V containtt;'S that are, in general, not constant vectors, so they cannot be removed to theoutside of these integrals. For these cases, we can transform the u; 's in terms ofX, y, and Zin the form

u; = cos a; x+ cos fi; y + cos "(; Z, i = 1, 2, 3. (3.28)

Then

where

3 3

f= Lfiu; = Lfi(cosa;x+cosfi;y+cOSYiZ);=1 ;=1

= /xx + hy+ !zz,

3

Ix = L.Ii cos u.,;=1

(3.29)

(3.30)


3

fy = L.Ii COS J3;,;=1

3

fz = L.Ii cosy;.;=1

(3.31)

(3.32)

Afterwards, the unit vectors X, y, and zcan be removed to the outside of theintegrals, and the remaining scalar integrals can be evaluated by the parametricmethod. In a later section, we will introduce a relatively simple method to find thetransformation of the unit vectors from one system to another like (3.28).

3-3 Classification of Surface Integrals

A surface in a three-dimensional space, in general, is characterized by a governingequation

F (x, y, z) = 0, (3.33)

in which we can select any two variables as independent and the remaining one willbe a dependent variable. We assume that we can convert (3.33) into the explicitform

s: z = f(x, y). (3.34)

For two neighboring points located on S, the total differential of the displacementvector between the two adjacent points can be written in the form

dR p = dx x+ dy Y+ dz z. (3.35)

Only two of the Cartesian variables are independent because of the constraint statedby (3.34). If the same surface can be described by the coordinates (VI, V2, V3) withunit vectors (UI' U2, U3) and metric coefficients (hI, h2, 1) in a Dupin system, thenthe total differential of a displacement on the surface can be written as (Fig. 3-2)

(3.36)

(3.37)

(3.38)

with dV3 = o.The partial derivatives of (3.35) and (3.36) with respect to VI and V2 are

aRp h A ax A ay A az A

--= IU)=-X+-y+-z,av) aVI aVI av)

aRp h A ax A ay A az A-- = 2U2 = -x + -y + -z.aV2 aV2 aV2 8V2

Sec. 3-3 Classification of Surface Integrals 49

x

o

Q(x + dx,y + dy, Z + dz);

dR p =hI dVI ul + h2 dV2"2=dxx+dyy+dzz

Figure 3-2 Total differential of the position vector on a surface z = !(x t y) whereV3 =constant.

Let the vector differential area of the surface be denoted by dS; then

dS = h, du, Ut x h2dv2u2

= h , h i dVI dV2 U3

x y zax ay az- -

du, dV2 = J du, dV2.aVt aVt aVIax By az- -BV2 aV2 aV2

(3.39)

The determinant in (3.39), denoted by J, results from hI UI x h z U2. Forconvenience, we will call it the vector Jacobian of transformation between (x, y, z)and (Vt, V2). If we write

(3.40)

(3.41)

then

oy 8z

aVt OVI

oy OZOV2 OV2

is the Jacobian of transformation between (y, z) and (VI, V2). Alternatively, it maybe denoted by

(3.42)

50

Similarly,

Line Integrals, Surface Integrals, and Volume Integrals Chap. 3

and

(3.43)

(3.44)

Now, let us consider the case where the rectangular variables (x, y) areselected as (VI, V2); then

x y z0

azoz A az A A1 -

JI = ax =--x--y+z. (3.45)DZ ox ay

0 -ayThe subscript 1 attached to J 1 means that this is our first choice or first case. From(3.45), we can determine the unit normal vector U3, namely,

A JIU3 = - = -------------~

IJtI

The directional cosines of U3 are therefore given by

8z_ ax

COS£X3 - [(:;y + (:;y + IJ/2 '

OZ

COS~3 = oy[(:;y + (:;y + IJ/2 '

1

(3.46)

(3.47)

(3.48)

(3.49)

Sec. 3-3 Classification of Surface Integrals 51

(3.53)

Based on (3.39) and (3.45), we find

dS= IdSI = IJll dVldv2

= [(:;y + er + IJ/2 dxdy (3.50)

1=--dxdy.COSY3

Equation (3.50) can be used to find the area of a surface. As an example, let thesurface be a portion of a parabola of revolution described by

1 1S : z = 2 (x 2 + y). 2 2: z 2: O. (3.51)

Then

[ (:;)

2

+ (:;)2

+ 1] 1/2 = (x 2 + y + 1)1/2 • (3.52)

Hence

s =fIs. d S =fIs. (x2 + Y + 1)1/2 dx dr

where 81 denotes the domain ofintegration with respect to (x, y), covering the projection of S in the x - y plane. For this particular example, it is convenient to convert(3.53) into an integral with respect to the cylindrical variables rand <1>, that is,

S=III(1 + r 2f /2 r dr de =12ft11

(1 + r2) 1

/2 r dr d. = 4~1t. (3.54)

Here, we have used the transformation

dx d = a(x, y) dr dey a (r, ep) 'fI'

where

ax oy-o (x, y) or or

= =ra(r, ,) ax ayact> act>

with

(3.55)

(3.56)

x = r cos e, y = r sin e.

Equation (3.55) is a special case of (3.39) when it is applied to a plane surfacecorresponding to the x-y plane.


Returning now to the expression for J defined by (3.39), we can select either(y, z) or (z, x) as two alternative choices for (VI, V2); then

xaxByaxBz

y z1 0

o 1

" ax " ax A=x - -y- -z.By Bz

(3.57)

In this case, x is the dependent variable. The expression for the unit vector U3 isnow given by

(3.58)

and

Similarly, we have

x y z0

By1 By" " ay"

J3 = az =--x+y--z,ax az1

By0

ax

dS = J3dxdz = IJ31 dxdzU3 = ~dxdzU3.COS..,3

(3.59)

(3.60)

(3.61)

The directional cosines of U3, therefore, can be expressed in several different forms.Our three different choices of (VI, V2) yield

Sec. 3-3 Classification of Surface Integrals

az-- 1

cos u, = [(:=Y + (:;Y + IJ/2 = [(:;Y + (::Y + IJ/253

ay_ ax

-[(::Y + (:~y + IJ/2 •

az ax

(3.62)

1

8y_ az

-[(::Y + (:~Y + IJ/2 •

We remind the student that

dy = IjdXdx dy

for functions of single variables, such as y = f'tx), but

ay ¥ 1 jaxax ay

(3.63)

(3.64)

(3.65)

(3.66)


for functions of multiple variables. As an example, we consider the relations

Then

x = r cos e,

(2 ..2)1/2r=x+y ,

y = r sin o.

so that

and

That is,

ax- = cos ear

or x- = ]/2 = cos e.ax (x 2 + y2)

ax ar-=-,or ax

while

so that

ax .ac!> = -rslDc!>,

a~ sin epax = --r-

ax 2a.-=r-.aCl> ax

One must therefore be very careful to distinguish between the dependent andindependent variables.

Like the line integrals, there are five types of surface integrals. From now on,functions of space variables (x, y, z) or (VI, V2, V3) will be denoted by F(Rp) orF(Rp) , where Rp denotes the position vector. The five types of surface integralsare as follows:

Type I:

Type II:

Type Ill:

I) = lis F(Rp)dS,

12 = lis F(Rp)dS,

(3.67)

(3.68)

(3.69)

Sec. 3-3 Classification of Surface Integrals

Type IV:

/4 =fl F(Rp) · dS,

Type \1:.

Is = fl F(Rp ) x dS,

55

(3.70)

(3.71)

where F(Rp) is a scalar function of position and F(R p) denotes a vector function.We assume that the surface S can be described by a governing equation of

the form

z = f(x, y). (3.72)

The same surface can always be considered as a normal surface (V3 = 0) in a properDupin system with parameters (VI, V2), (UI' U2) and metric coefficients (h I, h2).Treating VI and V2 as two independent variables, we can write

x = fi (VI, V2), (3.73)

Y = h (VI, V2) , (3.74)

z = f(x, y) = f[fi (VI, V2), h (v], V2)] = h (VI, V2). (3.75)

The functions F (rp) and F (rp ) contained in (3.67)-(3.71), therefore, can bechanged into functions of (VI, V2) for the scalar function, and of (VI, V2) as well as(u I, U2, U3) for the vector function. An integral of Type I can be transformed into

II = fl F (VI, V2) IJI dVI dV2, (3.76)

which can be evaluated by the parametric method. Thus, if we let (VI, V2) = (x, y),(3.76) becomes

II = for rt». y) _1_ dxdy, (3.77)JS3 COSY3

where S3 denotes the domain of integration on the x-y plane covered by theprojection of S on that plane. The execution to carry out the integration is verysimilar to the problem of finding the area of a curved surface, except that theintegrand contains an additional function.

An integral of Type II is equivalent to

12 = Xff Fx (VI. V2) IJI dVI d V2

+ yff r, (VI. V2) IJI dVI dV2 (3.78)

+ zff r. (VI. V2) IJI dVI dV2.


The three scalar integrals in (3.78) are of Type I. However, it is not necessary touse the same set of (VI, 1J2) for these integrals.

An integral of Type III, in view of (3.39) and (3.40), is equivalent to

13 = ff F (Vlt 112) J dv, d112 = Xff J x F (Vlt 112) dv, dV2

+ Yff s,F (Vlt 112) dv, d112 (3.79)

+ zff J z F (Vlt 112) du, d112.

The three scalar integrals in (3.79) are of Type I with different integrands.An integral of Type IV is equivalent to

/4 =ff (Jx r; + J y r; + J zFz) ss. (3.80)

which belongs to Type I. Here, we have omitted the functional dependence ofthese functions and the Jacobians on (VI, V2).

Finally, an integral of Type V is equivalent to

Is = xff (Jz r; - i; Fz) d V l d v2

+ Yff o. r; - J z Fx ) du, dV2 (3.81)

+ zff (Jy r; - Jx F y ) du, dV2.

All of the scalar integrals in (3.81) are of Type I. In essence, an integral of Type Iis the basic one; all other types of integrals can be reduced to that type. The choiceof (VI, V2) depends greatly on the exact nature of the problem. Many integralsresulting from the formulation of physical problems may not be evaluated in aclosed form. In these cases, we can seek the help of a numerical method.

3-4 Classification of Volume Integrals

There are only two types of volume integrals:

Type I:

(3.82)

Type II:

(3.83)

Sec. 3-4 Classification of VolumeIntegrals 57

where V denotes the domain of integration, which can be either bounded orcovering the entire space. We now have three independent variables. In anorthogonal system, they are (VI, V2, V3). An integral of Type I, when expressed inthat system, becomes

II =IIIv F(vlt 1J2. V3)hi h2 h3dVI d1J2 dV3. (3.84)

The choice of the proper coordinate system depends greatly on the shape of V.From the point of view of the numerical method, we can always use a rectangularcoordinate system to partition the region of integration.

An integral of Type n is equivalent to

12 =xIIIFxdV + yIIIFydV +zIIIFzdV. (3.85)

The three scalar integrals in (3.85) are of Type I. We will not discuss the actualevaluation of(3.84), as the method is described in many standard books on calculus.

Chapter 4

Vector Analysis in Space

4-1 Symbolic Vector and Symbolic Vector Expressions

In this chapter, the most important one in the book, we introduce a new method intreating vector analysis called the symbolic vector method. The main advantagesof this method are that (1) the differential expressions of the three key functionsin vector analysis are derived based on one basic formula, (2) all of the integraltheorems in vector analysis are deduced from one generalized theorem, (3) thecommonly used vector identities are found by an algebraic method withoutperforming any differentiation, and (4) two differential operators in the curvilinearcoordinate system, called the divergence operator and the curl operator anddistinct from the operator used for the gradient, are introduced. The technicalmeanings of the terms "divergence," "curl:' and "gradient" will be explainedshortly. Note that the nomenclature for some technical terms introduced in thischapter differs from the original one used in [6].

Because vector algebra is the germ of the method, we will review severalessential topics covered in Chapter 1. In vector algebra, there are various products,such as

ab, a vb,

c· (a x b),

a x b, c(a · b), c(a x b),

c x (a x b), (a x b) · (c x d).(4.1)

All of them have well-defined meanings in vector algebra. Here, we treat thescalar and vector quantities a, a, b, b, c, c, d as functions of position, and they are

S8

Sec. 4-1 Symbolic Vector and Symbolic Vector Expressions 59

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

assumed to be distinct from each other. For purposes ofidentification, the functionslisted in (4.1) will be referred to as vector expressions. A quantity like ab is nota vector expression, although it is a well-defined quantity in dyadic analysis, asubject already introduced in Chapter 1. For the time being, we are dealing withvector expressions only. In one case, a dyadic quantity will be involved, and itsimplication will be explained. All of the vector expressions listed in (4.1) arelinear with respect to a single function, that is, the distributive law holds true. Forexample, if e = Cl + C2, then

C • (a x b) = (CI + C2) • (8 x b) = Cl • (8 X b) + C2 • (8 X b). (4.2)

The important identities in vector algebra are listed here:

ab = ba,

a- b =b·a,

a x b = -b x a,C • (8 X b) = b · (c x a) = a · (b x c),

C x (a x b) = (c · b)a - (c · a)b.

The proofs of (4.6)-(4.7) are found in Section 1-2.Now, if one of the vectors in (4.1) is replaced by a symbolic vector denoted

by \7, such as aV, V· b, a- V, V x b, a x V, cV· b, ea- V, c· (V x b), andso on, these expressions will be called symbolic vector expressions or symbolicexpressions for short. The symbol \7 is designated as the symbolic vector, orS vector for short. Besides V, a symbolic expression contains other functions,either scalar orland vector. Thus, c(V x b) contains one scalar, one vector, andthe S vector. In general, a symbolic expression will be denoted by T (V). Morespecifically, ifthere is a need to identify the functions contained in T (V) besides V,we will use, for example, T(V, a, b), which shows that there are two functions, aand b, besides V. These functions may be both scalar, both vector, or one each.We use a tilde over these letters to indicate such options. The symbolic expressionso created is defined by

T(V) = lim Li T(ni) ASi , (4.8)aV~O I1V

where I1Sj denotes a typical elementary area (scalar) of a surface enclosing thevolume !1V of a cell and n; denotes the outward unit vector from I1S;. Therunning index i in (4.8) corresponds to the number of surfaces enclosing ~V.For a cell bounded by six coordinate surfaces, i goes from 1 to 6. Because thedefinition of T (V) is independent of the choice of the coordinate system, (4.8)is invariant to the coordinate system. The expression on the right side of (4.8),from the analytical point of view, represents the integral-differential transform ofthe symbolic expression T(V), or simply the functional transform of T(V). Bychoosing the proper measures of t!.S; and ~V in a certain coordinate system, onecan find the differential expression of T (V) based on (4.8).

60 Vector Analysis in Space Chap. 4

There are several important characteristics of (4.8) that must be emphasized.In the first place, because all symbolic expressions are generated by well-definedvector expressions, they are, in general, involving simple multiplication, scalarproduct, and vector product. For example, if we replace the vector function d inthe vector expression ab · (e x d) by V, we would create a symbolic expressionof the form

T(V) =ab · (c x V), (4.9)

where a is a scalar function, b and c are vector functions; then, we have amultiplication between a and the. rest of T(V); a scalar product between band(c x V); and finally a vector product between c and V, or the S vector. The vectorproduct c x V is not a true product because V is a symbolic vector, or a dummyvector, not a function. However, the expression leads to a function T (n }) in (4.8)in the form

(4.11)

(4.12)

(4.17)

(4.13)

(4.14)

(4.15)

(4.16)

T(nj) =ab· (c x n}). (4.10)

This isa well-defined function in which c x n} is truly a vector product. Thisfunction is used to find the differential expression of T(V) based on the rightside term of (4.8). This description consists of the most important concept in themethod of symbolic vector: the multiplication, the scalar product. and the vectorproduct contained in T(V) are executed in the function T(nj).

Another characteristic of (4.8) deals with the algebraic property of T(V).Because T(n) is a well-defined function for a given T(V), the vector identitiesapplicable to T(ni) are shared by T(V). In the previous example,

T(ni) =ab · (e x nil = -b · (ni x c)a = an; · (b x c),which implies

T(V) =ab · (c x V) = -b · (V x c)a = aV · (b x c).This property can be stated in a lemma:

Lemma 4.1. For any symbolic expression T (V) generated from a validvector expression, we can treat the symbolic vector V in that expression as a vectorand all of the algebraic identities in vector algebra are applicable.

For example, we have the vector identities listed in (4.3) to (4.7); then,according to Lemma 4.1, the following relations hold true:

aV = Va,

V·a=a·V,

V x a= -a x V,

b · (8 X V) = V · (b x a) = a · (V x b),

V x (8 X b) = (b x a) x V

= (V · b)a - (V · a)b=(b · V)a - (a · V)b.

(4.19)

Sec. 4-2 Differential Formulas in the Orthogonal Curvilinear Coordinate System 61

4-2 Differential Formulas of the Symbolic Expressionin the Orthogonal Curvilinear Coordinate Systemfor Gradient, Divergence, and Curl

Orthogonal coordinate systems are the most useful systems in formulating problems in physics and engineering. We will therefore derive the relevant differentialexpressions in OCS first based on the method of symbolic vector and treat theformulation for the general curvilinear system later. Actually, it is more efficientto do GCS first and consider OCS as special cases. For expository purposes,however, it is desirable to follow the proposed order because most readers arelikely to have some acquaintance with vector analysis in orthogonal systems froma course in college physics or calculus. The mathematics in GCS would distractfrom the main feature of the new method at this stage.

To evaluate the integral-differential expression that defines T (V) in (4.8) inthe OCS, we consider a volume bounded by six coordinate surfaces; then therunning index in (4.8) goes from i = 1 to i = 6. Because T(n;) is linear withrespect to ni, we can combine it; with Ii.S i to form a vector differential area. Ifthat area corresponds to a segment ni 60S on a coordinate surface with Vi beingconstant, then

A SAO An, ti. =h jh/c /),.Vj /),.v/c u, = hi /),.v j /),.V/c u., (4.18)

where i, j, k = 1,2,3 in cyclic order and 0 = hihzh». The hi'S are the metriccoefficients of any, yet unspecified, orthogonal curvilinear system. For a cellbounded by six surfaces located at Vi ± /lVi /2, its volume, denoted by Ii.V , is

6.V = n aVi /lVj ~Vk.

We can separate the sum in (4.8) into two groups:

L:=l [r (~ Ui) - r (~ Ui) ] !:1Vj 6.vkT(V) = lim I V;+!:.Vi/2 I Vi-!:.Vi/2 •

!:.v-+o n !:1Vi AVj !:1vk

The limit yields

1"a (0 A)T(V) = - L.J- T -U; ·Q t aVi hi

Equation (4.19) is the differential expression for T(V) in any OCS. In (4.19), thesummation goes from i = 1 to i = 3 and the metric coefficients and hence theparameter n are, in general, functions of Vi, the coordinate variables. Equation(4.19) is perhaps the most important formula in the method of symbolic vectorwhen it is applied to an OCS.

We now consider the three simplest but also most basic forms of T(V),in which T (V) contains only one function besides V. There are only three

62

possibilities, namely,

Vector Analysis in Space Chap. 4

(4.20)

Tl(V) = VI or IV,T2(V) = V · F or F· V,

T3 (V) = V x F or - F xV,

where I denotes a scalar function and F, a vector function. They are the sameas the ones in (4.13), (4.14), and (4.15). The differential expressions of thesefunctions can be found based on (4.19).

1. Gradient of a scalar functionWhen T(V) = T1(V) = VI = IV, (4.19) yields

Vf = ..!- E ~ (0 itd ). {1 i av; h;

=..!- E [0 iti af + f~ (0 iti ) ] .{1; h, av; av; h,

The last sum vanishes because of the closed surface theorem stated by(2.62) in OCS, namely,

Hence

E ~ (0 iti) = O.t av; h;

(4.21)

(4.25)

Vf = L iti af . (4.22)i hi av;

The differential expression thus derived is called the gradient of the scalarfunction I. It is a vector function and it will be denoted by VI, where Vis a differential operator defined by

V - " iti ~ (4.23)- ~h; av;·,

It will be called a gradient operator; thus,

Vf= L iti af . (4.24)i hi aVt

The linguistic notation used by some authors is grad I. The symbol V isalso called del or nabla or Hamilton operator.

2. Divergence of a vector functionWhen T(V) = T2(V) = V · F =F· V, (4.19) yields

V·F= ..!- E ~ (Q it · F) = ..!- E ~ (Q F;) .Q i av; h, Q i av; b,

Sec. 4-2 Differential Formulas in the Orthogonal Curvilinear Coordinate System 63

(4.28)

(4.29)

The differential expression thus derived is called the divergence of thevector function F. There is another functional form of this function thatcan be obtained from the second term of (4.25). We split the derivativewith respect to Vi into two terms, just as the splitting in (4.20), that is,

!..L~(OU.F)=!..L[QUi' aF +F.~(QUi)].n i (JVi hi nih; Bu, (JVi hi(4.26)

The second term vanishes as a result of (4.21); hence

V·F= L Ui . aF . (4.27)i h; av;

Now, we introduced a divergence operator, denoted by V , and defined by

'"' u- aV=£...J-'!".-.i hi av;

Then,

LUi. aF =VF.i hi av;

It can be verified that by evaluating the derivatives of F with respect to V;,

taking into due consideration that the unit vectors associated with Farefunctions of the coordinate variables and they can be expressed in terms ofvarious £Ii's according to (2.58) and (2.61), the function at the left of (4.29)reduces to (4.25) as it should be. The operand of a divergence operator mustbe a vector. Later OD, we will show that it can also be applied to a dyadic.Comparing the divergence operator with the gradient operator defined by(4.23), we see that there is a dot, the scalar product symbol, between £Ii / h,and the partial derivative sign. It is this morphology that prompts us to usethe symbol V for the divergence operator. We must emphasize that there isno analytical relation between the gradient operator V and the divergenceoperator V. They are distinct operators. In the history of vector analysis,there are two well-established notations for the divergence. One is thelinguistic notation denoted by div F. The connotation of this notationis obvious. Another notation is V · F, which was due to Gibbs, one ofthe founders of vector analysis. Unfortunately, some later authors treatV · F as the scalar product or "formal" scalar product between V and Fin the rectangular coordinate system, which is not a correct interpretation.The contradictions that resulted from the improper use of V are discussedin the last chapter of this book. The evidence and the logic describedtherein strengthens our decision to adopt VF as the new notation for thedivergence. In the original edition of this book [7], we kept Gibbs'snotations for the divergence and the curl, a function to be introduced

(4.31)

(4.32)

(4.33)


shortly. The use of Gibbs's notations does impede the understanding ofthe symbolic expressions V · F and V x F. We have therefore made a boldmove in this edition by abandoning a long-established tradition. With thismuch discussion of the new operational notation for the divergence, weconsider the last case of the triad.

3. Curl of a vector functionWhen T(V) = T3(V) = V x F = -F x V, (4.19) yields

1'" a (0" )v x F = - L.J - - U; x Fn t av; h,

1[3 " a hh"= S1 8Vt (h2h3Ut x F) + 8112 ( t 3U2 X F)

+ 8~ (h th2U3 x F)]l[a " " a hh " A= - - h2h3( F 2U3 - F3 U2) + - t 3(-FtU3 + F3UI)Q aVt aV2

+ a~3 ht h2( F tU2 - F2Ut)] ·

(4.30)Each term in (4.30) can be split into two parts; for example,

.!.. ~ (h h F " ) - .!.. [h F B(h3U3) h" B(h2F2) ]n a 2 3 2U3 - 1"'\ 2 2 B + 2U3 B ·~~ VI i)li VI VI

As a result of (2.56), one finds that all the terms containing the derivativesof hi"; cancel each other. The remaining terms are given by

V x F = .!.. Lh;u; [B(hkFk) _ B(h)FJ ) ] ,o ; av) aVk

where i, j, k = 1, 2, 3 in cyclic order. This function is called the curlofF. Like the divergence, we can find an operational form ofthis function.Using the first line of (4.30), we have

1'" B (0" ) 1"[0 "BF a (0,,)]- L.J - - U x F = - L..J - Uj x - - F x - - U; •Q i av; h; n I hi av; av; h;

The last term vanishes because of (4.21); hence

"Il; 8FVXF=L..J-x-.i h; av;

Now we introduce a curl operator, denoted by V, and defined by

"Ui aV=L..J-x-.i h, av;

Sec. 4-3 Invariance of the Differential Operators 65

(4.34)

(4.35)

The operand of this operator must be a vector. The curl of a vectorfunction F therefore can be written in the form

L UI x aF = 'f F.t b, av;

It can be verified that by evaluating the derivatives of F with respect to VI,

we can recover the differential expression of ., F given by (4.31). Thelinguistic notation for the curl is simply curl F, used mostly in Englishspeaking countries, and rot F in Germany. Gibbs's notation for thisfunction is V x F. As with the divergence, some authors treat V x F as avector product or a "formal" vector product between V and F, which is amisinterpretation. The new notation avoids this possibility. The locationof the cross sign (x) in the left side of (4.34) suggests to us to adopt thenotation V for this function. In summary, we have derived the differentialexpressions of three basic functions in vector analysis in the OCS basedon a symbolic expression defined by (4.8); they are

Vf -- "u; af d·LJ (gra lent),; h, ov;

VF =L UI • aF =.!. L ~ (n F;)i h, av; n; av; h,

'fF= L UI x aFi h, oV;

(divergence), (4.36)

(4.37)

(4.38)

(4.39)

where g = h 1h 2h 3 • In the rectangular system, all the metric coefficientsare equal to unity; hence

f "A afV = L..JX; -,i ax;

"A of "oF;VF= L..Jx;· - = L..J-'i OX; i OX;

" A aF "A (OFk OFj )V F = L..J X; x - = L..J Xi - - - ·i ax; i aXj aXk

4·3 'nvarlance of the Differentia' Operators

(4.40)

From the definition of T(V) given by (4.8), the differential expressions evaluatedfrom that formula should be independent of the coordinate system in which thedifferential expressions are derived, such as the gradient, the divergence, and the


curl. It is desirable, however, to show analytically that such an invariance is indeedtrue.

We consider the expressions for the divergence in two orthogonal curvilinearsystems:

and

VF= LUi. aFi hi av;

(4.41)

(4.42)V'F' = L U~ . a~ .j h j 8vj

All the primed functions and the operator V' are defined with respect to vj.Because F and F' are the same vector function expressed in two different systems,it is sufficient to show the invariance of the two divergence operators. The totaldifferential of a position vector is given by

dR p = LhiUi dv, = Lhjuj dvj. (4.43)i j

Thus,

h~ dv~ = Lhi(u; · u~) dv.,i

or

hj dv', = L lti(Ui · uj) dv.,i

Hence

av'. h.J , (A AI )-a-: ;::: hi u. - U j •v, j

Now,

k = 1,2,3

j=I,2,3. (4.44)

(4.45)

In view of (4.45), we obtain

but

'" A (A AI ) 1 AI AIL..JU; ui v u , = ·Uj=Uj,i

(4.46)

(4.47)

(4.48)

Sec. 4-3 Invariance of the Differential Operators 67

(4.49)

where I denotes the idemfactor introduced in Chapter 1, now in terms of the dyadsu;u;. Another interpretation of (4.48) is to write

'"' A (A AI) ~ A AI~ u, u, . U j = L...J u, cos a;j = U j'

i ;

where cos aU denotes the directional cosines between u; and uj. Substituting(4.48) into (4.47), we obtain

u'. 0W = L -1-. -, = V'. (4.50)

j h j oVi

The proofs of the invariance of V and V follow the same steps.The invariance of the differential operators also ascertains the nature of

these functions. To show their characteristics, let the primed and the unprimedcoordinates represent two rectangular coordinate systems rotating with respectto each other as the ones formed in Section 1-3. The invariance of the gradientoperator means

LX; :~ = Lxi :~ ·; x, j x j

By taking a scalar product of this equation with xk' we find

a~=Lakiaf, k=1,2,3.oXk i ax;

These relations show that the components of the gradient obey the rules oftransform of a polar vector.

The invariance of the divergence operator means

'"x. . of =" x'.. aF'~'o L..JJ a"i Xi j X j

or

La~ =LaFf.i ax, j aX j

The divergence of a vector function, therefore, is an invariant scalar.The invariance of the curl operator has a more intricate implication. In the

first place, the invariance of V'F means

LX; (OFt _ BFj ) =LX~ (a~ _aFf) ,i Bx, OXk p oXq aXr

where i, t. k = 1,2, 3 and p, q, r = 1,2, 3 in cyclic order. By taking the scalarproduct of the previous equation with x~, we obtain

a~~ _aF~ = Lat; (aFt _ OFj ) , l = 1,2,3,dXm oXn i ax j dXk


where l, m, n as well as i, j, k = 1, 2, 3 in cyclic order. If we denote

aF~ _ aF~ _ C'ax' ax' - t»

m "

aFk _ aFj =c/aXj OXk '

then

C~ = LQt;C;,i

which shows C~ and C; transform like a polar vector. On the other hand, if wedenote

aF~ _ aF~ _ C'ax' ax' - mn»

m "

aFk _ aFj = Cjk.aXj aXk

then

Hence

C~1I =L Qmj QlIkC j k .jtk

The preceding relation shows that C~n and Cjk transform according to the role ofan antisymmetric tensor in a three-dimensional space. The tensor is antisymmetricbecause

and

C~m =0, Cj j = O.

Sec. 4-4 Differential Formulas in the General Curvilinear System 69

In summary, the curl ofF is basically an antisymmetric tensor, but its three distinctvector components C; and C~ also transform like a polar vector. Its propertytherefore resembles an axial vector. However, one must not treat ~ F as the vectorproduct between V and F, which is a misleading interpretation; it is fully explainedin Chapter 8.

4-4 Differential Formulas of the Symbolic ExpressionIn the General Curvilinear System

The integral-differential expression that defines T (V) in (4.8) will now be evaluatedto obtain a differential expression for T (V) in GCS, which was introduced inChapter 2. We consider a volume bounded by six coordinate surfaces; then therunning index in (4.8) goes from i = 1 to i = 6. Because T(n;) is linear withrespect to n;, we can combine !!taS; with n; to form a vector differential area. Ifthat area corresponds to a.segment n; tiS; on a coordinate surface with V; beingconstant in GCS, then

n; tiS; = tiS; = PJ X Pk ~ooJ ~rok = Ari !!taro} ~OOk. (4.51)

For a cell bounded by six surfaces located at 00; ± (~co; /2) with i = 1,2,3, itsvolume would be equal to

~V = A ~co; !!taco} ~COk.

We can separate the surface sum in (4.8) into two groups:

. Ei=l [T(Ari)cd+4m'/2 - T(Ari)mi-4mi/2] !!taoo J !!taco k

T(V) = lim . k •~v-+o A !!taro E 11CJ)J !!taco

The limit yields

1 3 a .T(V) = - L -a. T(Ar'). (4.52)

A ;=1 co'

Equation (4.52) is the differential expression for T(V) in GCS. From now on, it isunderstood that the summation index i goes from 1 to 3 unless specified otherwise.We now consider the three symbolic expressions VI, V · F, and V· Fin GCS.

Because T(Ari) is linear with respect to Ari, there are also three possiblecombinations of Ari with the remaining part of T(Ari). In a rather compactnotation, we can write T(Ari) for the three cases in the form

T(Ari) = Ar; *1, (4.53)

(4.54)

where *represents either a null (absent) when lis a scalar function, or a dot (scalarproduct symbol), or a cross (vector product symbol) when 1 is a vector function.We list these cases in Table 4-1. Substituting (4.53) into (4.52), we obtain

1 La. - 1 L [ . a1 a(Ari) -]T(V)=- -.(Ar'*f)=- Ar'*-a' +-a-·-*I ·A . a00' A . CO' 00', ,


Table 4-1: The Three Simplest Forms of T(Ari)

Case T(V, 1> T(Ari) * 11 V/=fV Aril null I, scalar2 V·F=F·V Ari·F F, vector3 V x F= -F x V Ari x F x F, vector

The second term in (4.54) vanishes as a result of the closed surface theorem statedby (2.27), hence

alT(V) = ~rI * ami ·,

(4.55)

We now treat the cases listed in Table 4-1 individually.

1. The gradient of a scalar functionIn this case, we have

T(V) = VI = IV, 1 = f,and * is null or absent in (4.55); then

Vf =L:rI :~, . (4.56)i

This function is the gradient of I in GCS. The gradient operator is nowdefined by

Thus,

L · av = r' -. (gradient operator).. am',

L:ri aa l , = VI. m,

(4.57)

(4.58)

In summary, the gradient of a scalar function I in GCS is represented by

VI = lim Lj(njf) ASj = "rI a/. . (4.59)~v...o ~v ~ aro'

J

2. The divergenceIn this case, we have

T (V) =V · F =F · V, T(Ari) = Ari · F, *=., I=F.Substituting these quantities into (4.55), we obtain

L:. BF

V·F= r'.-. am; ·J

(4.60)

Sec. 4-4 Differential Formulas in the General Curvilinear System

The divergence operator now has the form

L · av = r' . - (divergence operator).. aroi,

Hence

L · aFr'.- =VF. aroi ·,

71

(4.61)

(4.62)

In summary, the divergence of a vector function F in GCS is representedby

(4.63)

(4.64)

(4.65)

It will be recalled that the operational form of this function wasoriginally derived from the following expression:

1 La.VF = - -.(Ar' ·F).A . aro',

According to (2.11),

where t denotes the reciprocal components of F in GCS; hence thedivergence of F can be written in the form

1 La.VF= - -a.(Ag).A . ro',

The differentiations are now applied to the scalar functions Ai; it is nolonger applied to the full vector function F as in (4.62) or (4.63).

3. The curlIn this case, we have

T(V) = VxF = -FxV,

Then

T(A~) = Ari xF, *= x, j=F.

L · 8FVxF= r'x-... aro',

The curl operator now has the form

L · av = r' x -. (curl operator).. 8ro',

Hence

L · aFr' x -. = "F.. aro',

(4.66)

(4.67)


The curl of a vector function in GCS is therefore represented by

VF = lim L;(n; x F) AS; ="rJ x aF.. (4.68)~v~o ~V c: am',

The operational form of curl F was originally derived from the following expression:

1 La.1f'F = - -. (Ar' x F).A . am',

(4.69)

The vector product in (4.69) can be expressed in terms of the primary componentsof F in GCS. We start with

F=Lfjr i , (4.70)i

where fi denotes a primary component ofF as stated by (2.10); then

ri xF= Ljjri x r i , i = 1,2,3. (4.71)i

More specifically,

1 1 -2 1 _3 1r x F = hr x r + hr x r = A (hp3 - hp2)'

r x F = fir x r1+ hr'l x .-J = ~(-fip3 + hPl),A

__3 3 1 3 -2 1r x F = fir x r + hr x r = A (fip2 ..,.. hpt),

where Pi and f; denote the primary vectors and the primary components, respectively, of F. Substituting these expressions into (4.69), we have

1[8 a a ]1f' F = A 8ro l (hp3 - !3P2) + 8m2 (/3Pl - lip3) + 8ro3 (Jip2 - hpl) .

(4.72)The derivative of the first term in (4.72) consists of two parts:

a ap3 a12aool (hp3) = h amI + P3 amI .

The derivative of the last term in (4.72) gives

a apI af28003 (-hpI) = - 12 8m3 - PI am3 ·

According to (2.2),

aRp an,P3 = aoo3 and PI = aco l ;

(4.73)

(4.74)

Sec. 4-4 Differential Formulas in the General Curvilinear System

hence

73

(4.75)

82Rp 8P3 8Pt

8m l 8m 3 = oro t = 8m 3 .

The first two terms at the right sides of (4.73) and (4.74) are therefore equal andopposite in sign. Six terms in (4.69) involving the derivatives ofthe primary vectorscancel each other in this manner; the net result yields

V F = ~ [PI (:~2 - :~~)+ P2 (:~~ - :~I )+ P3 (:; - :~~)lor

"F = .!.." .(8fk _ Ojj)A c: P, aro} CJro k

I

with (i, j, k) = (1,2,3) in cyclic order. The expressions for VI, VF, andV F given by (4.60), (4.65), and (4.75) have previously been derived by Stratton[5, p. 44] [based on the variation of I (total differential of j) for Vf], Gauss'stheorem for VF, and Stokes's theorem for V F. Wehave not yet touched upon thesetheorems. Our derivation is based on only one formula, namely, the differentialexpression of the symbolic expression T(V) as stated by (4.8). In summary, thethree differential operators in GCS have the forms

L · av = r' -. (gradient operator),· oro',

L · av= r'·-· aro i,

L · av= r'x-· oro i,

(divergence operator),

(curl operator).

The three operators can be condensed into one formula:

L · 0..,= r'*-.,. 8ro'I

(4.76)

(4.77)

where * represents a null, a dot, or a cross. We would like to emphasize thatthese operators are independent of each other; in other words, they are distinctlydifferent differential operators, and they are invariant with respect to the choice ofthe coordinate system. We leave the proof in GCS as an exercise for the reader.For the three functions, they can be written in a compact form

- L· a1~f= r'*-. oro i ',where 1 can be a scalar (for the gradient) or vector (for the divergence or curl).This completes our presentation of the differential expressions of the three keyfunctions in their most general form.


The expressions of the three key functions in OCS given by (4.35) to (4.37)can now be treated as the special case of the formulas in GCS. We let

Pi =h;u;, i = 1,2,3 (4.78)

(4.79)

(4.80)

according to (2.28), where hi and u; denote, respectively, the metric coefficientsand the unit vectors in OCS. The unit vectors are orthogonal to each other; thus,

{h~, i=j

Pi · Pj = 0: i # j

{h;h j U/c, i ¥= j ¥= k

Pi X Pj = 0 .., l=}

for (i, j, k) = (1, 2, 3) in cyclic order. The parameter A reduces to

A = Pi · (Pj x Pk) = h;hjh k = Q.

The reciprocal vectors r.i become

. 1 h jh/c " u;r' = -Pj X P/c = -- u; = -.A Q h;

The differential expression of the symbolic expression in OCS is then given by

1'" a (0" )T(V) = - L...J -. T - U; •Q i aro' h,

We still use (J)i to denote the coordinate variables in OCS. The operational formof the three key functions previously described by

- L' alv j « r'*-.. 8ro't

reduces to

or more specifically,

vr- LUi af "; h; dro'

VF= LUi. aF ,; hi aro;

VF= L Ui x aF,.i hi aro'

To find the component form of VF and V F in OCS, we let

F= Lfiu;.

(4.81)

(4.82)

(4.83)

(4.84)

(4.85)

(4.86)

Sec. 4-5 Alternative Definitions of Gradient and Curl 75

The primary and the reciprocal components of F, in view of (2.10) and (2.11), arerelated to Pi by

Ji=h;F;, i=I,2,3, (4.87)1

gi = -Pi, i = 1,2,3. (4.88)hi

The component forms of VF and V'F as given by (4.65) and (4.75) respectivelybecome

(4.89)

(4.90)

(4.91)

In the special case ofan orthogonal linear system or the rectangular system Ui = Xi,roi = Xi, hi = 1 for i = 1,2,3, and 0 = 1. Equations (4.83), (4~89), and (4.90)are the commonly used formulas for these functions in devising physical problems.

4-5 Alternative Definitions of Gradient and Curl

To distinguish these functions in a conceptual manner, we propose some namesfor the integral-differential expressions of these functions based on a "physical"model. Thus, the quantity Ei nifb. Si in the second term of(4.59) will be identifiedas the total directional radiance of I from the volume cell !:l V, or radiance forshort; the gradient is then a measure of radiance per unit volume. The quantityL; ni .F f:,.S; in the second term of (4.63) has a well-established name used bymany authors as the total flux of F from 8 V, or flux for short; the divergence isthen a measure of flux per unit volume. For the vector quantity L;n; x F 6.S i

in the second term of (4.68), we propose the name of the total shear of F aroundthe enclosing volume !:l V , or shear for short; the curl is then a measure of shearper unit volume. From the mathematical point of view, there is no need to invokethis physical model. It is proposed here merely as an aid to distinguish thesefunctions.

The expressions for the gradient and the curl that have been derived bythe method of symbolic vector can be derived alternatively by two differentapproaches.

In the defining integral-differential expression for VI, let the shape of t1Vbe a flat cell of uniform thickness b.s and area 8A at the broad surfaces, as shownin Fig. 4-1. The outward normal unit vector is denoted by s. By taking a scalarproduct of the third term of (4.59) with s, we obtain

"VII. Ei I(s · ni) 8Si 1· L;!(s · ni) 8S;S· = 1m = 1m .

~V-+O ~V 6V-.O t1A!:ls6s-.0


as

~ --.LT

Figure 4-1 Thin flat volume of uniform thickness b.s and area ~A.

(4.92)

The scalar products·;,; vanishes for all the side surfaces because sis perpendicularto ;, i therein. The only contributions result from the top and the bottom surfaceswhere nj = ±s and as; = aA; thus, we obtain

A • [/(S+ ~S) - I(S+ ~S)] als·Vf= 11m =-,

~s-+o t1s aswhere s ± I1s /2 correspond to the locations of the broad surfaces along s, and thecenter of the flat cell is located at s. Equation (4.92) can be treated as an alternativedefinition of the component of the gradient in an arbitrary direction s.

By the rule of chain differentiation,

(4.93)

where 00; denotes one of the coordinate variables in GCS. Equation (4.93) can beinterpreted as the scalar product between

V/="~ al.c: 8m',and

where we have used the relation

sds = LPjdroj

(4.94)

(4.95)

(4.96)

Sec. 4-5 Alternative Definitions of Gradient and Curl 77

to obtain (4.95). Pj and r' are, respectively, the primary and reciprocal vectors inGCS. Equation (4.93), therefore, is the same as

of Aos = So Vf,

previously derived by means of (4.91).The same model can be used to find a typical component of the curl of F in

GCS. Let us assume that s represents the unit normal in the direction of r! in GCS,that is,

A r 1A

s = = rio (4.97)(r! . r! )1/2

By taking the scalar product of the second term of (4.68) with "1, we obtain

A mF 10 Li"l · (ni x F) as; 10 Li(rt x nil · F aSj (4.98)rt 0 v = 1m = 1m 0

~v-+o ~V ~A-+O ~A ~s~s-+o

The area of the broad surface, ~A, corresponds to

[ ]1/2 2 3

~A = (P2 X P3) · (P2 X P3) ~ro ~ro 0

Because

1 1r = - P2 X P3'

A

it is evident that

(4.99)

In (4.98), the only contributions come from the side surfaces where

"1 x ni = t.j and ~Si = l!:.s /).(, l :

i j at j represents a segment of the contour around the periphery of the broadsurface. Equation (4.98) then reduces to

LoFo se,rIo VF = lim J. (4.100)~A-+O aA

By considering a contour formed by P2 tJ.ro2 and P3 ~ro3 with center located at 002

and 003, we obtain

A tit7 F l' [(F 0 P2)0)3_~r03 /2 - (F · P2)C03+~r03 /2rl' v = rm

~m2-+0 a 003~ro3-+0

+ (F· P3)m2+~r02/2 - (F· P3)ro2_~ro2/2]' 1~ro2 A(r l

0 r! )1/2

1 [O(P3 · F) o(P2 · F)]= A(r 1 • r 1) 1/2 a002 - aro 3

78

or


r 1 • VF = ..!- (ah _ ah ) , (4.101)A a0)2 a0)3

where hand h denote two of the primary components of F in GCS. In general,we find

r' . VF = ..!- ( 8ik. _ afj), (4.102)A a O)} a O)k

where (i, j, k) = (1,2,3) in cyclic order. Equation (4.102) represents onecomponent of (4.75) because

'" i '" Pi (afk afj )~Pi(r . VF) = \VF = ~ A ami - aO)k ., ,

4-6 The Method of Gradient

Once the differential expressions of certain functions are available in terms ofdifferent coordinate variables, we can derive many relations from them by takingadvantage of the invariance property of these functions. The method of gradientis based on this principle. We will use an example to illustrate this method.

It is known that the relationships between (x, y), two of the rectangularvariables, and (r, ~), two of the cylindrical variables, are

x = r cos e,y = r sin e,

( 2 ..2)1/2r=x+y ,

(4.103)

(4.104)

(4.105)

(4.110)

and~ = tan-I (;) . (4.106)

By taking the gradient of (4.103) and (4.104) in the rectangular system for x and yon the left sides of these two equations, and in the cylindrical coordinate systemon the right sides, we obtain

x= cos e r- sinq>~, (4.107)

y = sinq> r+ cos 4> ~. (4.108)

By doing the same for(4.105) and (4.106), but in reverse order, we obtain

A x A Y A tf\ A • tf\ A (4 109)r = 1/2 x + 1/2 Y = COSy X + sm e y, ·(x 2 + y2) (x2 + y2)

~ -y A X A

; = x 2 + y2 x + x 2 + y2y,

~ = - sin x+ cos 4> y.

Sec. 4-6 The Method of Gradient 79

These relations can be derived by a geometrical method, but the method ofgradientis straightforward, particularly if the orthogonal system is a more complicated onecompared to the cylindrical and spherical coordinate systems. Equations (4.107)(4.110), together with the unit vector z, can be tabulated in a matrix form, as shownin Table 4-2. The table can be used in both directions. Horizontally, it gives

r = cos q> X+ sin y, (4.111)

~ = - sinq> x+ coset> y, (4.112)which are the same as (4.109) and (4.110). Vertically, it yields

x = cos r- sin ~, (4.113)

Y= sin q> U r + cos ~, (4.114)

which can be derived algebraically by solving for x and yfrom (4.109) and (4.110)in terms ofrand~. Each coefficient in Table 4-2 corresponds to the scalar productof the two unit vectors in the intersecting column and row; thus, r .x = cos <1>,

<t> . x = - sin <1>, and so on. For this reason, the same table is applicable to thetransformation of the scalar components of a vector in the two systems. Because

f = fxx + fyy + h Z= j,.r + h~ + hz,it follows that

fx = x . f = (x . r)f,. + (x . ~)h =cos f,. - sin h (4.115)

andI» = y · f = (y. r) f,. + (y .~) h = sin cl> f,. + cos cl> h. (4.116)

These relations are of the same form as (4.107) and (4.108). The transformationsof the unit vectors of the orthogonal system reviewed in Section 2-1, and theunit vectors in the rectangular system, are listed in Appendix A, including thecylindrical system just described.

Table 4-2: Transportationof Unit Vectors

i y Z

r cos e sincf» 0~ - sincf» coset> 0z 0 0 1

As another example, let us consider the problem of relating (R, e,~) to

(R, a,~) of another spherical system in which the polar angle ex is measuredfrom the x axis, and the azimuthal angle ~ is measured with respect to the x-yplane; thus,

x = R sin ecos = R cos a,y = R sin esin = R sin a cos ~,

z = R cos e = R sin a sin ~.

(4.117)

(4.118)

(4.119)


We are seeking the relationships between (a, 13) and (9, ~). The metric coefficientsof the two systems are (1, R, R sin 9) and (1, R, R sin ex). By taking the gradientof sin 9 cos ~ =cos a in the two systems, we obtain

1 a A 1 a A 1 a A

R ao (sino cosell) 0 + R sinO aell (sin 0 cos ell) ell = R aa (cosa) a.

Hence - sin ex a = cos 9 cos «I> a- sin ep ~, or

-1 ( A A)a= 1/2 cos 9 cos ep 9 - sin ep ep •(1 - sin2 9cos2 «1»

By taking the gradient of

cot~ = tan 0 sin ell = (;) ,

we obtain

(4.120)

(4.121)

(4.122)

(4.124)

(4.123)A -1 (A A)P= 1/2 sin e 9 + cos Bcos o ep .(1 - sin2a cos2 ep)

From (4.121) and (4.123), we can solve for eand ~ in terms of a and ~.alternative method is to use (4.119) and the relation

tanell=tanacos~(= ~),

and repeat the same operations; we then obtain

An

A -1 ( A)a= 1/2 cos ex sin Pex + cos P J3(1 - sin2 exsin2 P)

and

(4.125)

(4.126)A 1 (A A)~ = 1/2 cos J3 a - cos a sin P f3 .

(1 - sin2 a sin2~)

The reader can verify these expressions by solving for eand ~ from (4.121) and(4.123) at the expense of a tedious calculation.

These relations are very useful in antenna theory when one is interested infinding the resultant field of two linear antennas placed at the origin, with oneantenna pointed in the z direction and another one pointed in the x direction. Inorder to calculate the resultant distant field, the individual field must be expressedin a common coordinate system, say (R, a, ~) in this case. Because the field of thex-directed antenna is proportional to a, (4.121) can be used to combine it with thefield of the z-directed antenna, whose field is proportional to e. In fact, it is thistechnical problem that motivated the author to formulate the method of gradientmany years ago.

Sec. 4-7 Symbolic Expressions with Two Functions and the Partial Symbolic Vectors 81

(4.127)

(4.128)

The method of gradient can also be used effectively to derive the expressionsfor the divergence operator and the curl operator in the orthogonal curvilinearsystem from their expressions in the rectangular system. In the rectangular system,the divergence operator and the curl operator are given respectively by

"A av = L....Jx;·_,i ax;

"A av = L....Jx; x -.; ax;

Upon applying the method of gradient to the coordinate variables Xi with i =(1,2,3), we obtain

A " Uj ax;Xi = Vx; = L....J - - ,

j h j aVj

and by the chain rule of differentiation,

a "aVk aax; = L.r aXi aVk •

(4.129)

(4.130)

i = j,i # j.

(4.131)

(4.133)

(4.132)

Upon substituting (4.129) and (4.130) into (4.127) and (4.128), and making use of(4.131), we find

'"A a "" Ui av = L.J x; · - = L.-t - . - ,i ax; i h; av;

"A a "U; av = L..Jx; x - = L..J - x -.i ax; ; h, Bu,

This exercise shows again that the divergence operator and the curl operator, aswith the del operator for the gradient, are invariant to the choice of the coordinatesystem, a property we have demonstrated before.

4-7 Symbolic Expressions with Two Functionsand the Partial Symbolic Vectors

Symbolic expressions with two functions are represented by T(V, 'ii, b), wherea and b both can be scalars, vectors, or one each. In this section, we willintroduce a new method for finding the identities of these functions in terms ofthe individual functions aand b without using the otherwise tedious method indifferential calculus.


Because of the invariance theorem, it is sufficient to use the differentialexpression of T (V, a, b) in the rectangular system to describe this new method.In the rectangular system,

_- ~o ,.._-T(V, a, b) = c:~ ru; a, b).

i X,(4.134)

(4.136)

(4.135)

(4.138)

We now introduce two partial symbolic vectors, denoted by V a and Vb, which aredefined by the following two equations:

_- ~o ,.._-T(Va , a, b) = z: ~ re; a, b)jj=constant'

i X,

_- ~a ,.._-T(Vb, a, b) = z: ~ T(Xi, a, b)a=constant.

i X,

In (4.135), b is considered to be constant, and in (4.136), ais considered to beconstant. The process is similar to the partial differentiation of a function of twoindependent variables, that is,

of(X, Y) = [df(X, Y)J. (4.137)ox dx y=constant

The name partial symbolic vector was chosen because of this analogy. It is obviousthat Lemma 4.1 is also applicable to symbolic expressions defined with a partialsymbolic vector, because, in general,

T(V - b-) - I" Li T(nj, a, b)iJ=c ~Sja, a, - 1m V

6V~O t!1

and

(4.139)

We now introduce the second lemma in the method of symbolic vector.

Lemma 4.2. For a symbolic expression containing two functions, thefollowing relation holds true:

(4.140)

The proofof this lemma follows directly from the definition ofthe expressions(4.135) and (4.136) or (4.138) and (4.139). This lemma may also be called thedecomposition theorem.

Let us now apply both Lemma 4.1 and Lemma 4.2 to derive various vectoridentities without actually performing any differentiation. Because the stepsinvolved are algebraic, most of the time we merely write down the intermediatesteps, omitting the explanation.


The symbolic expression with two functions besides the symbolic vector arelisted here; there are only eight possibilities.

Two scalars:

Vab = aVb = bVa.

One scalar and one vector:

v · ab = aV . b = Va · b,

V x ab = aV x b = Va x b.

Two vectors:

V(a· b) = (a- b)V = (b- a)V,

V . (8 X b) = a · (b x V) = b · (V x a),

(V · a)b = (a · V)b = V · ab,V x (a x b) = (V. b)a - (V· a)b = V· ba - V· ab,

(V x a) x b = (b · V)a - (a . b)V = V . ba - V(a . b).

1.

Vab = Va (ab) + Vb(ab) [Lemma 4.2]

= bVaa +aVbb; [Lemma 4.1]

hence

V(ab) = bVa + aVb.

2.

V . (ab) = Va . (ab) + Vb· (ab)

= (Vaa) . b + a Vb · b;

hence

V (ab) = b· Va + aVb.

3.

v x (ab) = Va X (ab) + Vb x (ab)

= (Vaa) X b + a Vb X b;

hence

.V (ab) = -b x Va + aVb.

4.

V(a· b) = v,« b) + Vb(a· b).

(4.141)

(4.142)

(4.143)

(4.144)

(4.145)

(4.146)

(4.147)

(4.148)

(4.149)

(4.150)

(4.151)


In view of Lemma 4.1,

Vb(a· b) = a x (Vb x b) + (a· Vb)b

= a x Vb + a . Vb.

By interchanging the role of a and b, we obtain

Va(a . b) = b x V'a + b· Va:

hence

V(a· b) = a· Vb + b· Va + a x Vb + b x Va. (4.152)

(4.153)

We would like to point out that the first two terms in (4.152) involve twonew functions in the form of Vb and Va. They are two dyadic functionscorresponding to the gradient of two vector functions. In the rectangularsystem, Vb is defined by

Vb= LX; ab; ax;

",,,,,,a "-= L." L..Jxi-.(xjbj)

; j x,

'" '" ,. ,. ab j= L-, L..JX;Xj- .t j ox;

Then,

a · Vb = L La; ab j xj = La; ab .i j ax; i OX;

In an orthogonal curvilinear system,

a.Vb= La; ~i h, av;

""aja '" "= L." - -. L..JbjUj. hi av; .I J

_ "'" a, "'" [b au j ab j " ]-L."-L.,, j-+-Uj.. h;. av; av;I }

(4.154)

(4.155)

(4.156)

The derivatives of Uj with respect to Vi can be expressed in terms of theunit vectors themselves and the derivatives of the metric coefficients withthe aid of (2.59) and (2.61). The result yields

"'""a; ob j "-a . Vb = L." L." - - U j + A x b,i j hi au;


where

1 ,,( ahk 8h j) AA = - L.J ak- - «rz:: h;u;n i aVj 8Vk

with (i, j, k) = (1,2,3) in cyclic order, and Q = h th 2h3. To obtain(4.156) by means of (4.154) through coordinate transformation would bea very complicated exercise.

5.

v . (a x b) = Va . (a x b) + Vb . (a x b)

=b·(Va xa)+a·(bx Vb);

hence

6.

hence

V (a x b) = b . Va - a . Vb.

(V . a)b = (Va· a)b + (Vb· a)b;

V (ab) = bVa + a- Vb.

(4.157)

(4.158)

7.

It is seen that (V -ajb is not equal to (Va)b; rather, it is the divergenceof a dyadic function abo

V x (a x b) = Va X (a x b) + Vb x (a x b)

= (Va· b)a - (Va· alb + (Vb· b)a - (Vb· a)b;

hence

8.V (a x b) = b· \7a - bVa + aVb - a- Vb, (4.159)

(V x a) x b = (Vax a) x b + (Vb X a) x b

= (VaX a) x b + a(V b . b) - Vb (a . b)

= (Va X a) x b+a(Vb· b) -a x (Vb x b) - (a- Vb)b;

hence

(V x a) x b = -b x Va + a(Vb) - a x V'b - a . Vb, (4.160)

9.

Two vector identities can be conveniently derived by means ofpartialsymbolic expressions.

Hence

a x Vb = (Vb) . a - a . Vb, (4.161)

86

10.


Hence

(8 x Vb) x b = (Vb) · a - aVb= a- Vb - aVb + 8 x Vb.

The second line of (4.162) results from (4.161).

(4.162)

(4.163)

(4.165)

It is interesting to observe that the partial symbolic expression (8 x Vb) x binvolves the products of a with the gradient, divergence, and curl of b. Theconvenience ofusing the method ofsymbolic vector in deriving the vector identitiesis evident.

4-8 Symbolic Expressions with Double SymbolicVectors

When a symbolic expression is created with a vector expression containing twovector functions gl and g2 and a third function 1 that can be scalar or vector, wecan generate a symbolic expression of the form T(V, g2' 'j) after gl is replacedby a symbolic vector V. In the rectangular system,

- ,"",8 -T(V, 12' f) = L.J a:- ra; g2' j).

i X,

Now, if g2 is replaced by another symbolic vector V' in (4.163), we wouldobtain a symbolic expression with double S vectors whose differential form inthe rectangular system will be given by

, - a2A. A. -

T(V, V , f) = L L a T(Xi, Xj, f). (4.164)i j X; aXj

It is obvious that Lemma 4.1 also applies to T (V, V', J). Several distinct caseswill be considered.

1. Laplacian of a scalar function

T(V, V', 'j) = (V · V')/,

where f is a scalar function; then

'f " " a2

" "f " a2

f fV·V = LJLJ---(x; ·Xj) = LJ-2 =VV .i j ax; ax j ; aX i

Although we have arrived at this result using functions defined in therectangular system, the result is applicable to any system because of the

Sec. 4-8 Symbolic Expressions with Double Symbolic Vectors 87

(4.167)

(4.166)

(4.168)

(4.169)

invariance theorem of the differential operators. The function V '\lf iscalled the Laplacian of the scalar function I, in honor of the Frenchmathematical physicist Pierre Simon Laplace (1749-1827). In the past,many authors used the notation V2 f for this function, which is a compactform of the original notation V · Vf used by Gibbs. We have completelyabandoned Gibbs's notations in this new edition so that the "formal" scalarand vector products, discussed in Chapter 8, will not appear and interferewith the operations involved in the method of symbolic vector.

2. Laplacian of a vector function

T(V, V', h = V· V'F,

where F is a vector function; then,

, ~ '" a2~" ~ a2

FV· V F = LJLJ --(Xi ·Xj)F = L...J -2 = VVF.i ) ax; ax} ; ax;

In this case, we encounter a dyadic function corresponding to the gradientof a vector function. The divergence of a dyadic is a vector function. Inthe rectangular coordinate system, we can write

a2F a2

VVF= E-2 = LE-2 FjXj = L(VVFj)Xj.; ax. ; j ax; j

In the orthogonal curvilinear system,

VVF-"'u; .~",Uj aF- 7' hi au; 7 hj avj ,

where the operational form of the divergence and the gradient have beenused. The derivatives of the dyadic function VF can be simplified asfollows:

Hence

VVF- '"'~ u; . [Uj~ +~ (Uj) aF]- 7' 7 hi h , av; au} au; h , aVj

"'"' {(Ui Uj) a2F u; a (Uj) aF }

= 7' 7 h;· h j av; avj + h; · au; h j au j

" 1 a2F "''' [Ui a (Uj) aF]

= 7' h~ av; + 7' 7 h j • au; h j avj •

The expression of V VF will be used later to demonstrate an identityinvolving this function.


3. The gradient of the divergence of a vector function

T(V, V', 'j) = V(V' · F).

Then,

, ,,~a2 A A

V(V . F) = L.JL.J a "a . x;(Xj· F)i j X, Xl

a2

-~"" s.»,- ~~ ax; ax)" I JI J

""" a ""aFj=L.Jx; - L.J - = VVF.i ax; j aXj

InOCS,

VVF ="Ui ~~ Uj . 8FLrh;OViLrh j OVj

"" u; [a (Uj) aF Uj a2F]

=Lr Lr hi aVi h j · av j + h j · aViav j ·

We will leave it in this form for the time being.

4. The curl of the curl of a vector function

T(V, v', 1> = V x (V' x F).

Then,

(4.170)

(4.171)

(4.173)

, '""~ 82

A AV X (V x F) = L..JL.J Xi X (x j x F)i j ax; Bx]

=L s. [Xi X L ~(Xj x F)] (4.172); ax; j aXj

=L~[xixVF]=vvF.i ax;

InOCS,

"u; 8(" Uj OF)""F=L.J- x - L.J-x-; h; av; j h j av j

LL u; [a ("j) aF Uj a2F]- -x - - x-+-x--- i j hi av; h j BVj h j BVjovj·

Because T (V, V', 1> obeys Lemma 4.1, we can also change V x (V' x F)to

v x (V' x F) = (V .F)V' - (V· V')F. (4.174)

Sec. 4-8 Symbolic Expressions with Double Symbolic Vectors 89

In view of (4.166), (4.170), and (4. J72), we have

~~F=VVF-VVF. (4.175)

This identity has been derived using functions defined in the rectangularcoordinate system. However, in view of the invariance theorem, it is validin any coordinate system. The misinterpretation of VVF in (4.175) inOCS has troubled many authors in the past (see Chapter 8); it is thereforedesirable to prove (4.175) analytically to confirm our assertion that (4.175)is an identity in any coordinate system. By taking the difference between(4.173) and (4.171) and rearranging the terms, we find

VWF-'V'VF=VVF+LLa~ x[U~ x~(hU~)], (4.176); j v} h, av, }

where V VF corresponds to the function given by (4.169). The last termin (4.176) vanishes because

L Ui x !.-. (Uj) = VVVj = O.i h; av; h j

The curl of the gradient of any differentiable scalar function vanishes byconsidering V Vf in the rectangular coordinate system that yields

~Vf=LXi[.!-(af)_.!-(af)]=o. (4.177)i aXj aXk aXk aXj

This theorem is treated later from the point of view of the method ofsymbolic vector. We have thus shown that (4.175) is indeed valid in OCS.The identity can be proved in GCS in a similar manner.

5. The curl of the gradient of a scalar or a vector function

T(V, V', 1> = V x v'].Then,

~ " a [~A a1] r= £..JX; x - £..JXj- = ~v ·; ax; j aXj

But

hence

A A {O,X; X Xj = A "

-Xj X Xi,

vvl=o,

i = j,i:/; j;

(4.178)

where 1can be a scalar or a vector. When 1is a scalar, we have alreadyproved it as stated by (4.177).


6. The divergence of the curl of a vector function

T(V, V', 1> = V· (V' x F).

Then,

, "'" a2

" "V· (V x F) = £...J £...J --[Xi· (Xj x F)]i j ax; OXj

" " a [""' A aF ]= £...JX;· - .L..JXj X-i ax; j aXj

= VV'F.

But

Hence

VV'F = o.

i = j,i =1= j.

(4.179)

(4.180)

When a symbolic expression consists of double S vectors and twofunctions, its definition in the rectangular system is

T(V. V'. ii. b) = L L a ~: . T(Xi. Xj. ii, b).; j x, X J

To simplify (4.180), we can apply Lemma 4.2 repeatedly to an expressionwith a single S vector, that is,

T(V, V', a, b) = T(V, V~, a, b) + T(V, V~, a, b)=T(Va , V~, a, b) + T(Vb , v~, a, b) (4.181)

+ T(Va , V~, a, b) + T(Vb, V~, a, b).As an example, let

T(V, V', a, b) = (V· V')ab. (4.182)

According to (4.165), this is equal to VV(ab). Equation (4.181) yields

VV(ab) = bVVa + 2(Va)(Vb) + aVVb. (4.183)

The same answer can be obtained by applying (4.149) and (4.150) toV V(a b) in succession.

Sec. 4-9 Generalized Gauss Theorem in Space 91

(4.185)

(4.187)

4-9 Generalized Gauss Theorem in Space

The principal integral theorem involving a symbolic expression can be formulatedbased on the very definition of T(V), namely,

T(V) = lim Li T(nj) I:i.Sj . (4.184)~v~o ~V

Equation (4.184) can be considered as a limiting form of a parent equation

T(V) = Lj T(ni) I:i.S j + Eav '

where e ~ 0 as li. V ~ O. If li. Vj denotes a typical cell in a volume V with anenclosing surface S, then for that cell, we can write

T(V) sv, = L T(nij) ss; + £jli. Vj, (4.186)

where li.Sij denotes an elementary area of li. Vj, nij being an outward normal unitvector. By taking the Riemann sum of (4.186) with respect to i. we obtain

LT(V)li.Vj = LLT(nij)li.Sij+ L£jL\Vj.j j j

Then, as li. Vj ~ 0, £j ~ O. By assuming T(V) to be continuous throughout V,we obtain

ffi T(V)dV =ffs T(n)dS. (4.188)

The sign around the double integral means that the surface is closed. It is observedthat the contributions of T (n;j) li. Sij from two contacting surfaces of adjacent cellscancel each other. The only contribution results from the exterior surface wherethere are no neighboring cells. In (4.188), ndenotes the outward unit normal vectorto S. The same formula can be obtained by integrating the symbolic expression inthe rectangular system

ffi T(V)dV =ffi~ a:;~j) dV.,(4.189)

The integral involving the partial derivative of T(x;) with respect to X; can bereduced to the surface integral found in (4.188). The linearity of T(x;) withrespect to Xi is a key link in that reduction. In a later section, we will give adetailed treatment ofa two-dimensional version ofa similar problem for a surface todemonstrate this approach. The formula that we have derived will be designated thegeneralized Gauss theorem, which converts a volume integral of T (V), continuousthroughout V , to a surface integral evaluated at the enclosing surface S. Many oftheclassical theorems in vector analysis can be readily derived from this generalizedtheorem by a proper choice of the symbolic expression T (V).


1. Divergence theorem or Gauss theorem. Let T(V) = V . F = VF.Upon substituting these quantities into (4.188), we obtain the divergencetheorem, or the standard Gauss theorem, named in honor of the greatmathematician Karl Friedrich Gauss (1777-1855):

ffiVFdV =!£(n ·F)dS. (4.190)

2. Curl theorem. Let T(V) = V x F =VF; then T(n) =;, x F. By meansof (4.188), we obtain the curl theorem:

ffiVFdV =!£(n x F)dS. (4.191)

3. Gradient theorem. This theorem is obtained by letting T(V) = Vf =VI; then T(n) = h f'; hence

ffiVfdV =!£fn d S =!£fdS. (4.192)

4. Hallen'sformula. IfweletT(V) = (V·a)b, then T(n) = (n-ajb, BecauseT (V) consists of two functions a and b, we can apply Lemma 4.2 to obtain

(V · a)b = (Va · a)b + (Vb · a)b = b(Va · a) + (a- Vb)b.

The rearrangement of the various terms follows Lemma 4.1; thus,

(V· a)b = b(Va) + (a- V)b. (4.193)

By substituting (4.193) into (4.188), we obtain

ffi£b v a +a . Vb]dV =1£(;, ·a)bdS. (4.194)

Equation (4.194), with b equal to etr, where r = (x2+ j2 + z2)1/2 andc is a constant vector that can be deleted from the resultant equation,was derived by Hallen [8], based on differential calculus carried outin a rectangular system. We designate (4.194) as Hallen's formula forconvenient identification.

The three theorems stated by (4.190)-(4.192) are closely related. In fact,it is possible to derive the divergence theorem and the curl theorem based onthe gradient theorem. The derivation is given in Appendix D. The relationshipbetween several surface theorems to be derived in Chapter 5 is also shown in thatappendix.

With the vector theorems and identities at our disposal, it is of interest togive an interpretation of the closed surface theorem (2.62) based on the gradienttheorem, and to identify (2.63) as a vector identity.

According to (4.19), when T (V) = V1 and f = constant, we have

L ~ (0 Ui) = 0, (4.195)t av; h;

Sec. 4-10 Scalar and Vector Green's Theorems 93

which is the same as (2.62), originally proved with the aid of the relationshipsbetween the derivatives of the unit vectors. From the point of view of the gradienttheorem given by

IIIVfdV =# fdS

when f =constant, we obtain

(4.196)

which is the closed surface theorem in the integral form. Equation (4.195) cantherefore be considered as the differential form of the integral theorem for a closedsurface.

In view of the definition of the curl operator given by (4.85), (2.63) isrecognized as

Now,

v (:~) = 0, j=(1,2,3). (4.197)

u·z: -Vv··- J'hj

hence (4.197) is equivalent to

VVVj = 0, (4.198)

which is a valid identity according to (4.178). By applying the curl theorem to thefunction F = Vvj, we obtain

IIIV VvjdV = #;, x VvjdS =#it x n~~ dS = O. (4.199)

Hence (4.198) may be considered as the differential form of the integral theoremstated by (4.199).

4-10 Scalar and Vector Green's Theorems

There are numerous theorems bearing the name ofGeorge Green (1793-1841). Wefirst consider Green's theorem involving scalar functions. In the Gauss theorem,stated by (4.190), if we let

F=aVb,

where a and b are two scalar functions, then

VF = aVVb + (Va) . (Vb),

(4.200)

(4.201)


(4.204)

which is obtained by (4.150). Upon substituting (4.201) into (4.190) with;' . F =a(n . Vb), we obtain

III[aVVb + (Va) · (Vb)] dV =Iia (Ii · Vb) dS. (4.202)

Because n· Vb is the scalar component of Vb in the direction of the unit vector;',it is equal to ab/tJn; (4.202) is often written in the form

ffl[aVVb + (Va)· (Vb)] dV =Iia :: dS. (4.203)

For convenience, we will designate it as the first scalar Green's theorem.If we let F = aVb - bVa, then it is obvious that

III(aVVb - bVVa) dV =Ii(a :: - b ::) dS

=IfIi · (aVb - bVa) dS.

Equation (4.204) will be designated as the second scalar Green's theorem. Both(4.203) and (4.204) involve scalar functions only.

Two theorems involving one scalar and one vector can be constructed from(4.202) and (4.204). We consider three equations of the form (4.202) with threedifferent scalar functions hi (i = I, 2, 3)..Then, by juxtaposing a unit vector Xi toeach of these equations with i = 1, 2, 3 and summing the resultant equations, weobtain

flf[aVVb + (Va)· (Vb)]dS =Ifa(li. Vb)dS. (4.205)

Similarly, by moving the function b to the posterior position in (4.204) andfollowing the same procedure, we obtain

111 [aVVb - (VVa)b]dV =IfIi · [aVb - (Va)b]dS. (4.206)

Equation (4.205) is designated as the scalar-vector Green's theorem of the firstkind and (4.206) as the second kind. Because of the invariance of the gradient andthe divergence operators, these theorems, derived here using rectangular variables,are valid for functions defined in any coordinate system, including GCS. However,one must be careful to calculate Vb, a dyadic, in a curvilinear system. In OCS,

Vb =L Uj ab =L Uj [ab j Uj + bj aUj] , (4.207). h; av; . . hi av; av;I I,)

which has appeared before in (4.155).There are two vector Green's theorems, which are formulated, first, by letting

F=axVb. (4.208)

Sec. 4-11 Solenoidal Vector, Irrotational Vector, and Potential Functions 9S

In view of (4.157), we have

VF=Vb·V'a-a·VVb

and

(4.209)

It · F =n· (8 X 'rb). (4.210)

Upon substituting (4.209) and (4.210) into the Gauss theorem, we obtain

f f [[Vb · Va - a· V Vb]dV =Iin· (a x Vb)dS. (4.211)

Equation (4.211) is designated as thefirst vector Green's theorem. By combining(4.211) with another equation of the same form as (4.211) with the roles of 8 andb interchanged, or by starting with F = 8 X Vb - b x V a, we obtain the secondvector Green's theorem:

ff[[b. VVa - a· VVbjdV =Iin· (a x Vb - b x Va)dS. (4.212)

The continuity of the function F imposed on the Gauss theorem is now carriedover, for example, to the continuity of 8 x V b in (4.212), and similarly for theother theorems.

4·11 Solenoidal Vector, Irrotatlonal Vector, and PotentialFunctions

The main purpose of this book is to treat vector analysis based on a new symbolicmethod. The application of vector analysis to physical problems is not coveredin this treatise. However, there are several topics in introductory courses onelectromagnetics and hydrodynamics involving some technical terms in vectoranalysis that should be introduced in a book of this nature.

When the divergence of a vector function vanishes everywhere in the entirespatial domain, such a function is called a solenoidal vector, and it will be denotedby F, in this section. If the curl of the same vector function also vanisheseverywhere, it can be proved that the function under consideration must be aconstant vector. Physically, when both the divergence and the curl of a vectorvanish, it means that the field has no source. In general, a solenoidal field ischaracterized by

v . F, = 0, (4.213)

VFs = r, (4.214)

where we treat f as the source function responsible for producing the vector field.When the curl of a vector vanishes but its divergence is nonvanishing, such

a vector is called an irrotational vector, and it will be denoted by F j • Such a fieldvector is characterized by

VFj=O,

VFj = f,(4.215)

(4.216)


(4.217)

(4.218)

(4.219)

(4.220)

(4.221)

where the scalar function f is treated. as the source function responsible forproducing the field. In electromagnetics, F s corresponds to the magnetic fieldin magnetostatics and Fi to the electric field in electrostatics. In hydrodynamics,F, corresponds to the velocity field of a laminar flow, and F, to that of a vortex.

In electrodynamics, the electric and magnetic fields are coupled, and theyare both functions of space and time. Their relations are governed by Maxwell'sequations. For example, in air, the system of equations is

aHVE = -J.1o-,at

aEVH=J+£o-,at

v (£oE) = p,

V (JloH) = 0,ap

VJ=--at 'where J and p denote, respectively, the current density and the charge densityfunctions responsible for producing the electromagnetic fields E and H, and J.,1{)

and Eo are two fundamental constants. It is seen that the magnetic field H is asolenoidal field, but the electric field is neither solenoidal nor irrotational, that is,VE :j:. 0 and VE =F O.

The theoretical work in electrostatics and magnetostatics is to investigatethe solutions of (4.213)-(4.214) and (4.215)-(4.216) under various boundaryconditions of the physical problems. In electrodynamics, the theoretical workis to study the solutions of the differential equations such as (4.217)-(4.221) forvarious problems. In the case of electrostatics, in view of the vector identity(4.178), the electric field, now denoted by E, can be expressed in terms of a scalarfunction V such that

E= -VV. (4.222)

The negative sign in (4.222) is just a matter of tradition based on physicalconsideration; mathematically, it has no importance. The function V is calledthe electrostatic potential function. As a result of (4.219), we find that

VE = -vvv = £.., (4.223)Eo

where we have replaced the function f by P/Eo, with p denoting the densityfunction of a charge distribution and Eo, a physical constant. The problem is nowshifted to the study of the second-order partial differential equation

VVV = _E- , (4.224)Eo

which is called Poisson's equation. The operator V V, or div grad, is the Laplacianoperator we introduced in Section 4.8.

Sec. 4-11 Solenoidal Vector, Irrotational Vector, and Potential Functions 97

In the case of magnetostatics, in view of identity (4.179), the magnetic field,now denoted by H, replacing Fs , can be expressed in terms of a vector function Asuch that

H = V A. (4.225)

A is called the magnetostatic vector potential. The function f in (4.214)corresponds to the density of a current distribution in magnetostatics, commonlydenoted by J. By taking the divergence of (4.214), we find that 'WC = 'WJ = 0,which is true for a steady current. Upon substituting (4.225) into (4.214) with F,and f replaced by Hand J, respectively, we obtain

VVA=J. (4.226)

According to the Helmholtz theorem [9], in order to determine A, one must imposea condition on the divergence of the vector function A in addition to (4.225).Because

if we impose the condition

then (4.226) becomes

v V A = - 'W VA + VV' A,

VA=O,

VVA = -J.

(4.227)

(4.228)

(4.229)

The condition on the divergence of A so imposed upon is called the gaugecondition. This condition must be compatible with the resultant differentialequation for A, (4.229). By taking the divergence of that equation, we observethat VA must be equal to zero because V'J = O. Thus, the gauge condition soimposed is indeed compatible with (4.229). The analytical work in magnetostaticsnow rests on the study of the vector Poisson equation stated by (4.229) for variousproblems.

To solve the system of equations in electrodynamics such as the ones statedby (4.217)-(4.221), we let

floH = VA, (4.230)

because H is a solenoidal vector. The function A is called the dynamic vectorpotential. Upon substituting (4.230) into (4.217), we obtain

V (E+ ~~) =0. (4.231)

Hence E + (aAjat) is irrotational, so we can express it in terms of a dynamicscalar potential <t> such that

aAE+ - = -vep.

at(4.232)

(4.233)

Upon substituting the expressions for Hand E given by (4.230) and (4.232) into(4.218), we obtain

1 (B2A Bel»)'f'fA=fJ{)J-- -+V- ,c2 at2 at

where c = (J.lo£o)-1/2 is the velocity of light in free space.In view of identity (4.227), we can impose a gauge condition on A such that

Then, (4.233) reduces to

1 a~VA=---.

c2 at (4.234)

1 a2AVVA + c2 Bt2 = -J..loJ, (4.235)

which is called the vector Helmholtz wave equation. By taking the divergence of(4.235) and making use of (4.221) and (4.234), we find that are known, the electromagnetic field vectors E and H can be foundby using the following relations:

~H = 't' A,

BAE= -- -vep.at

(4.237)

(4.238)

The method of potentials in electrodynamics is a classical method. Anotherapproach is to deal with the equations for E and H directly. Thus, by eliminatingE or H between (4.217)-(4.218), we obtain

1 a2E aJV 'f E + c2 at2 = -Jloat ' (4.239)

1 B2H"'fH + c 2 at2 = V x J. (4.240)

These are two basic equations that can be solved by the method of dyadic Greenfunctions [3] or the vector Green's theorem [5].

Chapter 5

Vector Analysis on Surface

5-1 Surface Symbolic Vector and Symbolic Expressionfor a Surface

Vector analysis on a surface has previously been treated by Weatherbum [10]. Hisworks are summarized by Van Bladel [II]. Most books on vector analysis donot cover this subject. The approach taken by Weatherburn is to define a twodimensional surface operator similar to the del operator in space. Some key differential functions analogous to gradient, divergence, and curl are then introduced.

In this work, the treatment is different. We approach the analysis based on asymbolic vector method similar to the one found in Chapter 4 for vector analysisin space. A symbolic expression for a surface is defined in terms of a surfacesymbolic vector. Afterwards, several essential functions in vector analysis for asurface are introduced. They are different from the ones defined by Weatherburn.The relationships between the set introduced in this book and Weatherburn's set willbe discussed later. Finally, it will be shown that there is an intimate relationshipbetween the symbolic expression for a surface and the symbolic expression inspace. In fact, the former can be deduced from the latter without an independentformulation, However, it is more natural to treat the vector analysis on a surfaceas an independent discipline first, and then point out its relationship to the vectoranalysis in space.

Following the symbolic method discussed in Chapter 4, we will introduce asymbolic surface vector, denoted by Vs, and the corresponding symbolic vector

99

100 Vector Analysis on Surface Chap. 5

expression T (Vs) for a surface that is defined by

T(Vs) = lim L; T(mj) !l.l; , (5.1)

~s~o l!J..Swhere Sl, denotes an elementary arc length of the contour enclosing l:1S, and miis the unit vector tangent to the surface and normal to its edge. The running index icovers the number ofsides of ~S. For a cell with four sides, i goes from 1 to 4. Thesymbolic expression is generated by replacing at least one vector in an algebraicvector expression with VS. For example, Vs x b is created by replacing the vectora in a x b with Vso The expression defined by (5.1) is invariant to, or independentof the choice of, the coordinates on the surface in the general Dupin system. It isrecalled that the choice of (VI, V2) and the corresponding tangential unit vectors(UI, U2) is quite arbitrary. To find the differential expression based on (5.1) in thegeneral orthogonal Dupin system, let the sides of the surface cell be located atVI ± (l:1vl/2) and V2 ± (l:1v2/2), with the corresponding unit normal vectors ±Uland ±U2 located at these positions. The value of UI evaluated at VI + (l:1vI/2) isnot equal to the value of the same unit vector evaluated at VI - (l:1VI /2). The sameis true for the metric coefficients hi and h2 • The area of the elementary surfacel:1S is equal to h lh2l:1VI l:1v2. Figure 5-1 shows the configuration of the cell. Bysubstituting these quantities into (5.1) and taking the limit, one finds

T(Vs) = hllh2

{a~l [h2T(ut>] + a~2 [h 1T (U2)] } · (5.2)

For a plane surface located in the x-y plane in a rectangular system,

a A a A

T(Vs) = ax T(x) + ay T(y). (5.3)

This is the only case where T (Vs) can be expressed conveniently in a rectangularsystem. In general, rectangular variables are not the proper ones to describe the

o

......

Figure 5·1 Cell on a surface in the general Dupin coordinate system.

Sec. 5-2 Surface Gradient, Surface Divergence, and Surface Curl 101

(5.5)

function T(Vs) for a curved surface. From the definition of T(Vs) given by (5.1)and its differential form stated by (5.2), it is obvious that Lemma 4.1, introducedat the end of Chapter 4, Section 4-1, is also applicable to T(Vs) because T(mj),T(Ul), and T(U2) all have the proper form of a vector expression. By means of(5.2), it is now possible to derive the differential expression of some key functionsin the vector analysis for a surface, analogous to the gradient, the divergence, andthe curl in space.

5·2 Surface Gradient, Surface Divergence, and SurfaceCurl

For convenience, we repeat here the differential expression for T (Vs) expressedin the general Dupin system:

T(Vs) = h)Ih2

{8~) [h2T(u)] + O~2 [hI T(U2)]} · (5.4)

5-2-1 Surface Gradient

If we let T (Vs) = Vsj, where f is a scalar function of position, then T (u 1) =jUI, T(U2) = jU2. Upon substituting these quantities into (5.4), we obtain

T(Vs) = h)~2 [O~I (h2ful) + O~2 (ht/U2)]

= _1_ [h2f OU) + (f

Oh2+h2 Of) UI

h th 2 aVt aVt aVt

+ h)f::: + (f:~~ + hI :~) U2].

By making use of (2.61), with h3 = 1, we can express the derivatives of Ul and U2in terms of (Ut, U2, "3), which yields

T(Vs) = _1_ [-h 21 (..!.- oh) U2 + oh) U3) + (f oh2 + h2 8f)

UIb ih« b: aV2 aV3 aVt aVt

- hl/(~ 8h2UI+Oh2ih ) + (f~ +h l Of) U2].

h, aVI aV3 aV2 aV2

Some of the terms cancel each other. The net result is

1 at ,. 1 ej • ( 1 ah l 1 ah 2 ) A

VsI = --u) + --U2 - -- + -- IU3.hI aVt h2 aV2 hI aV3 h2 aV3

The coefficient in front of IU3 can be written in several different forms. If wedenote the product b ib: by H, which is equal to n (= h1h 2h3) with h3 = 1, then

~ :: = hI) ::~ + :2 ::: = - (~) + ~2) = -J, (5.6)


where R I and R2denote the principal radii of curvature of the surface as stated by(2.73) and (2.76), and J is the surface curvature defined by (2.84). The differentialfunction we have just derived is designated as the surface gradient of I, and itwill be denoted by Vs[ , which can now be written in the form

"I af "2 al Avsf= - - + - - + JU3! (5.7)hi (JVI h: (JV2

Equation (5.7) shows that the symbol Vs is indeed a differential-algebraic operator,defined by

"1 a "2 (J AVs = - - + - - + JU3.hI (JVl h2 aV2

The operation of "3 on f is a simple multiplication.

5-2-2 Surface Divergence

If we let T(Vs) = Vs • F, then T(";) = u; ·F; hence

Vs · F = h 1h [-!-(h2F1) + ~(hlF2)J1 2 QVI aV2

1 2 a (H )--E- -R- H ;=1 av; h; I •

(5.8)

(5.9)

(5.10)

The function so obtained is designated as the surface divergence of F, and it willbe denoted by VsF; thus,

1 2 a (H )VsF = - E - - F'; .H 1=1 av; h,

Equation (5.10) can be converted to an operational form as follows:

1 2 a (H )VsF= - E- -u;·FH ;==1 av; h,

= t [Ui . BF +.!- ~ (H Ui) ·FJ;=1 h; Bu, H av; hi

~ [u; (JF A ]=L.J -. - + J U3 • F ·;=1 h; av;

(5.11)

(5.12)

This expression shows that Vs is another differential-algebraic operator, defined by

[

2 U; a A]VS = E - .- + JU3· .

;=1 h, av;

When this operator is applied to F, it yields the surface divergence of F.

Sec. 5-2 Surface Gradient, Surface Divergence, and Surface Curl

5-2-3 Surface Curl

If we let T(Vs) = Vs x F, then T(u;) = u; x F; hence

'C7 F 1 [8" a,,]Vs X = -hh -a (h2 U I x F) + -(h 1u2 X F)

I 2 VI aV2

1 2 a (H )=-E- -ujxFH ;=1 av; h,

~ [u; aF 1 a (H,,) ]= £..J - x - + - - - '" x F;=1 h, av; H av; h,

~ [u i aF " ]= £..J -h. x -. + JU3 X F .;=1 ' av,

103

(5.13)

The function so created is called the surface curl of F and it will be denoted by'WsF; thus,

[

2 Ui a A]VsF = E - x - + JU3X F.

;=J hi av;

It is evident that Vs is another differential-algebraic operator, defined by

(

2 U; a A)V S = E-

h. x -. + JU3X •

;=1 I OV,

(5.14)

(5.15)

When it is operated on F, it yields the surface curl ofF. As with the vector analysisin space, we have three independent surface operators; they are partly differentialand partly algebraic, a special feature of the surface operators. By evaluating thederivatives ofF with respect to Vi in (5.14) and simplifying the result with the aidof (2.59) and (2.61), with h3 = 1, one finds

1 I[ aF3 ah t ] AVsF= - h 1 - +hIF2- UI

H OV2 OV3

(5.16)

The U3 component of VsF is the same as the corresponding component of the threedimensional 'W F in a Dupin coordinate system, but the two transversal componentsare different.


(5.17)

(5.20)

(5.21)

5-3 Relationship Between the Volume and SurfaceSymbolic Expressions

Although the surface symbolic vector Vs and the symbolic expression T (Vs)

involving Vs as defined by (5.1) and its differential form by (5.2) appear to beindependent of V and T(V), actually they are intimately related. If we expressT(V) in the general Dupin system (h3 = 1), then (4.80) becomes

1 3 a [H ]T(V) = - L - -T(u;) ,H ;=1 au; h,

where H = h 1h 2 • The differential expression of T(Vs) as given by (5.2) can bewritten in the form

1 2 a [H ]T('v's) = H t; OVj h j T(uj) · (5.18)

It is obvious that the first two terms of (5.17) are exactly the same as T (Vs) ; hence

T(V) = T(Vs) + HI~ [HT(U3)] . (5.19)aV3

Equation (5.19), therefore, can be used to find T(Vs) once T(V) is known orT (Vs) can be defined as the sum of the first two terms of T (V). From this point ofview, T (Vs) is not an independent function, and Vs is not an independent symbolicvector.

The last term of (5.19) can be written in the form

1 a [HT(A)] aT(U3) 1 en T(A )-- U3 = +-- U3H aV3 aU3 H aU3

= OT(U3) _ J T(U3).aV3

Equation (5.19) is therefore equivalent to

aT(U3) A

T(V) = T(Vs) + a - J T(U3).U3

By using the expression of VI, VF, and V F in the Dupin system and theexpressions of Vs/, VsF, and VsF given by (5.7), (5.10), (5.14), it can beeasily verified that (5.21) is indeed satisfied when we let T(V) equal VI, VF,V x F, respectively.

5-4 Relationship Between Weatherburn's SurfaceFunctions and the Functions Defined In the Methodof Symbolic Vector

In the classic work ofWeatherburn [10], he defines the surface gradient, the surfacedivergence, and the surface curl by retaining the two transversal parts of the threedimensional functions. Our notations for his functions are ~/, VtF, and VtF.

Sec. 5-4 Weatherbum's Surface Functions lOS

Weatherbum originally used the same notations as the three-dimensional functionsVI, V· F, and V x F. The surface functions defined by Brand [1] in an orthogonalDupin system are the same as Weatherbum's. Van Bladel [11] uses three linguisticnotations for these functions, namely, grad.j", div.F, and curl.F, They are definedby

2 U; afgrads! = Vr.1 = L -h -,

;=1 ; av;

• ~Ui aFdlVsF = VtF = L....J -h .-,

;=1 ; av;

2 U; aFcurlsF = "tF = L -h x -.-.

;=1; 8v;

(5.22)

(5.23)

(5.24)

These three terms have appeared before in our surface functions defined by (5.7),(5.11), and (5.14). Thus, the relations between the two sets are

«r = Vr.1 + JU3f,

VsF = VtF + JU3 . F,

'WsF = VtF + JU3 x F.

(5.25)

(5.26)

(5.27)

We must emphasize that VtF and V tF are not scalar and vector products betweenvt and F. As with our~, Vs, and 'Ws, they are three independent operators.

We have so far presented the three key surface functions in orthogonal Dupincoordinate systems. By following the same procedure in the three-dimensionalGCS, it is not difficult to extend the formulation to nonorthogonal Dupin systems(UI · "2 =F 0, but "I ·"3 = "2·"3 = 0). By introducing the primary and reciprocalvectors on a curved surface, we can show the invariance of these surface functionsin the nonorthogonal Dupin coordinate systems.

As far as the surface functions are concerned, once the relationships betweenthe two sets are known, it is a matter of personal preference as to which set shouldbe considered as the standard surface functions. In a subsequent section dealingwith integral theorems, it will be evident. that the set derived from the presentmethod, that is, Vsf, VsF, and VsF, orin general, T(Vs), is much more convenientto formulate the generalized Gauss theorem for a surface. We may also injecta remark that in electromagnetic theory, the equation of continuity (the law ofconservation of charge) relating the surface current density Js and the time rate ofchange of the surface charge density Ps is described by

V J - - aps (5.28)s s - at

when the surrounding medium has no loss. Here, it is VsJ, not Vi · J or divsf, thatenters the formulation. On the other hand, for the rate of change ofa scalar function


(5.29)

on a surface in a direction tangent to the surface, both VsI and Vr./ produce thesame result:

af ~ I ~- = Us • ~ = Us • Vrf.asThe vector component (al/aV3)U3 in VsI does not affect the value of ai/as. Forthe curl function, one finds

U3 · Vsf = U3 · ~ x f = U3 · Vf,

an identity to be used later.

5-5 Generalized Gauss Theorem for a Surface

(5.30)

(5.31)

By integrating the differential expression for T(V's), (5.2), on an open surface Swith contour L, we have

for T(Vs)ds=fort~ [h~ T(Ui)] dVtdV2,is is ;=1 av, ,

where dS = hIhzdvtdvz = Hdvtdvz. We assume that T(u;) is continuousthroughout S. The integrals in (5.31) can be carried out as follows:

for ~[h2 T(Ut)]dVt dV2 =1\1;!-[h2 T(Ut)]~ dV2Js aVt V2min

(5.32)

= i h2 T(Ul)dv2,

where the locations (PI, Pz), the segments Lv, L2, and the two extreme valuesV2min, V2max are shown in Fig. 5-2a. Similarly,

for ~[hl T(U2)]dvl dV2 =l v,- [hl T(U2)]~ dv,is aV2 VI min

(5.33)

= - i h t T(U2)d vt ,

where the locations P3, P4 , the segments L3, L 4 , and the two extreme values VI min,

VI max are shown in Fig. 5-2b. Hence

JLT(Vs) dS =i [h2T(Ut) dV2 - h t T(U2) dvt]. (5.34)

Sec. 5-5 Generalized Gauss Theorem for a Surface 107

V2max - - - - - - - - - -:::: -<: ,,,,

PI ....-------1I

I

I

\v2 min ~

'---------------- VI

(a)

Vtmaxvrmin""-----------......,jl....--- VI

(b)

Figure 5·2 Domain of integration in the (VI, V2) plane of a simple region.

Because T(u;) is linear with respect to Ui with i = 1,2, the integrand in the lineintegral is proportional to

u1h2dv2 - U2hl dVI, (5.35)which can be simplified. Let us consider a segment of the contour L J, which isthe edge of S. In the tangential plane containing Ul and U2 at a typical point P,the four key unit vectors are shown in Fig. 5-3. All of them are tangential to thesurface at P. A three-dimensional display of these vectors and the normal vector"3 is shown in Fig. 5-4. In these figures, Ul is tangential to the edge of the surface,and Urn is normal to the edge, but tangential to the surface. The relations betweenthese unit vectors are

(5.36)

108 Vector Analysis on Surface

"2I

ittII

II,

Pa

, UI,

--

Figure 5-3 Four tangential vectors in the plane containing Ut and U2-

Chap. 5

(5.41)

(5.39)

(5.40)

Figure 5-4 Three-dimensional view of the unit vectors at the edge of an open surface.

The algebraic relations between them are

Ul = cosaUt + sinaum , (5.37)

U2 = sinaut - cosaum , (5.38)

where a is the angle between Ul and Ut. If we denote the total differential arclength of the contour at P by dI, then

hi dVI = cosadl,

h2dv2 = sinadl.

Upon substituting (5.39) and (5.40) into (5.35), we find

u1h2 dV2 - U2 h l dv, = Urn dl;

hence

(5.42)

Sec. 5-5 Generalized Gauss Theorem for a Surface

Equation (5.31) therefore reduces to

IiT(Vs)dS =i T(urn)dl.

109

(5.43)

(5.44)

(5.45)

(5.47)

(5.46)

(5.48)

The unit vector Urn is commonly denoted by m, in contrast to the notation fz usedfor U3; (5.43), therefore, will be written as

IiT(Vs)dS= t T(m)dl.

Equation (5.44) is designated as the generalized Gauss theorem for a surfaceor the generalized surface Gauss theorem. It converts an open surface integralinto a closed line integral, and it has the same significance as the generalizedGauss theorem in space, which converts a volume integral into a closed surfaceintegral. Various integral theorems can be derived by choosing the proper formfor T(Vs) .

1. Surface gradient theoremLet rcVs) = Vsf = V.~f; then T(m) = mf. Hence

IiVs/dS= i fm dl .

2. Surface divergence theoremLet T(Vs) = Vs • F = V · F; then rem) = Tn . F. Hence

IiW·FdS= i m · Fdl.

3. Surface curl theoremLet TC'vs) = Vs x F = V'sF; then T(rn) = Tn x F. Hence

Ii'VsFdS= i m x Fdl.

In view of the relationship between T (V) and T (Vs) as described by(5.19), the generalized Gauss theorem for a surface can be written in theform

f~ { {T(V) - ~~ [HT(U3)]} dS= J: T(m) dl.h Ha~ ~

If T (V) is proportional to U3 x V or n x V, then T (U3) = 0, and (5.48)becomes

IiT(V) dS= i T(m) .u,

Three cases are considered now.

T (fz) = o. (5.49)


4. Cross-gradient theoremLet T (V) = (Ii x v) f. As a result of Lemmas 4.1 and 4.2, we have

(Ii x V) f = (Ii x Vn ) f + (Ii x VI) f= - (V Ii) f + Ii x Vf = Ii x Vf,

because 1f n= 0, and

T (m) = (Ii x in) f = Ut!

Hence

Iinx VfdS = i rae. (5.50)

For identification purposes, we designate it as the cross-gradient theorem.

5. Stokes's theoremLet

T (V) = (Ii x V) ·F

= (Ii x v n ) • F + (Ii x vF ) • F

= - (v n) · F + Ii · (VF X F) = Ii . VF.

Then

T (m) = (Ii x in) · F = Ut . F.

Hence

(5.51)

which is the famous theorem named after George Gabriel Stokes (18191903).

6. Cross-V-cross theoremLet T(V) = (Ii x V) x F. By means of Lemma 4.2, we have

(Ii x V) x F = (Ii x V n) X F + (Ii x V F) x F

= -(V Ii) x F + (Ii x VF ) x F.

Now, 'tv Ii = 0 because Ii is a linear vector, not curvilinear, and T (in) =(Ii x m) x F = i x F; hence

II(n x VF ) x F dS = f(i x F) ae. (5.52)

The function (Iix VF) x F is given by (4.162) with a and b therein replacedby Ii and F, respectively.

Sec. 5-6 Expressions with a Single Symbolic Vector

5-6 Surface Symbolic Expressions with a SingleSymbolic Vector and Two Functions

111

A complete line of formulas and theorems can be derived covering these twotopics. However, we will present only the essential formulation without actuallygoing into detail. A third lemma dealing with symbolic expressions with a singlesurface S vector and two functions is one of the main subjects to be covered. Ascalar Green's theorem on a surface involving the surface Laplacian will also bepresented.

For a symbolic expression with two functions, its differential form in thegeneral Dupin system according to (5.2) is defined by

- - 1 2 a [H ~,..., - ]T(Vs , a, b) = H L -. -h. ro.. a, b) ,;=I ov, ,

(5.53)

where H = h 1h2. We now introduce two expressions with two partial surface Svectors, denoted by Vsa and Vsh ' as follows:

_ - 1 ~ a [H A"'" -]T(Vsa, a, b) = H LJ ~ -h. T(u;, a, b) _ ';=1 uV" b=c

_ - 1 ~ a [H A"'" -]T(Vsht a, b) = H LJ ~ -h. T(u;, a, b) _ .;=1 uV, I a=c

(5.54)

(5.55)

It is obvious that Lemma 4.1 also applies to (5.54) and (5.55). Equation (5.53) cannow be decomposed into three parts:

,..., - ,..., - ,..., ,..., 1 2 a [H A _,...,]

T(Vs, a, b) = T(Vsa , a, b) + T(Vsb , a, b) - H L ~ -h. T(u;, a, b) _ _ .;=1 aV" a,b=c

(5.56)Because T(u;) is linear with respect to "i, we can combine it; with H / h, to formone function, and examine its derivatives. According to (2.62), with h3 = 1,

1 ~ a (H A) 1 a A- LJ- -u; +--( HU3) =0.H ;=1 au; h; H aV3

Thus,

1 ~ a (H A ) 1 en ~ A- LJ - - U; = -- -(HU3) = JU3.H ;=1 au; h, H aV3

The last term in (5.56), therefore, can be written as

1 ~ a [H A"'" -] A _--- LJ - - T(u;, a, b) _ = -J T(U3, a, b).H ;=1 au; h, a,b=c

(5.57)

(5.58)


Lemma 5.1. For a symbolic expression defined with respect to a singlesurface S vector and two functions, the following relation holds true:

T(Vs, 'ii, b) = rc«: II, h) + T(Vsb, 'ii, b) - J T(U3, 'ii, b). (5.59)

The proof of this lemma follows directly from (5.56) and (5.58). By means ofthis lemma, we can derive all the possible surface vector identities similar to theidentities described by (4.140)-(4.151) and (4.156)-(4.160). We merely writedown these relations without detailed explanation.

1.

Vs(ab) = Vsa(ab) + Vsb(ab) - JU3ab;

hence

Vs(ab) = bVsa + aVsb - Jabii«. (5.60)2.

Vs(ab) = Vsa . (ab) + Vsb · (ab) - JU3 · ab;

hence

Vs(ab) = aVsb + b · Vsa - Jab · U3. (5.61)3.

Vs x (ab) = Vsa x (ab) + Vsb x (ab) - JU3 x ab;

hence

Vs(ab) = -b x Vsa +aVsb+ Jab x U3. (5.62)4.

Vs(a · b) = Vsa(a . b) + Vsb(a · b) - JU3(a · b)

= b x (Vsa x a) + b · Vsaa

+ 8 X (Vsb X b) + a . Vsbb - JU3(a · b);

hence

Vs(a· b) = b x Vsa + b· Vsa + a x Vsb + a· Vsbb - J(a· b)U3.

(5.63)We are making effective use of Lemma 4.1 in these exercises.

5.

Vs • (a x b) = Vsa • (a x b) + Vsb • (a x b) - JU3 · (a x b)

= b · V sa x a + a · (b x Vsb) - JU3 · (8 X b);

hence

(5.64)

Sec. 5-7 Expressions with Two Surface Symbolic Vectors

6.

(Vs . a)b = (Vsa . a)b + (Vsb • a)b - J("3 . a)b;

hence

7.

113

(5.65)

Vs x (a x b) = Vsa x (8 X b) + Vsb x (a x b) - JU3 x (a x b)

= (Vsa • b)a - (Vsa . a)b + (Vsb . b)a - (Vsb . a)b

- jU3 x (a x b);

henceVs(a x b) = b · Vsa - bVsa + aVsb - a · Vsb+ J(a x b) x "3.

(5.66)8.

(Vsx a) x b = (Vsa x a) x b + (Vsb x a) x b - J(U3 x a) x b

= (Vsa x a) x b + a(Vsb· b)-Vsb(a· b)-J(U3 x a) x b

= (Vsa x a) x b + a(Vsb · b) - a x (Vsb x b) - (a · Vsb)b

- J(U3 x a) x b;

hence(Vs x a) x b = -b x 'fsa+aVsb - axVsb - a- Vsb - Jb x (a x "3).

(5.67)As with (4.159), the surface symbolic expression involves the vectorproduct between b and Vsa and the products of a with the surface gradient,the surface divergence, and the surface curl of b.

5-7 Surface Symbolic Expressions with Two SurfaceSymbolic Vectors and a Single Function

In presenting the symbolic expressions with two symbolic expressions in a threedimensional space, we define these functions in the rectangular system and thentheir relations. The rectangular system is not a convenient coordinate system fora curved surface. For this reason, the subject will be presented using the basicdefinition of the surface symbolic expression as stated by (5.1), namely,

T(v') = lim L;T(m) /!ii; . (5.68)s AS-+O I1S

For an expression consisting of two surface symbolic vectors and a single function,we can apply the same formula twice, that is,

I - • • Li Lj rt«. in',1> l1ltl1ljT(Vs , Vs ' f) = lim hm S S' . (5.69)

AS'-+O AS-+O 11 11Several cases will be considered.


(5.70)

1. Surface Laplacian of a scalar functionLet T(Vs,V;, j) = Vs · V;I; then T(m, m', j) = m.m' /. Thus,

L' L·m om'l se srV . V'I = lim lim _' 1 _

s s ~S-+O ~S'-+O ~s ~S'

. L;moVs/~l= hm = VsVsf~s-+o ~S

The double operator VsVs is designated as the surface Laplacian.special notation will be introduced for this operator.

No

2. Surface Laplacian of a vector functionLet T(Vs, V;, j) = (Vs 0 V~)F; then ris, m', j) = (m .m')F. Followingthe same procedure as the previous case, we obtain

Vs 0 V;F = VsVsF. (5.71)

3. The surface gradient of the surface divergence of a vector functionLet T(Vs, V~, j) = v,V~ 0 F; then T(m, m', j) = m(m' .F). The doublelimit of this function yields

Vs • V;F = VsVsF. (5.72)

4. The double surface curl of a vector functionLet T(Vs , V~, j) = Vs x (V: x F); then T(m, m', F) = m x (m' x F).We find

Vs x (V; x F) = VsVsF.

As a result of Lemma 4.1,

Vs x (V~ x F) = (Vs · F)V~ - (Vs · V;)F

= V;(Vs • F) - Vs . V;F.

We obtain the identity

VsVsF = VsVsF - VsVsF= (VsVs - VsVs)F.

This is analogous to the three-dimensional identity

VVF=VVF-VVF

= (VV - VV)F.

(5.73)

(5.74)

Finally, we would like to present a scalar Green's theorem for a curved surface.This theorem can be obtained by applying the generalized surface Gauss theorem,(5.44), with

(5.75)

Sec. 5-7 Expressions with Two Surface Symbolic Vectors 115

As a result of Lemmas 4.1 and 5.1 or by means of (5.61), we find

T(Vs) = aVsVsb - bVsVsa

and

T(rn) = m . (aVsb - bVsa)

(5.76)

(5.77)ab aa=a--b-.am am

Substituting (5..76) and (5.77) into (5.44), we obtain the scalar surface Green'stheorem of the second kind:

II(aVsVsb - bVsVsa)ds = f (a :~ - b::) dl.

Other theorems can be derived following similar procedures.

(5.78)

Chapter 6

Vector Analysis of TransportTheorems

6-1 Helmholtz Transport Theorem

Thus far, we have been dealing only with functions of position, that is, functionsthat are dependent on spatial variables only. In many engineering and physicalproblems, the quantities involved are functions of both space and time. Examplesare the induced voltage in a moving coil of an electric generator and the transportof fluid in a channel. The mathematical formulation of these problems requires aknowledge of vector analysis involving a moving surface or a moving body. One ofthe fundamental theorems in this area is the Helmholtz transport theorem, namedafter the renowned German scientist Hermann Ludwig Ferdinand von Helmholtz(1821-1894). The theorem deals with the time rate of change of a surface integralof Type IV stated by (3.70), in which the domain of integration and the integrand arefunctions of both space and time. The quantity under consideration is defined by

1= dd f·[ F(R,t)·dSt }S(R,/)

(6.1)

= lim _1 [f~[ F(R, t + ~t) . dS - f·( F(R, t) . dS] ,6/~O ~t J~(R, 1+6/) JS. (R,t)

where F(R, t) is an abbreviated notation for F(Xl, X2, X3, t). The Xi'S denotethe coordinate variables of the position vector where the function F is defined,and t is the time variable. The domain of integration changes from St (R, t) to~ (R, t + ~t) in a small time interval ~t, as shown in Fig. 6-1. To evaluate the

116

Sec. 6-1 Helmholtz Transport Theorem 117

tlS 3 =dt X v ~t

dS 2 =dS at t + ~t

dS l =dS at t

Figure 6-1 Moving surface at two different instants.

limiting value of the difference of the two surface integrals contained in (6.1), wefirst expand the integrand F(R, t + t:1t) in a Taylor series with respect to t:

aF(R t) 1 a2F(R t)F(R, t + l:1t) = F(R, t) + at' l:1t + 2" at2 ' (M)2 + .". (6.2)

Upon substituting (6.2) into (6.1), we have

. 1 {fl [ aF(R, t) JI = lim - F(R, t) + t:1t + ... . dS~t-+O t:1t . ~(R, t+~t) at

- 1"( F(R, t) . dS}]SI(R,t)

(6.3)

11 8F(R, t) . I [11= ·dS+ 11m - F(R,t) ·dSS(R,t) at L\t-+O Ilt S2(R, t+L\t)

- I"( F(R, t) . dSJ .]SI(R,t)

In (6.3), SI (R, t) is the same as S(R, r); the subscripts 1 and 2 are used toidentify the location of the surface at time t and at a later time t + t:1t, respectively,as shown in Fig. 6-1. A point PI at the contour of SI is displaced to a point P2at the contour of ~ during the time interval t:1t. The displacement is equal tov!!it, where v denotes the velocity of motion at that location, which may varycontinuously from one location to another around the contour. For example, whena circular loop spins around its diagonal axis, the linear velocity varies aroundits circumference. The two surface integrals within the brackets of (6.3) can be

118 Vector Analysis of Transport Theorems Chap. 6

written in the form

f'f F(R,t) .as-.ff F(R,t) ·dS)S2(R, t+M )SI(R,t) (6.4)

=J[ F(R, t) · dS - f" f F(R, t) · dS 3 ,

~.+~+~ ~where 83 denotes the lateral surface swept by the displacement vector v li.t as 81moves to ~. It is observed that dS1 is pointed into the volume bounded by Sl, ~,and S3, while dS 2 is pointed outward. The closed surface integral in (6.4) can betransformed to a volume integral, that is,

In (6.4) and (6.5),

!Is F(R, t) · dS =IIIv V F(R, t) dV.

dS3 = dl x (v 6.t) ,

dV = (v 6.t) · dS.

(6.5)

(6.6)

(6.7)

(6.8)

(6.9)

By making use of the mean-value theorem in calculus, the surface integral in (6.4)evaluated on 83 and the volume integral in (6.5) can be written in the followingform:

-ILF(R, t) · d S 3 = l:!:.t t F(R, t) • (v x dl:)

= -M t[V x F(R, -n ae,

IIIv V F(R, t) dV = l:!:.t Il [vV F(R, t)] · dS.

Equation (6.4) now becomes

f"f F(R, t) · dS - fre ( F(R, t) · dSJS2(R, t+~t) JS. (R,t)

= l:!:.t {/l[VVF(R, .n dS - t[V x F(R, or--l (6.10)

= l:!:.t {Ii {vV F(R, t) - V [v x F(R, t)]} · -l.In (6.10), the line integral has been converted to a surface integral by means of theStokes theorem. Equation (6.3), after taking the limit with respect to li.t, yields

dd f" f F(R, t) · dS = f~( {aF~R, t) + vV F(R, t)t JS(R,t) JS(R,t) t

- V [v x F(R, t)]} ·dS,

Sec. 6-2 Maxwell Theorem and Reynolds Transport Theorem

or simply,

:1 IiF·dS =Ii[~~ +vV F- V(v x F)] ·dS,

119

(6.11 )

(6.14)

(6.13)

which represents the Helmholtz transport theorem [12] using the modem notationof vector analysis first formulated by Lorentz [13], and reiterated by Sommerfeld[14]. Equation (6.11) can be cast in a different form by making use of identity(4.159) for V (v x F), which yields

~ IiF·dS =Ii[~~ +v· W+FVv-F· vvJ ·dS. (6.12)

This version of the Helmholtz transport theorem is used by Candel and Poinsot[15] in formulating a problem in gas dynamics.

Because the material derivative ofF or the total time derivative ofF is definedby

dF = aF + t: aFaxj = aF + v . VF,dt at i=1 ax; at at

(6.12) can be written in the form

~ IiF· dS =Ii[~~ +FVv - F· vvJ .os.

This form of the Helmholtz theorem is found in the treatment by Truesdell andToupin [16].

6-2 Maxwell Theorem and Reynolds Transport Theorem

Two related theorems can now be derived from the Helmholtz transport theorem,although in the original works these two theorems were formulated independentlyof the Helmholtz theorem. In the Helmholtz theorem, if we let

F=vr (6.15)

(6.16)

and then convert the surface integral into a line integral, we find, noting that V F = 0in view of (4.178),

!!- 1. f . dl = J (ar - v x V r) .se.dt rs X at

This is the statement of the Maxwell theorem originally found in his great workon electromagnetic theory [17, 18].

In the Helmholtz theorem, if the surface is a closed one, we obtain

:1 #F·dS =# (~~ +vV F) .ss. (6.17)

120 Vector Analysis of Transport Theorems Chap. 6

The closed surface integral of V (v x F) vanishes because it is equal to a volumeintegral of V W(v x F), which vanishes identically because of (4.179).

As a consequence of the Gauss theorem, (6.17) can be changed into the form

:t IIIWFdV = III[:t WF+ W(VWF)] dV. (6.18)

Now, if we let VF = p, a scalar function, then we obtain the Reynolds transporttheorem [19], namely,

:t IIIpdV =III[~~ + W(PV)] dV,

where we identify v as the velocity of the fluid with density p. Because

V (pv) = pVv + v Vp,

and the total time derivative, or the material derivative, of p is given by

dp apdt = at + v · Vp,

(6.19) can be written in the form

:t IIIpdV =III[~~ +PWv] dV.

(6.19)

(6.20)

(6.21)

It should be mentioned that in the original work of Reynolds, (6.19) was derivedby evaluating the total time derivative of JfJ P d V ,

!!- ff! P dV = lim ~ [ff"( p(t + M) dV - ff"( p(t) dVJ 'd t Dot-+O 6.1 Jv (t+Dot) JV (t)

in a manner very similar to the derivation of the Helmholtz theorem.From the preceding discussion, it is seen that the Helmholtz transport theorem

can be considered as the principal transport theorem; both the Maxwell theoremand the Reynolds theorem can be treated as lemmas of that theorem.

(7.1)

Chapter 7

Dyadic Analysis

7-1 Divergence and Curl of Dyadic Functionsand Gradient of Vector Functions

In Chapter 1, the definition of dyadic functions in a rectangular system and thealgebra of these functions were introduced. In this chapter, the calculus ofdyadics,or dyadic analysis, will be developed. _

The divergence of a dyadic function 1:., expressed in a rectangular system by

(1.82) in Chapter 1, will be denoted by V F, and is defined by

. = ,"". " ""aFij"V F = ~(VFj)xj = c: c: --.xi,j i j ax,

which is a vector function. _ _The curl of a dyadic function of the form F, denoted by ., F, is defined by

V F = L(VFj)xj = L L(Vr;j x Xj)Xj, (7.2)j j

which is also a dyadic function. Here, we have used the vector identity

'WFj = V LF[jX; = L(V'F[j XXi) (7.3)

to convert the single sum in (7.2) to a double sum.

121

122 Dyadic Analysis Chap. 7

Sometimes we need the gradient of a vector function in dyadic analysis,denoted by VF. In a rectangular coordinate system, its expression is given by(4.153), that is,

In OCS, it is defined by

""'Uj of "Uj 0 '"' AVF = L...J - - = L...J - - L...J Fju ji h, oV; i h, oU; j

"u; (aFjA F aUj)=L...J- -Uj+ i i:': .

. . h; OV; OViI,}

(7.4)

(7.5)

The derivatives of the unit vector Uj in this equation can be expressed in terms ofthe other unit vectors Ui,Uk with the aid of (2.59) and (2.61), which yields

VF ""'u; [( aF; r, ah; Fk ah;) A= L...J- -+--+-- u,; b, aUi h j aUj hk aVk

(OFj F; ah;) A (OFk F; ah;) ,.. ]+ ---- U·+ ---- UkaVi h j aUj ] av; hk aVk

with i, I, k = 1,2, 3 in cyclic order. Expression of VF in GCS can be derived ina similar manner. _

When a dyadic function is formed by a scalar function f with an idemfactor Iin the form of

/1= Llx;x;, (7.6)

the divergence of this dyadic function is then a vector function, and it is given by

= afV(fI) = L-Xi = VI

; ax;

In the OCS, I is defined by

1= LU;u;,i

and the divergence of fl is defined by

f = ""' u; a '"fA ,..V ( I) = L...J - · - L...J U jUj; h; av; j

",u; "[of,, A aUjA f"auj]=L...J-0L...J -UjUj+!-Uj+ Uj- .t h, j au; OU; av;

(7.7)

(7.8)

(7.9)

Sec. 7-1 Divergence and Curl of Dyadic Functions and Gradient of Vector Functions 123

In (7.9), for j = i,

" aUi 0u, · - = 0, (7.1 )av;

which can be proved by taking the derivative of

Ui · Ui = 1with respect to Vi, or any variable for that matter. With the aid of (2.59) and (2.61),the last two terms in (7.9) cancel each other; hence

= 1 af A

V(ll) = L - -U; = Vi (7.11); hi av;

Equation (7.11), therefore, is invariant to the coordinate system. We demonstratehere once more the invariance of V. By following the same approach, we find

V'(/h = L(V'lxj)xj = L(VI x Xj)Xj = VI x J. (7.12)j j

which is a dyadic. This identity is valid in any coordinate system.To find the divergence of a dyadic in an OCS, one can transform all the

functions defined in a rectangular system to the functions in a specified DeS. Welet

F = L Fjx; = L Fjuj = F', (7.13)i j

where F; and Fi (with i, j = 1, 2, 3) denote, respectively, the components of thefunction For F' in the two systems; then

Fj =L F;x; · Uj = Lej;fi, (7.14)i i

where C ji denotes the directional cosines between X; and Ui: These coefficientscan be found by the method of gradient in Section 4-5. The inverse transform is

F; = LCji F; . (7.15)j

Equation (7.15) also applies to the transformation of the unit vectors. For example,

Xi = LCjiUj. (7.16)j

By definition,

V F = L(VFj)xj = L(VF'j)Xjj j

(7.17)

which is a vector function.


For mixed dyadics made of two independent vector functions of the form

F = M(R) N(R'), (7.18)

where Rand R' represent two indepe~dent position vectors in some coordinate

system, the divergence and the curl of Fwith respect to the unprimed coordinatesare defined as

v F = [VM(R)] N(R'),

V F= [VM(R)] N(R').

(7.19)

(7.20)

The divergence and the curl of F with respect to the primed coordinates are notdefined, but

and

V'[F]T = [V'N(R')] M(R)

V'[FlT = [V'N(R')] M(R).

(7.21)

(7.22)

These functions are found in the application of dyadic analysis to electromagnetictheory [3].

A vector-dyadic identity to be quoted in the next chapter is derived here toshow its origin. We have previously derived an identity, (4.158), showing thatwhen a dyadic is formed by two vectors in the form of ab,

Similarly,

V (ab) = (Va)b + a . Vb.

V (ba) = (Vb)a + b· Va.

(7.23)

(7.24)

The dyadic ba is the transpose of abe By taking the difference between the lasttwo equations, we obtain

V(ba - ab) = (Vb)a - (Va)b + b- Va - a· Vb.

The right side of (7.25), according to (4.159), is equal to V (a x b); hence

V (ba - ab) = V (a x b).

(7.25)

(7.26)

This identity was listed in Appendix B of reference [20] without derivation.

7-2 Dyadic Integral Theorems

There are several integral theorems in dyadic analysis that can be derived bychanging the vector functions in the vector Green's theorems to dyadic functions.

Sec. 7-2 Dyadic Integra) Theorems 125

1. First vector-dyadic Green's theoremThe first vector theorem stated by (4.211) will be written in the

following form:

I I1[cw P) · (V Q) - P · V V QJ dV = in. (P x V Q) dS.

(7.27)We have purposely placed the function Q in the posterior position in (7.27),a practice that is used to change a vector to a dyadic. Consider now threedistinct Qj with j = 1, 2, 3 so that we have three identities of the sameform as (7.27). By juxtaposing a unit vector i j at the posterior positionof each of the three equations and summing them, we obtain

Ill[(Vp). (V Q) -po VV QJdV = in. (P x V Q)dS,

(7.28)where, by definition,

and

V Q= L(VQj)Xjj

(7.29)

(7.30)vv Q= L(VVQj)Xj.j

Equation (7.28) is designated as the first vector-dyadic Green's theore.!TIof Type A because it involves a vector function P and a dyadic function Q.By interchanging P with Q in (7.27) and then raising the level of Q to adyadic, we obtain

III [(VP)· (V Q) - (V VP)· QJdV = - in. [(VP) x QJ dS.(7.31)

Equation (7.31) is designated as the first v~ctor-dyadic Green's theoremof Type B. Except for the term (1fP) · (V Q), which is common to (7.28)and (7.31), the rest are different.

2. .Second vector-dyadic Green's theoremBy subtracting (7.28) from (7.31), we obtain the second vector

dyadic Green's theorem:

III [p·vV Q-(VVP).QJdV = - i n·[(PxV Q)+(VP)x QJdS.(7.32)


This theorem is probably the most useful formula in the application ofdyadic analysis to electromagnetic theory [3].

3. First dyadic-dyadic Green's theoremEquation (7.28) can be elevated to a higher level by moving P and

V P into the posterior position and transposing the dyadic terms into theanterior position so that

III{rv Q]T. (VP) - rvv Q]T. p} dV = trv Q]T. (0 x P)dS.

(7.33)The vector function P can now be elevated to a dyadic level that yields thefirst dyadic-dyadic Green's theorem:

III{rv Q]T. (V 'F) - rvv Q]T. p} dV =trv Q]T. (0 x P)dS,

(7.34)By doing the same thing with (7.31), we obtain

IIIv {[V Q]T. (V h- rQ]T. (V V P)} dV = - t[Q]T-m x V P)dS.

(7.35)

4. Second dyadic-dyadic Green's theoremBy taking the difference between (7.34) and (7.35), we obtain the

second dyadic-dyadic Green's theorem:

II1{[V V Q]T ·P - [QlT · (V V P)} dV

= - i {[V O]T · (0 X P) + [OlT · (0 X V P)} dS.

(7.36)The two dyadic-dyadic Green's theorems involve two dyadics; hence wehave the name. They can be used to prove the symmetrical property ofthe electric and magnetic dyadic Green's functions [3]. We have nowassembled all of the important formulas in dyadic analysis, with the hopethat they will be useful in digesting technical articles involving dyadicanalysis, particularly in its application to electromagnetic theory.

Chapter 8

A Historical Study of VectorAnalysis

8-1 Introduction I

In a book on the history of vector analysis [21], Michael J. Crowe made a thoroughinvestigation of the decline of quatemion analysis and the evolution of vectoranalysis during the nineteenth century until the beginning of this century. Thetopics covered are mostly vector algebra and quaternion analysis. He had fewcomments to offer on the technical aspects of the subject from the point of view ofa mathematician or a theoretical physicist. For example, the difference betweenthe presentations ofGibbs and Heaviside, considered to be two founders ofmodemvector analysis, is not discussed in Crowe's book, and less attention is paid to thehistory of vector differentiation and integration, and to the role played by the deloperator, V.

Brief descriptions of the history of vector analysis from the technical pointof view are found in a few books. For example, in a book by Burali-Forti andMarcolongo [22] published in 1920, there are four historical notes in the appendixentitled, respectively, "On the definition of abstraction," "On vectors," "On vectorand scalar (interior) products," and "On grad, rot, div," In a book by Moon andSpencer [23] published in 1965, there is a brief but critical review of the history

I This chapter is based on C. T. Tai, "A historical study of vector analysis," Technical ReportRL 915, The Radiation Laboratory, Department of Electrical Engineering and Computer Science, TheUniversity of Michigan, May 1995.

127

128 A Historical Study of Vector Analysis Chap. 8

of vector analysis from a technical perspective; this book will receive furtherdiscussion later. In their introduction, Moon and Spencer firmly state an importantreason why they present vector analysis by way of tensor analysis [23, p. 9]:

The present book differs from the customary textbook on vectors in stressingthe idea of invariance under groups of transformations. In other words,elementary tensor technique is introduced, and in this way, the subjectis placed on the finn, logical foundation which vector textbooks havepreviously lacked.

Further, in Appendix C [23, p. 323], Moon and Spencer write:

In reading the foregoing book [referring to their book], one may wonder whynothing has been said about the operator V, which is usually considered suchan important part of vector analysis. The truth is that V, though providing thesubject with fluency, is an unreliable device because it often gives incorrectresults. For this reason-and because it is not necessary-we have omittedit in the body of the book. Here, however, we shall indicate briefly the useof the operator V ....

These two quotations are sufficient to indicate that after decades ofapplicationof vector analysis, there seems to be no systematic treatment of the subject thatcould be considered satisfactory according to these two authors. This observationis also supported by the fact that we have so far no standard notations in vectoranalysis. Many books on electromagnetics, for example, use the linguisticnotations for the gradient, divergence, and curl-grad u, div f, and curl f-whilemany others prefer Gibbs's notations for these functions-Vu, V-f',and V x f. Canwe offer a better explanation to students as to why we do not yet have a universallyaccepted standard notation than to say it is merely a matter of personal choice? Inregard to Moon and Spencer's comments about the lack ofa firm, logical foundationin previous books on vector analysis, there has been no elaboration. They do givean example of an incorrect result from using V to form a scalar product with f tofind the expression for divergence in an orthogonal curvilinear coordinate system,but they do not explain why the result is incorrect. In fact, the views expressed bythese two authors are also found in many books treating vector analysis. Thesewill be reviewed and commented upon later.

In this chapter, we assume that the reader has already read the previouschapters of this book, particularly the method of symbolic vector in Chapter 4.The aim of this chapter is to point out the inadequacy or illogic in the treatments ofsome basic topics in vector analysis during a period of approximately one hundredyears. The mistreatments are then rectified by the proper tools introduced inthis book.

Sec. 8-2 Notations and Operators 129

8-2 Notations and Operators

8-2-1 Past and Present Notations in Vector Analysis

In a book on advanced vector analysis published in 1924, Weatherbum [24]compiled a table of notations in vector analysis that had been used up to that time.The authors represented in that table are Gibbs, Wilson, Heaviside, Abraham,Ignatowsky, Lorentz, Burali-Forti, and Marcolongo. A table of notations is alsogiven in Moon and Spencer's 1965 book Vectors (from which the quotationsin the previous section were taken). The authors represented in that table areMaxwell, Gibbs, Wilson, Heaviside, Gans, Lagally, Burali-Forti, Marcolongo,Phillips, Moon, and Spencer. Among these authors, Gibbs, Wilson, Phillips,Moon, and Spencer are American; Maxwell and Heaviside belong to the Englishschools; while Abraham, Ignatowsky, Gans, and Lagally belong to the Germanschools. Lorentz was a Dutch physicist, Burali-Forti and Marcolongo were Italian,and Ignatowsky was a native of Russia but was trained in Germany, For ourstudy, we have prepared another list representing several contemporary authorsand some additional notations; this is given in Table 8.1. The dyadic notationis added because we need it to characterize the gradient of a vector, which is adyadic function. In perusing this table, the reader will recognize the linguisticnotations grad u, div A, curl A, or rot A for the three key functions. The reader isprobably familiar with Gibbs's notations Vu, V· a, and V x a, except that here theperiod in V.a has been replaced by a raised dot (.) as in Wilson's notations, andhis Greek letters for vectors are now commonly replaced by boldface Clarendon(or equivalent fonts), while the linguistic notations are used by many authors inEurope and a few in the United States. Gibbs's notations have been adopted inmany books published in the United States. We will later quote from two bookson electromagnetic theory, one by Stratton and another by Jackson. Their treatisesare well known to many electrical engineers, as well as physicists.

Historically, vector analysis was developed a few years after Maxwellformulated his monumental work in electromagnetic theory. When he wrotehis treatise on electricity and magnetism [25] in 1873, vector analysis was notyet available. Its forerunner, quaternion analysis, developed by Hamilton (18051865) in 1843, was then advocated by many of Hamilton's followers. It is probablyfor this reason that Maxwell wrote an article in his book (Article 618) entitled"Quatemion Expressions for the Electromagnetic Equations." Maxwell's notationsin our list are based on this document. Actually, he made little use ofthese notationsin his book and in his papers published elsewhere.

The notation used by Heaviside is unconventional from the present point ofview. His notation for the scalar product and the divergence does not have a dot andhis notation for the curl is of the quatemion form, as is Maxwell's. The notationsused by Burali-Forti and Marcolongo are obsolete now. Occasionally, we still see

~ ~

Tab

le8.

1:L

ist

ofN

otat

ions Sc

alar

Vec

tor

Dya

dic

Tens

orG

radi

ento

fG

radi

ento

fD

iver

genc

eof

Cur

lorR

otL

apla

cian

ofL

apla

cian

ofA

utho

r(s)

Vec

tors

Prod

uct

Prod

uct

in3-

spac

ein

3-sp

ace

aSc

alar

aV

ecto

ra

Vec

tor

ofa

Vec

tor

aSc

alar

aV

ecto

r

Max

wel

l[2

5]a,

pSa

pV

ap-

-V

u-

-SV

pV

vpv

2u

A,B

Gib

bs(4

]a,p

a,p

ax

pap

Vu

Va

V·a

Vx

aV

·Vu

V·V

aW

ilson

[30]

A,B

A·B

Ax

BA

D-

Vu

VA

V·A

Vx

AV

·Vu

V·V

AH

eavi

side

[26]

A,D

AD

VA

B-

-V

uV

·AV

A;d

ivA

VV

A;c

uriA

V2u

V2 A

Gan

s[37

]A

,B(A

,B)

[A,B

]-

-V

u;gr

adu

-V

·A;d

ivA

Vx

A;r

otA

6u

~A

Bur

ali-F

orti!

A,B

Ax

BA

AD

--

grad

u-

divA

rotA

62U

~;A

Mar

colo

ngo

[22]

Stra

tton[

5]A

,DA

·BA

xD

-T

;jV

uV

AV

·AV

xA

V2u

V2 A

...V

2 uV

2A]a

ckso

n(4

4]A

,BA

·BA

xB

TT

;jV

uV

AV

·AV

xA

Moo

n!A

,BA

·BA

xD

-T

;jgr

adu

-di

vAcu

rlA

V2u

t1.A

Spen

cer[

23]

Upp

erca

sesc

ript

sym

bols

are

used

here

inpl

aceo

fcap

italG

erm

anle

tters

orig

inal

lyus

edby

Max

wel

land

Gan

s.

the notation A /\ B for the cross product in the works of European authors. On thewhole, we now have basically two sets of notations in current use: the linguisticnotation and Gibbs's notation. The names of Moon and Spencer are included inour list primarily because these two authors considered the use of V to be unreliableand they frequently emphasize their view that the rigorous method of formulatingvector analysis is to follow the route of tensor analysis. In addition, their newnotation for the Laplacian of a vector function will receive detailed examinationin the section on orthogonal curvilinear systems.

8-2-2 Quaternion Analysis

The rise of vector analysis as a distinct branch of applied mathematics has itsorigin in quatemion analysis. It is therefore necessary to review briefly the laws ofquatemion analysis to show its influence on the development ofvector analysis, andalso explain the notations in the previous list. Quatemions are complex numbersof the form

q = w + i x + jy + kz, (8.1)

(8.2)

where ui, x, y, and z are real numbers, and i, I. and k are quaternion units, orquaternion unit vectors, associated with the x, y, and z axes, respectively. Theseunits obey the following laws of multiplication:

ij = k, jk = i, ki = j,

ji = -k, kj = -i, ik = - j,

ii=jj=kk=-I.

We must not at this stage associate these relations with our current laws ofunit vectors in vector analysis. We consider the subject as a new algebra, which isindeed the case. The product of the multiplication of two quaternions <1 and p inwhich the scalar parts wand w' are zero is obtained as follows:

We let

0' = i D, + jD2 + kD3,

p=iX+jY+kZ.

Then,

op = -(D1X + D2Y + D3Z)

+ i(D2Z - D3Y) + j(D3 X - D)Z) + k(DtY - D2X) .(8.3)

InThe resultant quaternion, op, has two parts, one scalar and one vector.Hamilton's original notation, they are

S.ap = -(D)X + D2Y + D3Z),

V.ap = i(D2Z - D3Y) + j(D3X - D) Z) + k(Dt Y - D2X).

(8.4)

(8.5)


The period between S or V and cp can be omitted without any resulting ambiguity.When one identifies (J as V, Hamilton's del operator defined with respect to thequatemion unit vectors, that is,

then,

8 a aa=V=i-+j-+k-,ax ay az(8.6)

(8.7)

(8.8)

(ax ar az)

SVp=- -+-+- ,ax ay 8z

VVp=i (az _ ay)+j(8X _ az.) +k(ay _ ax)..az az 8z ax ax 8y

Maxwell used the quatemion notation SVp for the negative of the divergence of p,which he termed the convergence. He used the quatemion notation VVp for thecurl of p, The term curl, now standard, was coined by Maxwell. According toCrowe [21, p. 142] the term divergence was originally due to William KingdomClifford (1845-1879), who was also the first person to define the modem notationsfor the scalar and vector products. However, his original definition of the scalarproduct is the negative of the modem scalar product. In the list of notations, wenotice that Heaviside used the quatemion notation for the curl even though he wasopposed to quatemion analysis. In one of his writings [26, p. 35], he concurredwith Gibbs's treatment of vector analysis but criticized Gibbs's notations withoutoffering a reason; we discuss this comment of Heaviside's later in Section 8-5.Before we discuss the works of these various authors, a review of the meaning ofthe algebraic and differential operators is necessary.

8-2-3 Operators

For our convenience, we would like to discuss in sufficient detail theclassification and the characteristics of a number of operators appearing in thisstudy. We will focus on unary and binary operators and consider such operators incascade or compound arrangements as the complexity of the case at hand requires.

A unary operator involves only one operand. A binary operator needs twooperands, one anterior and another posterior. A cascade operator could be unaryor binary. As an example, we consider the derivative symbol a/ax to be aunary operator. When it operates on an operand P, it produces the derivative,8P/ax. In some writings, the operator a/ax is denoted by Dx • The operandunder consideration can be a scalar function of x and other independent variablesor a vector function or a dyadic function; that is,

ap aa aA aft'ax ax' ax' ax

are all valid applications of the unary differential operator. The partial derivative


of a dyadic function in a rectangular system is defined by

of = L oF j Xj = L L oFij XjXj. (8.9)ax . ax .. ax

} I }

We list in Table 8-2 several commonly used unary operators and their possibleoperands. The function a in the weighted differential operator a (8/ax) is assumedto be a scalar function. A vector operator such as A(8/ax) can operate on a dyadicthat would yield a "triadic"-a quantity that is not included in this study. The lastoperator in Table 8-2 is the del operator, or the gradient operator. It can be appliedto an operand that is either a scalar or a vector.

Table 8-2: Valid Application of Some Unary Differential Operators

Operator

aaxa

a axa

A-ax"A av = L...JX;-

i ax;

Type of Operand

b, B, iJ

b. B, iJ

b, B

b. B

Results

ab aa aiJax' ax' ax

ab aB aiJa ax' a ax' a ax

Aab, A aBax ax

v». VB

A binary operator requires two operands. In arithmetic and algebra, wehave four binary operators: + (addition), - (subtraction), x (multiplication), and-+- (division). In these cases, we need two operands, one anterior and anotherposterior, as in 2 + 3, 4 - 3, 5 x 3, and 6 -:- 3. Note that the symbols + and- are also used to denote "plus" and "minus" signs. For example,-a = 1a Iwhen a is negative. In this case, the minus sign is not considered to be a binaryoperator in our classification, but rather as a unary "sign change" operator. Thetwo binary operators involved frequently in our work are the dot (.) and the cross(x). They appear in Gibbs's notations for the scalar and vector products, that is,a · b and a x b. We consider the dot and the cross as two binary operators, andtheir operands, one anterior and one posterior, must be vectors; that is,

A·B and A x B.The dot operator is not the same as the multiplication operator in arithmetic, nor isthe cross operator the same as the multiplication operator, although we use the samesymbol for both. According to the definitions of the scalar and vector products,

A· B = B· A = IAIIBIcos O, (8.10)

A x B = -B x A = IAIIBIsin9c, (8.11)


where 9 is the angle measured from A to B in the plane containing these twovectors, and cis the unit vector perpendicular to both A and B and is pointed in theright-screw advancing direction when A turns into B. The dot and the cross canalso be applied to operands where one of them or both are dyadics. Thus, we have

A·B, lJ·A, AxE, BxA, A·lJ, B·A. (8.12)

The first two entities are vectors and the remaining four are dyadics.The last group of operators are called cascade or compound operators. Of

particular concern in this study is the proper treatment of a pair of operators ofdifferent types, which are applied sequentially. When one of the operators is ascalar differential unary operator, and the other is a vector binary operator, therearise a number of hazards in their application which, if not properly treated, couldlead to invalid results. Several commonly used cascade operators are of the forms

a a· ay' ·V, x ay' xV. (8.13)

These operators also require two operands; the anterior operand must be a vectoror a dyadic and the posterior operand must be compatible with the part in front.Thus, we can have

(8.14)

a1lA·-·ay ,

A·VB;

aDA·ay ,A·Vu,

A x aD A x ail ·By , 8y ,

A x v«. A x VB.

In (8.13), the unary operators, a/8y and V, and the binary operators, · and x,are not commutative; hence the following combinations or assemblies are not validcascade operators:

(8.16)v ·11,

v x B,

V·A,

VxA,

a aay" V" ayx, Vx. (8.15)

These assemblies are formed by interchanging the positions of the symbols in(8.13). They are not operators in the sense that we cannot find an operand to forma meaningful entity. For example,

a a =ay ·A, ay' B,

a a =- xA - x Bay , 8y ,

do not have any meaningful interpretation. The reader has probably noticed thatthere are two assemblies, V . A and V x A, in (8.16) that correspond to Gibbs's

Sec. 8-3 The Pioneer Works of J. Willard Gibbs (1839-1903) 135

Chapter V.

Chapter I.Chapter II.

Chapter III.Chapter IV.

notation for the divergence and curl. This is true, but that does not mean that V . Ais a scalar product between V and A, nor is V x A a vector product between Vand A. In fact, this is a central issue in this study to be examined critically in thefollowing sections. We now have the necessary tools to investigate many of thepast presentations of vector analysis.

8-3 The Pioneer Works of J. Willard Gibbs (1839-1903)

8-3-1 Two Pamphlets Printed in 1881 and 1884

Gibbs's original works on vector analysis are found in two pamphlets entitledElements ofVector Analysis [4], privately printed in New Haven. The first consistsof 33 pages published in 1881 and the second of 40 pages published in 1884.These pamphlets were distributed to his students at Yale University and also tomany scientists and mathematicians including Heaviside, Helmholtz, Kirchhoff,Lorentz, Lord Rayleigh, Stokes, Tait, and J. J. Thomson [27, Appendix IV]. Thecontents are divided into five chapters and a note on bivectors:

Concerning the algebra of vectorsConcerning the differential and integral calculus

of vectorsConcerning linear vector functionsConcerning the differential and integral calculus

of vectors (Supplement to Chapter II)Concerning transcendental functions of dyadicsA note on bivector analysis

The most important formulations for our immediate discussions are coveredin Articles SO-54 and 68-71, which are reproduced here.

Functions of Positions in Space

50. Def.-If u is any scalar function of position in space (i.e., anyscalar quantity having continuously varying values in space), Vuis the vector function of position in space which has everywhere thedirection of the most rapid increase of u, and a magnitude equal tothe rate of that increase per unit of length. Vu may be called thederivative of u, and u, the primitive of VUe

We may also take anyone of the Nos. 51, 52, 53 for thedefinition of VUe

51. If P is the vector defining the position of a point in space,

du = Vu· dp.


52.

53.

du du duvu =i- +j- +k-.dx dy dz

(8.17)

du- =i· Vudx '

dudy = l :Vu,

dudz = k· VUe

54. Def.-If CO is a vector having continuously varying values in space,

dco dm droV·co=i· - +j. - +k·-,

dx dy dzdro dro dro

V x eo = i x dx + j x dy + k x dz '

V · co is called the divergence of ro and V x ro its curl.If we set

co = Xi + Y j + Zk,

we obtain by substitution the equation

dX dY dZV·co=-+-+-

dx dy dz

(8.18)

(8.19)

(8.20)

and

V x co = i (dZ _ dY) + j (dX _ dZ) + k (dY _ dX) ,dy dz dz dx dx dy

(8.21)which may also be regarded as defining V · ro and V x roo

Combinations of the Operators V, V·, and Vx

68. If mis any vector function of space, V · V x co = O. This may bededuced directly from the definition of No. 54.

The converse of this proposition will be proved hereafter.

69. If u is any scalar function of position in space, we have by Nos. 52and 54

(8.22)

(8.23)

70. Def.-If co is any vector function ofposition in space, we may defineV · Vro by the equation

(d2 d2 d 2

)V · Veo = dx2 + dy. + dz2 eo,


the expression V· V being regarded, for the present at least, as a singleoperator when applied to a vector. (It will be remembered that nomeaning has been attributed to V before a vector.) Note that if

ro=iX+jY+kZ,

then

V· Vro = iV· VX+ jV· VY +kV· VZ, (8.24)

that is, the operator V · V applied to a vector affects separately itsscalar components.

71. From the above definition with those of Nos. 52 and 54, we mayeasily obtain

v . VOl = VV · co - V x V x roo (8.25)

The effect of the operator V · V is therefore independent of thedirection of the axes used in its definition.

In quoting these sections, we have changed Gibbs's original notation for thedivergence from V.ro to V . ro, that is, the period has been replaced by a dot. Theequation numbers have been added for our reference later on.

After Gibbs revealed his new work on vector analysis, he was attacked fiercelyby Tait, a chief advocate of the quatemion analysis, who stated [28, Preface]:

Even Prof. Willard Gibbs must be ranked as one ofthe retarders ofquatemionprogress, in virtue ofhis pamphlet on vector analysis; a sort ofhennaphroditemonster, compounded by the notations ofHamilton and Grassman.

This infamous statement has been quoted by many authors in the past. Gibbs'sgentlemanly but finn response to Tait's attack [29]:

The merit or demerits of a pamphlet printed for private distribution a goodmany years ago do not constitute a subject of any great importance, butthe assumption implied in the sentence quoted are suggestive of certainreflections and inquiries which are of broad interest; and seem not untimelyat a period when the methods and results of the various forms of multiplealgebra are attracting so much attention. It seems to be assumed that adeparture from quatemionic usage in the treatment of vectors is an enormity.If this assumption is true, it is an important truth; if not, it would beunfortunate if it should remain unchallenged, especially when supportedby so high an authority. The criticism relates particularly to notations, but Ibelieve that there is a deeper question of notions underlying that ofnotations.Indeed, if my offense had been solely in the matter of notation, it wouldhave been less accurate to describe my production as a monstrosity, than tocharacterize its dress as uncouth.


Gibbs then went on to explain the advantage ofhis treatment of vector analysiscompared with quatemion analysis. In the final part of that paper he stated:

The particular form of signs we adopt is a matter of minor consequence. Inorder to keep within the resources of an ordinary printing office, I have useda dot and a cross, which are already associated with multiplication, whichis best denoted by the simple juxtaposition of factors. I have no specialpredilection for these particular signs. The use of the dot is indeed liableto the objection that it interferes with its use as a separatrix, or instead of aparenthesis.

Although Gibbs considered his choice of the signs or notations a matter ofminor importance, it was actually of great consequence, as will be shown in thisstudy. Before we discuss his notations, a comment from Heaviside, generallyconsidered by the scientific community as a cofounder with Gibbs of modernvector analysis, should be quoted. During the peak of the controversy betweenTait and Gibbs, Heaviside made the following remark [26, p. 35]:

Prof. W. Gibbs is well able to take care of himself. I may, however, remarkthat the modifications referred to are evidence of modifications felt to beneeded, and that Prof. Gibbs' pamphlet (not published, New Haven, 188184, p. 83), is not a quatemionic treatise, but an able and in some respectsoriginal little treatise on vector analysis, though too condensed and also tooadvanced for learners' use, and that Prof. Gibbs, being no doubt a littletouched by Prof. Tait's condemnation, has recently (in the pages of Nature)made a powerful defense ofhis position. He has by a long way the best of theargument, unless Prof. Tait's rejoinder has still to appear. Prof. Gibbs clearlyseparates the quatemionic question from the question of a suitable notation,and argues strongly against the quatemionic establishment of vector analysis.I am able (and am happy) to express a general concurrence of opinion withhim about the quatemion and its comparative uselessness in practical vectoranalysis. As regards his notation, however, I do not like it. Mine is Tail's,but simplified, and made to harmonize with Cartesians.

There are two implications in Heaviside's remark that are of interest to us.When he considered Gibbs's pamphlet to be too condensed, it implies that some ofthe treatments may not have been obvious to him. Secondly, he stated his dislike forGibbs's notations but without giving his reasons. The fact that Heaviside used someof Tait's quatemionic notations seems to indicate that he did not approve of Gibbs'snotations at all. We now believe that many workers, including Heaviside, did notappreciate the most eloquent and complete theory of vector analysis formulatedby Gibbs. For this reason, we would like to offer a digest of Gibbs's work so thatwe may have a clear understanding of his formulation.


8-3-2 Divergence and Curl Operators and Their NewNotations

The basic definitions of the gradient, divergence, and the curl formulated byGibbs are given by (8.18), (8.19), and (8.20). For convenience, we will make somechanges in symbols to allow the convenience of using the summation sign. Thesechanges are

x,y,z to XI,X2,X3,

i,j,k to XI,X2,X3.

(8.26)

The old total derivative symbols will be replaced by partial derivatives andthe Greek letters for vectors by boldface letters. Thus, Eqs. (8.17)-(8.19) become

~" auv« = L..JXi-,; ax;

(8.28)

(8.29)

(8.30)

(8.27)

(8.31)

~" aFV·F= L...Jx;·_,i ax;

~" aFv x F = L..Jx; x -.i ax;

It is understood that the summation goes from i = 1 to i = 3.The most important information passed to us by Gibbs concerns the nomen

clature for the notations in these expressions. In the title preceding Article 68quoted previously, he designated V, V·, and Vx as operators. If we examinethe expressions given by (8.26), (8.27), and (8.28) it is obvious that the gradientoperator, or the del operator, is unmistakably given by

""" aV'= £...JXi-.i OX;

For the divergence, Gibbs used two symbols, a del followed by a dot, todenote his divergence operator. For the curl, he used a del followed by a cross todenote the curl operator. If we examine the expressions for the divergence and thecurl defined by (8.27) and (8.28), it is clear that his two notations mean:

(V·)G - LX; .~ ,i ax;

""" a(Vx)G ~ L...Jx; X - .; ox;

We emphasize this point by labeling his two notations with a subscript G, and weuse an arrow instead of an equal sign to denote "a notation for."

According to our classification of the operators in Section 8.2, Gibbs's (V·)Gand (Vx)G are not compound operators; they are assemblies used by Gibbs asthe notations for the divergence and curl. On the other hand, the terms at the


(8.32)

(8.33)

right side of (8.30) and (8.31) are indeed compound operators, according to ourclassification. Because these operators are distinct from the gradient operator, wewill introduce two notations for them. They are

V ="x·· ~L.J I a 'i Xi

"A av = L.JXi X -.i ax;

They are called the divergence operator and the curl operator, respectively.Although these operators are so far defined in the rectangular coordinate system,we willdemonstrate later that they are invariant to the choice of coordinate system.One important feature of V and V is that both these operators are independent ofthe gradient operator V. In other words, V is not a constituent of the divergenceoperator nor of the curl operator. These two symbols are suggested by theappearance of the dot or the cross in between the unit vectors Xi and the partialderivatives a/ax; of the V operator as defined by (8.29). In Gibbs's notations,(V·)o and (Vx)o, V is a part of his notations for the divergence and the curlthat leads to a serious misinterpretation by many later users and is a key issue inour study. With the introduction of these two new notations, Eqs. (8.18)-(8.26)become

"A auVu = L.Jx;- ,; ax;

VF= LXi' aF ,i ax;

VF= L aFi ,i ax;

F "A ofV = L...JX; x -,; ax;

VF = LX; (8Fk _ aFi )

i Bx] aXk

with (i, j, k) = (1, 2, 3) in cyclic order,

a2uVVu=L-2 '; ax;

a2FVVF=L-2 '

i ax;

VVF = Lx;VVfj,i

VVF = VVF - V'rF.

(8.34)

(8.35)

(8.36)

(8.37)

(8.38)

(8.39)

(8.40)

(8.41)

(8.42)

Sec. 8-4 Book by Edwin Bidwell Wilson Founded upon the Lectures of J. Willard Gibbs 141

In these formulas, the del operator only enters in the gradient of a scalar, (8.34),or of a vector, (8.40)-(8.42). Except for the notations for the divergence and thecurl, we have not changed the content of Gibbs's work at all. These equations willbe used later in our study of other people's presentations.

8-4 Book by Edwin Bidwell Wilson Foundedupon the Lectures of J. Willard Gibbs

8-4-1 Gibbs's Lecture Notes

The first book on vector analysis by an American author was published in1901. The author was Edwin Bidwell Wilson [30], then an instructor at YaleUniversity. According to the general preface, the greater part of the material wastaken from Prof. Gibbs's lectures on vector analysis delivered annually at Yale. Arecord of these lectures is preserved at Yale's Sterling Memorial Library: it is aclothbound book of notes, handwritten in ink on 8! in. x llin. ruled paper, about290 pages long and consisting of 15 chapters [31]. The title page reads as follows:

Lectures Delivered

upon Vector Analysis

and its

Applications to Geometry and Physics

by

Professor J. Willard Gibbs 1899-90

reported by Mr. E. B. Wilson

The table of contents is given in Table 8-3.

8-4-2 Wilson's Book

Presumably, Wilson's book (436 pages) is based principally on these notes.The preface states, however, that some use has been made of the chapters on vectoranalysis in Heaviside's Electromagnetic Theory (1893) and in Foppl's lectures


Table 8-3: Table of Contents of E. B. Wilson's Lectures Delivered uponVector Analysis

Ch.lCh.2Ch.3Ch.4Ch.5Ch.6Ch.7Ch.8Ch.9Ch.l0Ch.llCh.12Ch.13Ch.14Ch.15

Fundamental Notions and OperatorsGeometrical Applications of Vector AnalysisProducts of VectorsGeometrical Applications of ProductsCrystallographyScalar Differentiation of VectorsDifferentiating and Integrating OperationsPotentials, Newtonians, Laplacians, MaxwelliansTheory of Parabolic OrbitsLinear Vector FunctionsRotations and StrainsQuadratic SurfacesCurvature of Curved SurfacesDynamics of a Solid BodyHydrodynamics

page

1112550627283

110125164200223234261276

on Maxwell's Theory of Electricity (1894). Apparently, Gibbs himself was notinvolved in the preparation of the body of the book, but he did contribute a preface,from which the following two paragraphs are taken:

I was very glad to have one of the hearers of my course on Vector Analysis inthe year 1899-1900 undertake the preparation of a text-book on the subject.

I have not desired that Dr. Wilson should aim simply at the reproductionof my lectures, but rather that he should use his own judgment in all respectsfor the production of a text-book in which the subject should be so illustratedby an adequate number of examples as to meet the wants of students ofgeometry and physics.

In the general preface, Wilson stated:

When I undertook to adapt the lectures ofProfessor Gibbs on Vector Analysisfor publication in the Yale Bicentennial Series, Professor Gibbs himselfwas already so fully engaged in his work to appear in the same series,Elementary Principles in Statistical Mechanics, that it was understood nomaterial assistance in the composition of this book could be expected fromhim. For this reason he wished me to feel entirely free to use my owndiscretion alike in the selection of the topics to be treated and in the mode


(8.44)

(8.45)

(8.46)

of treatment. It has been my endeavor to use the freedom thus granted onlyin so far as was necessary for presenting his method in text-book form.

The following passage from Wilson's preface is particularly significant for thepresent discussion:

It has been the aim here to give also an exposition of scalar and vectorproducts of the operator V, of divergence and curl which have gainedsuch universal recognition since the appearance of Maxwell's Treatise onElectricity andMagnetism, slope, potential, linear vector functions, etc. suchas shall be adequate for the needs of students of physics at the present dayand adapted to them.

We point out here that in Gibbs's pamphlets and in the lecture notes reported byWilson, there is no mention of the scalar and vector products of the operator V.We believe this concept or interpretation was created by Wilson, and unfortunatelyit has had a detrimental effect upon the learning of vector analysis within theframework of Gibbs's original contributions.

In explaining the meaning of the divergence of a vector function, Wilsonmisinterpreted Gibbs's notation for this function, namely V .F. After defining the\l operator for the gradient in a rectangular system as

o 0 BV=i-+j-+k-, (8.43)ox oy OZ

he stated in Section 70, p. 150, of Wilson's book [30]:

Although the operation VV has not been defined and cannot be at present,two formal combinations of the vector operator V and a vector function Vmay be treated. These are the (formal) scalar product and the (formal) vectorproduct of V into V. They are:

( a a a)V·V= i-+j-+k- ·V,ax ay oz

(0 0 a)VxV= i-+j-+k- xv.ox oy az

The differentiations :x' :y' iz, being scalar operators, pass by the dot andthe cross, that is

V . V = (i .av + i av + k- av) ,ax oy Bz

(av av BV)VxV= ix-+jx-+kx- .ox By Bz

They may be expressed in terms of the components VI, V2 , V3 0f V .

(8.47)


We have identified the equations with our own numbers. In order to comparethese expressions with Gibb's expressions now described by (8.34) to (8.38), weagain will change the notations V, x, y, z, i, j, k to F, Xl, X2, X3, XI, X2, X3; andV . V and V x V to VF and V F. Equations (8.43) to (8.47) then become

'"A aV= L.JX;-'i ax,

VF= (LXi~) ·F,; ax,

VF = (LXi~) x F,; ax,

F '" A aFv = L.JX; x -.i ax;

(8.48)

(8.49)

(8.50)

(8.51)

(8.52)

Equations (8.48), (8.51), and (8.52) are identical to Gibbs's (8.29), (8.35),and (8.37). However, (8.49) and (8.50) are not found in Gibbs's works. Wilsonobtained or derived (8.51) and (8.52) from (8.49) and (8.50). The derivationinvolves two crucial steps or assumptions. First, he considers Gibbs's notationsV . F and V x F as "formal" scalar and vector products between V and F. Inthe following, we will refer to this model as the FSP (formal scalar product)and FVP (formal vector product). He did not explain the meaning of the wordformal. Secondly, after he formed the FSP and FVP, he let the differentiationa/ax; "pass by" the dot and the cross with the argument that the differentiationsa/ax; (i = 1,2,3) are scalar operators. Wilson's statement appears to be quitefinn, but the standard books on mathematical analysis contain no proofs of anysuch theorem. Later on [30, p. 152], Wilson attempts to modify his position bysaying:

From some standpoints objections may be brought forward against treatingV as a symbolic vector and introducing V · V and V x V as the symbolicscalar and vector products of V into V, respectively. These objections maybe avoided by simply laying down the definition that the symbol V· and Vx,which may be looked upon as entirely new operators quite distinct from V,shall be

av av avV·V=i·-+j·-+k.-ax ay az (8.53)

Sec. 8-4

and

Book by Edwin Bidwell Wilson Founded upon the Lectures of J. Willard Gibbs 145

(8.55)

(8.57)

(8.56)

av sv avV x V = i x - + j x - + k x - . (8.54)

ax ay azBut for practical purposes and for remembering formulas, it seems by

all means advisable to regard

a a aV=i-+j-+k-

ax ay ozas a symbolic vector differentiator. This symbol obeys the same laws as avector just in so far as the differentiations a/ax, ajoy, a/az obey the samelaws as ordinary scalar quantities.

The contradictions between Wilson's statement above and his assertionconcerning FSP and FVP are evident. Equations (8.53) and (8.54), of course,are the same as Gibbs's (8.27) and (8.28) with V replaced by F and x, y, z, i, j, kreplaced by Xl, X2, x3, XI, X2, X3. The difference is that Gibbs never spoke of anFSP and FVP; Wilson introduced these concepts to derive the expressions for div Fand curl F by imposing. some nonvalid manipulations. What is the consequence?Many later authors followed his practice and encountered difficulties when thesame treatment was applied to orthogonal curvilinear coordinate systems. Beforewe discuss this topic, we must review Heaviside's treatment of vector analysis,particularly his handling of V.

We have pointed out that Gibbs's pamphlets were communicated to Heaviside.On the .other hand, Wilson also mentioned some use of Heaviside's treatmentof vector analysis in the preface to his book, Electromagnetic Theory (1893).The exchange between Heaviside and Wilson was therefore reciprocal. However,Heaviside goes his own way in presenting the same topics. Before we tum to thenext section, Wilson's FSP and FVP model will be analytically examined. If westart with one of Gibbs's definitions of divergence, without using his notation butrather by using the linguistic notation, that is,

d· F "aF;IV = L...J-'

i ax;

then, by substituting F} = XI . F into (8.55), we find

di F " a(x; · F) '"' [A aFax; FJIV = L.., = L.., Xi· - + - · .

; ax; i ax; ox;

Because ax; /ax; = 0, (8.56) reduces to

. F "A aFdiv = L..,X; . - ,; ax;

which is obviously not equal to

(L:Xia:J ·F,


or V . F. This is a proof of the lack of validity of the FSP. A similar proof can beexecuted with respect to the FVP. Another demonstration of the fallacy of an FSPis to consider a "twisted" differential operator of the form

A a A 0 A 0Vt =X2- + X3- + Xl- (8.58)

OXt OX2 aX3

and a "twisted" vector function defined by

F t = X2 Fl + X3 F2 + x1F3 • (8.59)

If the FSP were a valid product, then, by following Wilson's pass-by procedure,we obtain

V,. F, = aFt + aF2 + aF3 • (8.60)ox} OX2 OX3

In other words, div F is now treated as the formal scalar product between Vt and Ft.The result is the same as Wilson's FSP between V and F. Such a manipulationis not, of course, a valid mathematical procedure. We have now refuted Wilson'streatment of div F and curl F based on the FSP and FVP. The legitimate compounddifferential operators for the divergence and the curl are, respectively, V and 'W ,defined by (8.32) and (8.33). (V·)o and (Vx)G are merely Gibbs's notationssuggested for the divergence and the curl. They are not operators.

8-4-3 The Spread of the Formal Scalar Product (FSP)and Formal Vector Product (FVP)

Being the firstbook on vector analysis published in the United States, Wilson'sbook became quite popular. It received its eighth printing in 1943, and a paperbackreprint by Dover Publications appeared in 1960. Many later authors freelyadopted Wilson's presentation using the FSP and FVP to derive the expressionsfor divergence and curl in the Cartesian coordinate system. We have found over50 books [32] containing such a treatment. We now quote a few examples to showWilson's influence.

1. In the book by Weatherburn [24] published in 1924, we find the followingstatement:

To justify the notation, we have only to expand the formal productsaccording to the distributive law, then

V·f= [L(Xi~)]·f= L a~ = div f.i OX, i ax,

We remark here that any distributive law in mathematics must beproved. In this case, there is no distributive law to speak of becausethe author is dealing with an assembly of mathematical symbols and nota compound operator. Incidentally, Weatherburn's book appears to be


the first book published in England wherein Gibbs's notations, but notHeaviside's, have been used in addition to the linguistic notations, namely,grad u, div f, and curl f.

2. A book by Lagally [33] published in 1928 contains the following statementon p. 123 (the original text is in German):

The rotation (curl) of r is denoted by the vector product between Vwith field function r ... and

div grad! = V· Vf = (LXj~) ·(LXj af .) = V2 fi ax, i ax}

It is seen that a term like il(a/aXt)· Xt(a/aXt) is an assembly ofsymbols. It is not a compound operator.

3. In a book by Mason and Weaver [34, p. 336] we find the followingstatement:

The differential operator V can be considered formally as a vector ofcomponents a/ax, a/ay, a/az, so that its scalar and vector productswith another vector may be taken.

In comparison with Wilson's treatment, Mason and Weaver have usedthe word formally to be associated with V and then speak of scalar andvector products with vector functions.

4. In his book AppliedMathematics, Scheikunofffirst derived the differentialexpression for the divergence based on. the flux model [35, p. 126]; thenhe added:

In Section 6 the vector operator del was introduced. If we treat it asa vector and multiply it by a vector F, we find

(

" A a) (" A ) " aF; .v· F = L..J x;-. · L..JxjF j = £...J -. = div F.i AX, j i AX,

For this reason, v· may be used as an alternative for div; however,the notation is tied too specifically to Cartesian coordinates.

There are two messages in this statement: the first one is hisacceptance of the FSP as a valid entity. The second one is his implicationthat FSP only applies to the Cartesian system. Actually, the divergenceoperator, V , is invariant with respect to the choice ofthe coordinate system,a property shown in Chapter4, but V· is an assembly, not an operator. Onlyby means of an illegitimate manipulation does it yield the differentialexpression for the divergence in the Cartesian coordinate system.

5. In a well-known book by Feynman, Leighton, and Sands [36, pp. 2-7] wefind the following statement:

Let us try the dot product between V with a vector field that weknow, say f: we write

V· f = Vxfx + Vyfy + Vzh

or

The authors remarked on the same page before this statement:

With operators we must always keep the sequence right, so that theoperations make the proper sense ....

This remark is important. Our discussion and use of the operators inSection 8-2, particularly that related to the compound operators, closelyadheres to this principle. In the case ofGibbs's notation,V -f, we are facedwith a dot symbol after V, so that the differentiation cannot be applied tof; it is blocked by a dot in the assembly. Thus, the authors seem to haveviolated their own rule by trying to form a dot product or FSP.

6. In the English translation of a Russian book by Borisenko and Tarapov[2, p. 157], we find the following statement:

The expression V = L ik(B/Bxk) for the operator V implies thefollowing representation for the divergence of A:

. aAt adiv A = - = ;k- · A = V · A.

aXk aXk

A coordinate-free symbolic representation of the operator V is

V( ...) = lim VI i N(...) dSV-+O is (8.61)

where (...) is some expression (possibly preceded by a dot or across) on which the given operator acts. In fact, according to (4.31)and (4.29) [of their book],

grad <p = lim VI i cpNd S. (8.62)V-+O is

div A = lim .! i A · N as. (8.63)v-+o V is

From this passage, we see that the two authors believe the FSP is valid.Equation (8.61) also implies that they consider V to be a constituent of thedivergence and the curl in addition to comprising the gradient operator.The formula described by (8.61) appeared earlier in the book by Gans [37,

Sec. 8-5 V in the Hands of Oliver Heaviside (1850--1925) 149

p. 49], who used both Gibbs's notations and the linguistic notations in thisedition.

There are several authors presenting V as defined as L,(8/8x;)x;instead of L,x;(8/8x;); and the Laplacian, defined as div grad, is oftentreated as the scalar product between two nablas, presumably becauseGibbs used V · V as the notation for this compound operator. Thesepractices, including the use of an FSP and FVP, are not confined withinthe boundaries of the United States and continental Europe; some Chineseand Japanese books, for example, commit the same errors.

8-5 V in the Hands of Oliver Heaviside (1850-1925)

Although we have traced the concept of the FSP and FVP as due to Wilson,the same practice is found in the works of Heaviside. In Volume I of his bookElectromagnetic Theory [26, §127] published in 1893, Heaviside stated:

When the operand of V is a vector, say D, we have both the scalar productand the vector product to consider. Taking the formula along first, we have

div D = V1D1+ V2D2 + V3D3 •

This function of D is called the divergence and is an important function inphysical mathematics.

He then considered the curl of a vector function as the vector product betweenV and that vector. At the time of his writing, he was already aware of Gibbs'spamphlets on vector analysis but Wilson's book was not yet published. It seems,therefore, that Heaviside and Wilson independently introduced the misleadingconcept for the scalar and vector products between V and a vector function.Both were, perhaps, induced by Gibbs's notations for the divergence and thecurl. Heaviside did not even include the word formal in his description of theproducts. We should mention that Heaviside's notations for these two productsand the gradient are not the same as Gibbs's (see the table of notations in Section2.1). His notation for the divergence of f is Vf and his notation for the curl of f isV Vf (a quatemion notation), while his notation for the gradient ofa scalar functionf is V.f. Having treated V · r and V x f (Gibbs's notations for the divergence andthe curl) as two "products," Heaviside simply considered V as a vector in derivingvarious differential identities. One of them was presented as follows [26, §132]:

The examples relate principally to the modification introduced by thedifferentiating functions of V.

(a) We have the parallelopiped property

NVVE = VVEN = EVNV (176)

where V is a common vector. The equations remain true when V is vex,provided we consistently employ the differentiating power in the three forms,


Thus, the first form, expressing N component of curl E, is not open tomisconception. But in the second form, expressing the divergence of VEN,since N follows V, we must understand that N is supposed to remain constant.In the third form, again, the operand E precedes the differentiator; we musteither, then, assume that V acts backwards, or else, which is preferable,change the third form to VNV.E, the scalar product of VN"1 and E, or(VNV)E if that is plainer.

(b) Suppose, however, that both vectors in the vector product arevariable. Thus, required the divergence of V Eif, expanded vectorially.We have,

VVEH = EVHV = HVVE, (177)

where the first form alone is entirely unambiguous. But we may use eitherof the others, provided that the differentiating power of V is made to act onboth E and H. But if we keep to the plainer and more usual convention thatthe operand is to follow the operator, then the third term, in which E alone isdifferentiated, gives one part of the result, whilst the second form, or ratherits equivalent, -EVH, wherein H alone is differentiated, gives the rest. Sowe have, complete, and without ambiguity

div VEH = H curl E - E curl H, (178)

an important transformation.

First of all, in terms of Gibbs's notations, Heaviside's Eqs. (176), (177), and(178) would be written in the form

N· V x E = V· (E x N) = E· (N x V),

V . (E x H) = E · (H x V) = H · (V x E),

V . (E x H) = H . V x E - E · V x H.

(8.64)

(8.65)

(8.66)

According to the established mathematical roles, Heaviside's logic in arrivingat his (178) or our (8.66) is entirely unacceptable; in particular, present-day studentswould never write an equation (177) or (8.65) with V being the V operator. Thesecond term in (8.65) is a weighted operator, while the first and the third arefunctions and they are not equal to each other. His Eq. (178) or (8.66) in Gibbs'snotation is a valid vector identity but his derivation of this identity is not based onestablished mathematical rules. It is obtained by a manipulation of mathematicalsymbols and selecting the desired forms. The most important message passed onto us is his practice of considering V · f and V x r as two legitimate products, thesame as Wilson's FSP and FVP. Heaviside's "equations" will be examined againin a later section and will be cast in proper form in terms of the symbolic vectorand/or a partial symbolic vector.

Many authors in the past have considered Heaviside to be a cofounder withGibbs of modern vector analysis. We do not share this view. In Heaviside's

Sec. 8-6 Shilov's Formulation of Vector Analysis 151

treatment of vector analysis, he spoke freely of the scalar product and the vectorproduct between V and a vector function F, and he used Vas a vector in derivingalgebraic vector identities that incorporate differential entities. In view of thesemathematically insupportable treatments, Heaviside's status as a pioneer in vectoranalysis is not of the same level as Gibbs's. In the historical introduction of a 1950edition of Heaviside's book on Electromagnetic Theory [26], Ernst Weber stated:

Chap. III of the Electromagnetic Theory dealing with 'The Elements ofVectorial Algebra and Analysis' is practically the model of modem treatiseson vector analysis. Considerable moral assistance came from a pamphlet byJ. W. Gibbs who independently developed vector analysis during 1881-84in Heaviside sense-but using the less attractive notation of Tait; however,Gibbs deferred publication until 1901.

This statement unfortunately contains several misleading messages. In thefirst place, in view of our detailed study of Heaviside's works, his treatment wouldbe a poor model if it were used to teach vector calculus. Secondly, if Heaviside trulyreceived moral assistance from Gibbs's pamphlet, he would not have committedhimself to the improper use of V, and would have restricted his use of it to theexpression for the gradient. Most important of all, Gibbs did not develop histheory in the Heaviside sense. His development is completely different from thatof Heaviside. Finally, the book published in 1901 was written by Wilson, notby Gibbs himself. Even though it was founded upon the lectures of Gibbs, itcontained some of Wilson's own interpretations, which are not found in Gibbs'soriginal pamphlets nor in his lecture notes reported by Wilson. The two prefaces,one by Gibbs and another by Wilson, which we quote in Section 8-4-2, are proofsof our assertion. We were reluctant to criticize a scientist of Heaviside's status andthe opinion expressed by Prof. Weber. After all, Heaviside had contributed muchto electromagnetic theory and had been recognized as a rare genius. However, inthe field of vector analysis, we must set the record straight and call attention to theoutstanding contribution of Gibbs, who stood above all his contemporaries in thelast century. For the sake of future generations of students, we have the obligationto remove unsound arguments and arbitrary manipulations in an otherwise precisebranch of mathematical science.

8-6 Shilov's Formulation of Vector Analysis

A book in Russian on vector analysis was written by Shilov [38] in 1954, whoadvocated a new formulation with the intent of providing a rather broader treatmentof vector analysis. Shilov's work was adopted by Fang [39], who studied in theU.S.S.R. We were informed of Shilov's work through Fang. After a carefulexamination of the English translation of the two key chapters in that book, wefound the contradictions as described below:


Shilov defined an "expression" for V denoted by T (V) as

a a aT(V) = -T(i) + -T(j) + -T(k), (8.67)ax ay 8z

where i, j, k denote the Cartesian unit vectors and V (nabla) is identified as theHamilton differential operator, that is,

a 8 aV=i-+j-+k-.ax ay 8z

Equation (8.67) is the same as Shilov's Eq. (18) on p. 18 of [35]. We wantto emphatically call attention to the fact that the only meaningful expression forT (V) involving V are VI and vr, the gradient of f and f. In the case of VI,(8.67) is an identity, because the right side of (8.67) yields

~(if) + !-(jf) + !....(kf) = i at + ja t + k at ,ax ay az ax By az

which is VI.The most serious contradiction in Shilov's work is his derivation of the

expression for the divergence and the curl by letting T (V) equal to V · f andV x f, respectively. We have pointed out before that these two products do notexist. Shilov is defining a meaningless assembly to make it meaningful. It is likedefining 2 + x3 to be equal to 2 x +3 (= +6).

8-7 Formulations in Orthogonal Curvilinear Systems

After having revealed a number of "historical" confusions and contradictions invector analysis so far presented in the rectangular system, we now examine severalpresentations in curvilinear coordinate systems. We will show even more clearlythe sources of the various misrepresentations.

8-7-1 Two Examples from the Book by Moonand Spencer

In their book, Moon and Spencer write [23, p. 325]:

Let me apply the definition, Eq. (1.4) [of V in the orthogonal curvilinearsystem, our (7.20)], to divergence. By the usual definition ofa scalarproduct,

V . V = 1 a(V)t + 1 8(V)2 + 1 8(V)3.(gn)'/2 ax' (g22) '/2 ax2 (g33) '/2 ax3 (8.68)

But this is not divergence, which is found to be ....Similar inconsistencies are obtained with other applications ofEq. (1.4).

In (8.68), their (gu)I/2 correspond to our metric coefficients hi and their Xito our variables Vi.

Sec. 8-7 Formulations in Orthogonal Curvilinear Systems 153

In the first place, they have now applied the FSP to V and V in an orthogonalcurvilinear coordinate system without realizing that the FSP is not a valid entity inany coordinate system including the rectangular system. After obtaining a wrongfonnula for the divergence, (8.68), they did not offer an explanation of the reasonfor the failure.

In discussing the Laplacian of a vector function, Moon and Spencer state[23, p. 235]:

Section (7.08) showed that there are three meaningful combinations ofdifferential operators: div grad, grad div, and curl curl. Of these, the first isthe scalar Laplacian, V2 • It is convenient to combine the other two operatorsto form the vector Laplacian, I) :

I) = grad div - curl curl.

Evidently the vector Laplacian can operate only on a vector, so

I) E = grad div E - curl curl E.

(8.69)

(8.70)

Since the quantities on the right are vectors, (I E transforms as a univalenttensor or vector.

As noted in Table 1.01 [their table of notations on p. 10], the scalar andvector Laplacians are often represented by the same symbol. This is poorpractice, however, since the two are basically quite different:

V2 = div grad,

I) == grad div - curl curl.

(8.71)

(8.72)

This difference is evident also when the expression for the vector Laplacianis expanded....

Analytically, we have proved that in any general curvilinear system,

VVf=VVf-VVf, (8.73)

where Vf denotes the gradient of a vector function that is a dyadic function. Thedivergence of a dyadic function is a vector function. The use of V2 to denote theLaplacian is an old practice, but the use of V V is preferred because it shows thestructure of the Laplacian when it is applied to either a scalar function or a vectorfunction. By treating (8.69) as the definition for the Laplacian applied to a vectorfunction, the two authors have probably been influenced by a remark made byStratton [5, p. 50]:

The vector V · VF may now be obtained by subtraction of (85) [an expansionof V x V x F in an orthogonal curvilinear system] from the expansion ofVV . F, and the result differs from that which follows a direct application ofthe Laplacian to the curvilinear components of F.


As shown in our proof, VF is a dyadic, where the gradient operator mustapply to the entire vector function containing both the components and the unitvectors. When this is done, we find that (8.73) is indeed an identity. In view ofour analysis, it is clear that a special symbol for the Laplacian is not necessarywhen it is operating on a vector function. The same remark holds true for the twodifferent notations for the Laplacian introduced by Burali-Forti and Marcolongo,as shown in Table 8.1.

These two examples also show why Moon and Spencer thought that V is anunreliable device. The past history of vector analysis seems to have led them tosuch a conclusion. V is a reliable device when it is used in the gradient of a scalaror vector function, but not in any other application. We emphasize once more thatfor the divergence and the curl, the divergence operator, V, and the curl operator,V , are the proper operators. They are distinctly different from V.

8-7-2 A Search for the Divergence Operatorin Orthogonal Curvilinear Coordinate Systems

In a well-known book on the methods of theoretical physics [40, p. 44], theauthors, Morse and Feshbach, try to find the differential operators for the three keyfunctions in an orthogonal curvilinear coordinate system. They state:

The vector operator must have different forms for its different uses:

V - " Uj~ for the gradient- ~ hi av;

I

= !.. E Uj~ (Q) for the divergenceQ i oV; hi

and no form which can be written for the curl.

We have used Q to represent h 1h 2h 3 and have changed their coordinatevariables ~ to Vi and their symbols a, to Ui. It is obvious that the "operator"introduced by these two authors for the divergence can produce the correctexpression for the divergence only if the operation is interpreted as

(8.74)

Such an interpretation is quite arbitrary, and it does not follow the accepted rule ofa differential operator because the first term within the bracket is a function, so theentire expression represents the scalar product of [...] and f. One is not supposedto move the unit vector Uj to the right side of Q/ h and then combine Uj with -f', asshown in the right term of (8.74). It is a matter of creating a desired expression byarbitrarily rearranging the terms in a function and the position of the dot operator.A reader must recognize now that V can never be a part of the divergence operatornor the curl operator. The proper operators for the divergence and the curl are V

Sec. 8-8 The Use of V to Derive Vector Identities 155

and V ,respectively. We could have used any two symbols for that matter, such asDandC.

8-8 The Use of V to Derive Vector Identities

There are many authors who have tried to apply identities in vector algebra to"derive" vector identities involving the differential functions V f, V I, and V f.We quote here two examples. The first example is from the book by Borisenkoand Tarapov [2, p. 180], where a problem is posed and "solved":

Prob.7. Find V(A · B).Solution. Clearly

V(A . B) = V(Ae . B) + V(A . Be) (8.75)

where the subscript 'c' has the same meaning as on p. 170 [the subscript 'c'denotes that the quantity to which it is attached is momentarily being heldfixed]. According to formula (1.30)

c(A . B) = (A . c)B - A x (B x c). (8.76)

Hence setting

we have

A=A e , B=B, c=v,

(8.80)

\I(Ac • B) = (Ae . \I)B + Ae x (\I x B), (8.77)

and similarly,

V(A · Be) = V(B e · A) = (Be · V)A + Be X (\I x A). (8.78)

Thus, finally,

V(A · B) = (A . V)B + (B . V)A + A x curl B + B x curl A. (8.79)

As far as the final result, (8.79), is concerned, they have indeed obtained a correctanswer. But there is no justification for applying (8.76) with c replaced by \I.

The second example is found in the book by Panofsky and Phillips [41, p. 470].They wrote:

v x (A x B) = (\I . B)A - (\I . A)B

= (V . Be)A - (\I · B)Ae

- (\I . Ae)B - (V . A)Be

where the subscript 'c' indicates that the function is constant and may bepermuted with the vector operator, with due regard to sign changes if suchchanges are indicated by the ordinary vector relations.


It is seen that their (V · B)A in the first line is not (div B)A. Rather, it isequal to (V · Be)A + (V · D)Ac. Secondly, if Be is constant, the established rulein differential calculus would consider their V · Bc, (i.e., div Be) = O. The use ofalgebraic identities to derive differential identities by replacing a vector by V hasno foundation-the first line of (8.80). For the exercise in consideration, one wayto find the identity is to prove first that

'f' (A x B) = V(BA - AD), (8.81)

where AB is a dyadic and BA its transpose. Then, by means of dyadic analysis,one finds

Hence

V (BA) = (VB)A + B· VA,

V (AB) = (V A)B + A · VB.

V (A x B) = (VB)A + B . VA - (V A)B - A . VB,

(8.82)

(8.83)

(8.84)

(8.86)

where VA and VB are two dyadic functions. A simpler method of deriving(8.84) will be shown in a later section. It should be emphasized that one cannotlegitimately write

v x (A x B) = (V · B)A - (V · A)B

as the two authors did and then change (V · B)A to V · (BA), and similarly for(V . A)B, in order to create a desired identity.

A general comment on the analogy and no analogy between algebraic vectoridentities and differential vector identities was made by Milne [42]. He states onp.77:

The above examples [referring to nine differential vector identities expressedin linguistic notations such as grad (x . Y) = (grad x) · Y + (grad Y) . x,etc.] whilst exhibiting the relations between the symbols in vector or tensorform, conceal the nature of the identities. A little gain of insight is obtainedoccasionally if the symbol is employed. E.g., Example (9) [curl curl x =grad div x - V2x]may be written

V x (V x x) = V(V. x) - V2x, (8.85)

which bears an obvious analogy to

Q x (Q x x) = Q(Q · x) - Q2xwhere Q denotes a vector function.

On the other hand Example (5)

Curl (x x Y) =y. grad x - x .grad y.+ x div Y - Y div xmay be written

(8.87)

Sec. 8-9 A Recasting of the Past Failures by the Method of Symbolic Vector

which bears no obvious analogy to

Q x (x x Y) = x(Q . Y) - Y(Q . x)

To obtain a better analogy, one would have to write

Q x (x x Y) = Q . (YX - XV)

157

(8.88)

(8.89)

and replace Q by V.

We do not understand why (8.89) is a better analogy than (8.88) because, asalgebraic vector identities, they are equivalent. There is only one interpretation of(8.89), namely,

Q x (x x Y) = (Q . Y)x - (Q . x)Y, (8.90)

which is the same as (8.88).By replacing Q by V in (8.89), and treating the resultant expression as the

divergence of the dyadic XY - YX, the manipulation is identical to the one usedby Panofsky and Phillips. This short paragraph on the role played by del inan authoritative book on vectorial mechanics shows the consequence of treatingGibbs's notations for the divergence and the curl as two products, one scalar andone vector.

We have now shown the failures by several authors in trying to invoke V asan operator, not only for the gradient but also for the divergence and the curl. Therole is now filled in by the symbolic vector, to be discussed in the next section asintroduced in this book. Many of the ambiguities that have occurred in the pastpresentations covered in this paper will be recast correctly and unambiguously byour new method utilizing the symbolic vector.

8-9 A Recasting of' the Past Failures by the Methodof Symbolic Vector

If we replace Heaviside's "equations" (8.64)-(8.66) with

N . V x E = V(E x N) = E . (N x V),

V . (E x H) = E · (H x V) = H · (V x E),

V . (E x H) = V E • (E x H) - V H • (H x E)

= H · (VE X E) - E . (V'H x H),

then (8.93) yields

V (E x H) = H . VE - E . VH.

(8.91 )

(8.92)

(8.93)

(8.94)

Although (8.91) and (8.92) have the same form as Heaviside's except that hisV has been replaced by the symbolic vector V, yet there is a vast difference in

meaning between the two sets. For example, his H . V x E in (8.65) is interpretedas H . curl E, but our H . V x E is the same as V . (E x H) because of Lemma 4.2and it is equal to V (E x H).

Every term in (8.91) to (8.94) is well defined. Both Lemma 4.1 and Lemma4.2 are used to obtain the vector identity stated by (8.94).

Returning now to the problems posed by Borisenko and Tarapov, we startwith the symbolic expression V(A . B) for V(A . B); then, by applying Lemma4.2, we have

V(A· B) = VA(A· B) + VB(A· B).

Applying Lemma 4.1, we have

VA(A· B) = (B· VA)A - B x (A x VA)

and

Hence

VA (A · B) = B · VA + B x V A

and

Thus,

V(A · B) = A . VB + B . VA + A x VB + B x V A.

(8.95)

(8.96)

(8.97)

(8.98)

(8.99)

(8.100)

Our derivation of (8.1(0) appears to be similar to the derivation by Borisenkoand Tarapov in form, but the use of the FSP and FVP in their formulation and thetreatment of (8.77) as an algebraic identity is entirely unacceptable, while each ofour steps are supported by the basic pri<,nciple in the method of symbolic vector,particularly the two lemmas therein. -

The exercise posed by Panofsky and Phillips can be formulated correctly byour new method. The steps are as follows:

We start with V x (A x B), which is the symbolic expression of V (A x B);then by means of Lemma 4.2,

V x (A x B) = VA x (A x B) + V B x (A x B). (8.101)

By means of Lemma 4.1, we have

VA x (A x B) = (B· VA)A - (V A • A)B

= B· VA - BWA.

Similarly,

V B x (A x B) = (B· VB)A - (A· VB)B

= AWB -A· VB.

Sec. 8-9

Hence

A Recasting of the Past Failures by the Method of Symbolic Vector

V(B x B) = AVB -A· VB - BVA+B· VA,

159

(8.102)

which is the same as (8.84) obtained previously in Section 8-8 by a morecomplicated analysis. The convenience and the simplicity of the method ofsymbolic vector to derive vector identities has been clearly demonstrated in thelast two examples. All commonly used vector identities have been derived in thisway, as shown in Chapter 4.

8-9-1 In Retrospect

In this work, we have examined critically some practices of presenting vectoranalysis in several early works and in a few contemporary writings. We state withemphasis that the whole subject of vector analysis was formulated by the greatAmerican scientist J. Willard Gibbs in a precise and elegant fashion. Althoughhis original works are confined to formulations in a Cartesian coordinate system,they can be extended to curvilinear systems as a result of the invariance of thedifferential operators, as reviewed in this book, without the necessity of resortingto the aid of tensor analysis.

In spite of the richness of Gibbs's theory of vector analysis, his notations forthe divergence and the curl, in the opinion of this author, have induced severallater workers, including one of his students, Wilson, to make some inappropriateinterpretations. The adoption of these interpretations. has been worldwide. Wehave selected a few examples from the works of several seasoned scientists andengineers to illustrate the prevalence of the improper use of V.

As a result of this study, we have justified our adoption of the new operationalsymbols as the notations for the divergence and the curl to replace Gibbs's oldnotations. It seems that our move is reasonable from the logistical point of view.

We have examined a history covering a period of over one hundred years.It represents a most interesting period in the development of the mathematicalfoundations of electromagnetic theory. However, in view of the long-entrenchedand widespread misuse ofthe gradient operator Vas a constituent ofthe divergenceand curl operators, the obligation of sharing the insight presented here with manyof our colleagues in this field has been a labor fraught with frustration.

We hope that this historical study has been sufficiently clear to enable theserious workers in this subject to understand the issues, and that future students willnot have to ponder over contradictions and misrepresentations to learn this subject.

It may be proper to conclude this chapter and the book by quoting a remarkmade by E. B. Wilson 87 years ago [43]. In reviewing two Italian books on vectoranalysis by Burali-Forti and Marcolongo, Wilson concluded his article with thefollowing remark:

What the resulting residual system may be we will not venture to predict,but that there will be such a system fifty years hence we fully believe. And


whatever that system may be it should and probably will conform to tworequirements:

1. correct ideas relative to vector fields;

2. analytical suggestions of notations.

We sincerely believe that the method ofsymbolic vector together with the newoperational notations for the divergence and the curl have fulfilled Dr. Wilson'swish. Whether the new notations will be adopted by potential users we leave tofuture generations of students to decide. Our goal is to put a logical approach onrecord.

Appendix A

TransformationBetween Unit Vectors

A-1 Cylindrical System

(VI, V2, V3) = (r,~, z)

(h 1,h 2,h 3) = (l,r, 1)

x = r cos ~, y = r sin ~, z = z

x y zr cos e sin~ 0

~ - sin<t> cos e 0Z 0 0 1

A-2 Spherical System

(VI, V2, V3) = (R, 9, ~)(hI, h2 , h3) = (1, R, R sin 9)

x = R sin 9 cos <t>, y = R sin 9 sin ~, z = R cos 9

161

162 Appendix A

X Y zR sin 9 cos sin 0 sin cos Ba cos 9 cos cos 0 sin -sine

~ -sin~ cos 0

A-3 Elliptical Cylinder

(VI, V2, V3) = (11, ~, z)

[ (~2 2) 1/2 (~2 2) 1/2 ](h(,h 2,h3) = C 1~; • c 1;2-=-i ' 1

x=C11~, y=c[(1-112)(~2-1)]1/2, z=z

x y z

itc~ -C11

0hI h2

~C11 c~

0h2 hI

Z 0 0 1

A-4 Parabolic Cylinder

(VI, V2, V3) = (11,;, z)

(h 1. hz, h 3 ) = [(112+ 1;2)1/2. (112+ 1;2)1/2. 1]1

x = 2 (112_1;2) • Y = 111;. z = z

x y z

it 11 ~ 0h h

~-~ 11 0h h

Z 0 0 1

h = hi = hz = (112+ ~2)1/2

Appendix A

A-5 Prolate Spheroid

163

z ;. ~

11c~ -C11

0hi h2

~C11 c~ 0h2 hi

<t> 0 0 1

z = C11~

The unit vectors rand ~ can be expressed in terms of xand ycovered in thecylindrical case; the same applies to the oblate spheroid.

A-6 Oblate Spheroid

(VI, V2, V3) = (11, ~, ep)

z r ~

~c~ C11 0h2 hi

11-C11 c~ 0hi h2

~ 0 0 1

a sin 11y= ,

cosh ~ - cos II

164

A-7 Bipolar Cylinders

(v], V2, V3) = (11, ~, z)

(h 1,h 2,h 3 ) = ( h~a •cos - COS"

a sinh];x= ,

cosh ~ - cos 11z=z

Appendix A

x y z

it -h · h~ · h- sin sin 11 - (cosh~cos11 - 1) 0a a

~-h -h · h~ .- (cosh~cosll- 1) - sin slnll 0a a

Z 0 0 1

wherea

h = hI = h2 = .cosh ~ - cos 11

In all of these tables, the unit vectors are all arranged in the order of a righthanded system, that is, Xl X X2 = X3-

Appendix B

Vector and Dyadic Identities

Vector Identities

1. a- (b x c) = b · (c x a) = c- (a x b)2. a x (b x c) = (a . c) b - (a- b) c

3. V (ab) = aVb + bVa

4. V (ab) = aVb + (Va)b

5. V (ab) = aVb + b . Va

6. V (ab) = aV b - b x Va

7. V (8' b) = (Va) . b + a· Vb = a x Vb + b x V' a + a· Vb + b· Va

8. V (a x b) = b · ~ a - a . Vb

9. V (ab) = (Va)b + a· Vb

10. a x Vb = (Vb) · a - a- Vb

11. V (a x b) = V (ba - ab) = aVb - bVa - a- Vb - b · Va

12. VVa = VVa - "'iVa

13. VVa=O

14. VVa=O

165

166

Dyadic Identities

15. a· (b x c) = -b· (a x c) = (a x b) · c16. a x (b x c) = bta- c) - (a . b)c

17. V(ab)=aVb+(Va).b

18. V (ab) =aV b + (Va) x b19. V(a x b) = (Va)· b- a · Vb20. V V a = VVa - v Va

21. VVa = 022. VVa = 023. a- b= [b]T · a

24. a x b= - {[b]T x af

25. [e]T. (a x b) = -[a x c]T . b

Appendix B

2. Curl theorem:

3. Gradient theorem:

Appendix C

Integral Theorems

In this appendix, we use n to denote the normal unit vector to a surface that canbe either open or closed. The two tangential vectors of an open surface will bedenoted by i and m; i is tangential to the edge of the contour and mis normalto the contour. Both are tangential to the surface. The triad forms an orthogonalrelation mx i = nand dS = nds, dl = idle

1. Gauss theorem or divergence theorem:

IIIVFdv=lfn.FdS.

IffVFdV = If;, x FdS.

fff VldV =If srss.4. Surface divergence theorem:

ffVsFdS= fm .ru.5. Surface curl theorem:ffVsFdS= fm X r ae.

167

168 Appendix C

6. Surface gradient theorem:

II Vs/dS= fmfde.

7. Cross-gradient theorem:

II n x VfdS= ffdR..

8. Stokes's theorem:

II n·VFdS= fF.dR..

9. Cross-V-cross theorem:

II (n x V) x FdS= II[n x VF+n· VF-nVF]dS = - fF x «e.

10. First scalar Green's theorem:

111[avvb+ v«. Vb]dv =#n·aVbdS.

11. Second scalar Green's theorem:

I I I (aVVb - bVVa) dV = #;, ·(aVb - bVa) dS.

12. Second scalar surface Green's theorem:

f f (aV sVsb - bVs~a) dS = fm · (aVsb - bVsa) dS.

13. First scalar-vector Green's theorem:

Type 1:

Type 2:

fffuV VF+ vr VF)dV =#n· fVFdS,

Ilf (FVVf + vr VF)dV = #(n. Vf)FdS.

14. Second scalar-vector Green's theorem:

f f I UVVF - FVVf) dV = #;, ·[fVF - (Vf)F] dS.

15. First vector Green's theorem:

ffIHVP). (VQ) - p. VVQ] dV = #;,. (P x VQ) dS.

16. Second vector Green's theorem:

Iff(Q· VVP- p. VVQ) dV = #;,. (P x VQ - Q x VP) dS.

Appendix C 169

17. First vector-dyadic Green's theorem:

III[(VP) · V Q - p. V V Q] dV =#n· (P x V Q) dS.

18. Second vector-dyadic Green's theorem:

III[(V V P). Q - p. V V Q]dV = #n· [P x V Q+ (V P) x Q]dS.

19. First dyadic-dyadic Green's theorem:

III{[(nT. vv p- [V Qf· V p} dV =#[Q]T · (n x V P) dS.

20. Second dyadic-dyadic Green's theorem:

III{[Q]T . V V P- [V V Q]T . p} dV

=#{[Q]T . (n x V P) + [V Q]T · (n x P)} d S.

21. Helmholtz transport theorem:

!!-j.[ F.dS=j·{ [<JF +VVF-V(VXFl].dS.dt JS(t) J5(t) 8t

22. Maxwell's theorem:

!!- if. dl =1 (ar - v x V r) .ae.dt h(t) L(t) at

23. Reynolds's transport theorem:

!!- ff· ( p dV =jf·( [8P + V(PV)] dVdt }V(f) ]V(t) at

=11[</) (~~ +pvv) dV.

Appendix D

Relationships BetweenIntegral Theorems

The integral theorems stated by (1)-(3) and (7)-(8) in Appendix C are closelyrelated. By means of the gradient theorem, we can derive both the divergencetheorem and the curl theorem. From this point of view, we must first provethe gradient theorem, leaving aside its derivation by the symbolic method. Thetheorem states that

IIIVfdV = #Sf dS. (D.l)

(D.3)

In a rectangular system with coordinate variables (XI, X2, X3), the Xl component of (D.1) corresponds to

III:~ dx, dX2 dX3 = Iif dX2 dX3 - ILl f dX2 dX3. (D.2)

where SI and ~ denote the two sides of an enclosed surface S viewed in theXl direction. The negative sign associated with the surface integral evaluatedon 51 is due to the fact that the vector component of dS I is equal to -dX2 dX3 Xl.Equation (D.2) is a valid identity because the volume integral is given by

III:~ dx, dX2dx3 =11[f(P 2) - f(p\)] dX2 dX3

=Iif dx-i dx« - IL. f dx-i dx«,

170

Appendix D 171

where P2 and PI denote two stations located at opposite sides of the surface alongthe XI direction. The same procedure can be used to prove the remaining twocomponents of (D. 1). Having proved the validity of (D. 1), we can use it to deducethe divergence theorem (Gauss theorem) and the curl theorem.

We now consider three distinct sets of (0.1) in the form

fff VF; dV = Us F; dS, i=I,2,3. (D.4)

By taking the scalar product of (D.4) with Xi and summing the resultant equations,we obtain

Let

and because

we obtain

4: X; • fff VF; dV = 4: X; •is F; dS.I I

Xi· VF; = V· (F;xi),

fff VFdV =Us F·dS,

(0.5)

(0.6)

(D.7)

(0.8)

which is the divergence theorem. Similarly, by taking the cross product of Xi with(0.4), we obtain

~X; x fff VF;dV = ~Xi x ff F;dS.I I

Because

Xi x VF; = -V (F;x;),

(D.9) is equivalent to

fff VFdV =-ff F X dS,

(0.9)

(0.10)

(0.11 )

which is the curl theorem. The approach that we took can be applied to the othertwo theorems listed as (7)-(8) in Appendix C. In this case, we consider the crossgradient theorem as the key theorem that must be proved first. The theorem statesthat

ff;, x V/dS= f/dl. (0.12)

172

In a rectangular system, we can write

" ~"n = L...JnjXi,i

dl= LdxiXi.i

Appendix D

(0.13)

Then the Xl component of (0.12) reads

II (n2:~ -n3:~) dS= ff dX\ .

The surface integral in (0.13) can be written in the form

11 (af dx, dX3 + af dx, dX2) = 11 (a fdX2 + af dX3) dx,

aX3 aX2 aX2 OX3

=II d f dxi

=I[f(Pl) - f(p\)] dx,

=Lf(Pl) dx, - leI f(p\) dx,

=i f dX\ ,

where PI and P2 denote two stations on the closed contour, which consists oftwo segments Cl + C2. We have thus proved the validity of (0.13). The sameprocedure applies to the X2 and X3 components of (D.12). Once we have provedthe cross-gradient theorem, it can be used to deduce the Stokes theorem.

We consider three distinct sets of (D.12) in the form

II nx VF; dS = fF; se. ;=1,2,3. (D. 14)

By taking the scalar product of (D.14) with Xi and summing the resultant equations,we obtain

Because

and we let

4= Xi •II(n x VF;) d S = 4= Xi ·f F; ae., I

Xi· (n x VF;) = -n· (Xi X VF;) = n·V (F/Xi),

(D.15)

(D.16)

Appendix D

(D.15) can be written in the form

II;'·VFdS= !F'd/.,

173

(D.17)

which is the Stokes theorem. It is seen that in this analysis, the gradient theoremis considered the key theorem based on which the other three theorems can bereadily derived. The approach taken here has its own merit without consideringthe derivation of these theorems, independently, by the symbolic method.

The relationships among the gradient theorem, the divergence theorem, andthe curl theorem have previously been pointed out by VanBladel [11, Appendix I].Alternatively, we can use the divergence theorem and the Stokes theorem as thekey theorems to derive the other three theorems. The manipulations, however, aremore complicated.

Appendix E

Vector Analysisin the Special Theory

of Relativity

To study the theory of relativity, the most efficient mathematical tool is multidimensional tensor analysis with a dimension greater than three. Within the realmof the special theory of relativity, the subject can be treated by ordinary vectoranalysis in three dimensions. In fact, this is what Einstein did in his original workpublished in 1905. In this appendix, we shall first follow this approach and thenshow how the same result is obtained by a four-dimensional analysis.

Based on the experimental evidence that the velocity of light is independentof the status of source, moving or stationary, that emits the light signal, Einsteinpostulated the doctrine in his special theory of relativity that for two coordinatesystems in relative motion with a constant velocity of separation v = VZ, the spaceand the time variables in the two systems must obey the Lorentz transform statedby the following relations:

x = x',

y=y',

z = y(z' + vt'),

I= '1(I'+ ~ z') ;the reverse transforms of (E.3) and (E.4) are

z' = y(z - vt)

174

(E.1)

(E.2)

(E.3)

(E.4)

(E.5)

Appendix E

and

175

t' = y (t - ; z) , (E.6)

where v is the velocity of separation in the z or z' direction between the twosystems, 'Y = 1/(1 - (32)1 /2, J3 = vic, and c is the velocity of light in free space.Another principle contained in Einstein's theory is the invariance of Maxwell'sequations in the two coordinate systems, that is,

VE = _ aB , (E.?)at

aD'WH=J+-, (E.8)

atap

(E.9)VJ=--,at

VD=p, (E.IO)

VB=O, (E. I I)

and

'W'E' = _ aB' (E.12)at' ,aD'

'W'n' =J' +- (E.13)at' ,

V'J' = _ ap' (E.14)at' ,~'D'=p', (E.l5)

V'B'=O, (E.l6)

where the unprimed operators and the unprimed functions are definedwith respectto (x, y, z: t) and the primed ones with respect to (x', y', z'; t'). The Lorentztransform assures us the relation that when

then

,2 + ,2 +,2 2t,2 0X Y z -c =.

(E.l?)

(E.18)

(E. I?) or (E.18) corresponds to "equation of motion" of the propagation of thelight signal. In fact, these are two of the equations used to derive (E.3) and(E.4). With this background we can find the transform between the field vectors.Let us consider first the x-component of (E.?) pertaining to Faraday's law in therectangular systems, that is,

(E.19)

176 Appendix E

The derivatives with respect to y, z, and t can be converted to the derivatives withrespect to y', z', and t' with the aid of (E. I ) to (E.6). Thus, (E.19) can be writtenas

or

hence

aEz _ (aEy az' + aEy at') = _ (8Bx at' + aBx az')ay' Bz' az at' az at' at Bz' at (E.20)

(E.22)

(E.21)aEz_ ~ h (E + vB )] =-~ [1 (B + 3!..-E )].ay' az' r z at' x c2 y

The x-component of (E.12) reads

es: ee: es:z y xay' - az' = - at' .

The matching of (E.21) and (E.22) yields

E; = s.. (E.23)

E~ =1 (s, + vBx) , (E.24)

B~ = '1( s, + :2 Ey ) • (E.25)

By working, similarly, on the other equations and combining the resultantequations, we find

E' = 1· (E + v x B) ,

, : ( 1 )B =1· B- c2v x E ,

H' = Y· (H - v x D) ,

, : ( 1 )D=1· D+c 2

v x H ,

J' = 11-1(J - pv) ,

p' = '1(p - :2 v · J) .

where v = vi and the dyadics yand y-l are defined by

Y= 1 (xx + yy) + ZZ ,

:-1 1 (A A AA) AA1 = - xx + yy + zz:1

(E.26)

(E.2?)

(E.28)

(E.29)

(E.30)

(E.31)

Appendix E

so

and

:-1 A A A"" A'"

11 = xx + yy + 'Yzz

177

1·1-1= I = xx + yy + zz .

(E.26) to (E.29) are the transforms of the field vectors defined in the two coordinatesystems. These are the equations based on which the problems involving movingmedia can be formulated. When v2 / c2 « 1,

y == 1,

the following relations hold true

E'· =E+v x B,

B/•

=B,

H'· =H-v x D,

n/•

=D,

J'. = J - pv,

'.p =p.

(E.32)

(E.33)

(E.34)

(E.35)

(E.36)

(E.37)

The symbol * means that these expressions are approximate under the conditionv2/c2 « 1.

These transforms have been derived with a rather tedious procedure involvingaltogether nine scalar differential equations. Many of the details are not shownhere. A more elegant method is to recast the Lorentz transform into a pseudoreal orthogonal transform, a method due to Sommerfeld [45]. According to thatmethod, we let

then

Because

. ( 2 )1/2 i~sm 11 = 1 - cos 11 = 1/2 .(1 - ~2)

(E.38)

(E.39)

vf3 = - < 1,

c

11 must be an imaginary angle, hence the term "pseudo-real." Now, we introducethe four-dimensional coordinate variables (XI, X2, X3, X4) defined by

Xt=X, X2=Y, X3=Z, X4=;ct,

178 Appendix E

and similarly for xi with j = 1,2, 3, 4; then (E.I) to (E.6) can be written as

where

Xl = x~,

X2 = x~,

X3 = x~ cos 11 - x~ sin 11 = a33x i + a43x~ ,

X4 = x~ cos 11 + x~ sin 11 = a34x~ + a44x~ ,

(E.40)

a33 = cos 1'\,a43 = - sin 11,

a34 = sin 11,a44 = cos n.

(E.41)

The "directional cosines" between the two sets of axes can be tabulated as follows:

Xl X2 X3 X4

X' 1 0 0 0I

x' 0 1 0 0 (E.42)2

x' 0 0 a33 a343

x' 0 0 a43 a444

As with the three-dimensional coefficients, they satisfy the orthogonal relations

and

4

L a;jakj =a.j=l

(E.43)

la;jl = 1.

The field vectors in Maxwell's equations can now be formulated as vectorsor tensors in a four-dimensional manifold. We first define a four-potential vector,denoted by P, as

4

P= L~X;;=1

with PI = AI, P2 = A2, P3 = A3, P4 = i~/c, where Ai with i = 1,2,3 arethe components of the vector potential A and ~ is the dynamic scalar potential.The two functions have been introduced previously in Section 4-10. We define thecomponents of the curl of P as

~ p _ apn _ apmVmn - iJx

max" ' m, n = 1,2,3,4. (E.44)

They are functions with two indices and there are 12 of them, but because

(E.45)

Appendix E

and

~mmP = 0,

179

there are actually only six distinct components. These components or functionswill be denoted by Fmn and are designated as the six-vectors. The components ofthe field vectors Band E can now be treated as six vectors. For example,

8A3 8A2 8P3 8P2BI = - - - = - - - = ~23P = F23 , (E.46)

OX2 OX3 8X2 OX3

E 1 = _it _ aA 1 = ic (OP4 _ aPI) = ie~l4P = icFu;OXI at ax] OX4

or

(E.47)

The six functions (B}, -iEi/e) with i = 1, 2, 3 form the components of a 4 x 4antisymmetric tensor, which is shown here:

B3 -B2-i

0 -E1e

-B3 0 BI-i-E2

[Fij] = e (E.48)-i

B2 B1 0 -E3e

i i i-EI -£2 -E3 0e e e

The 3 x 3 minor in the upper left comer, or the adjoint of F44 , is recognized as theantisymmetric tensor of the axial vector B = V A.Another six-vector is defined by

G = (H, -jeD).

In free space,

G= ~F= ~"P.~ flo

The 4 x 4 antisymmetric tensor of G is shown here:

0 H3 -H2 -ieD)

[Gij] =-H3 0 HI - icD2

H2 HI 0 -ieo,teo, ie D2 «o, 0

(E.49)

(E.50)

(E.51)

It is possible to construct two four-dimensional dyadics using these tensors, butthey are not necessary in this presentation.

180 Appendix E

(E.52)

We can nowextend the rules of the transformation of3 x 3 tensors in Chapter 1for two orthogonal rectangular systems to the 4 x 4 antisymmetric tensors underconsideration, that is,

4 4

/;j = L La;majnfmn.m=l n=l

Applying (E.52) to F and G with aij given by (E.42), we find

E' =Y. (E + v x B) , (E.53)

B' = '1 · (B - ; x E) . (E.54)

H' = Y. (H - v x D) , (E.55)

, = ( v )D = 't : D + c2 X H . (E.56)

They are the same as (E.26) to (E.29) obtained before by the classical method. Thetransform of the current density function J and the charge density function p can befound by applying the four-dimensional rule to the components of a four-currentvector defined by

The transform is4

K; = EaijKj ,

j=)

i=I,2,3,4.

i = 1,2,3,4,

(E.57)

(E.58)

which yields

hence

or

hence

Ki = Ki; hence J{ = J),

K~ = K2; hence J~ = J2,

K~ = a33 K3 + a34K4;

J~ ="( (J3 - vp) ,K~ = a43 K3 + a44K4

, iyv ()icp = --J3 +1 icp ;c

(E.59)

(E.60)

(E.61)

p' = 'Y (p - ; J3). (E.62)

(E.53) to (E.62) are identical to (E.26) to (E.31). It is seen that the four-dimensionalanalysis to derive the transform of the field vectors is very elegant as long as we getused to the rather novel concept ofpseudo-real representation of Lorentz transformin a four-dimensional space or manifold.

Appendix F

Comparisonof the Nomenclatu res

and Notationsof the Quantities Used

in This Bookand in the Book

by Stratton [5]

Present Book (1) Quantities Stratton's Book (2)

aob scalar product aobaxb vector product axb

Pi (1) primary vectors a,(2) unitary vectors

ri (1) reciprocal vectors a'(2) reciprocal unitary vectors

Pi 0 P, = aij (1) scalar products of Pi a, 0 aj = gij

(2) scalar products of a,- (2) scalar products of a' a' 0 a j = s"

181

182 Appendix F

Present Book (1) Quantities Stratton's Book (2)

A = PI 0 (P2 x P3) a volume parameter in GCS gl /2 = 81 0 (a2 x 83)

F = L; Jiri vector function in F = L; Ji8;

=Ljgjpj component form in GCS = Lj fj8j

/; = P; 0 F (1) primary component /;=8;oF(2) covariant component

s' = r j0 F (1) reciprocal component fj = a j · F

(2) contravariant component

V; (1) coordinate variables u'(2) coordinate variables along 8;

- (2) coordinate variables along 8; U;

U; unit vectors in OCS i;h; metric coefficients in OCS h;

n = h lh 2h 3 product of metric coefficients -x,y,z;x; rectangular variables x,y,zx,y,z;x; unit vectors in rectangular system i;

r,9,z cylindrical variables r,9,z

r,9,z unit vectors in cylindrical ii, h, i3coordinate system

R, 9, ep spherical variables r,9,.

R,e,~ unit vectors in spherical i., h, i)coordinate system

dRp differential of position vector drV symbolic vector -

Vf gradient of a scalar function VfVF gradient of a vector function VFVF divergence of a vector function V·F~F curl of a vector function VxF

VVf' Laplacian of a scalar function V· Vf; V2f

VVF Laplacian of a vector function V· VF; V2F~~F curl curl F VxVxF

Vsf (1) surface gradient ofa scalar function -

VsF (1) surface divergence ofa vector function -

"sF (1) surface curl of a vector function -[1ij] tensor of rank 2 2T

T;j (2) component of a tensor of rank 2 T;j- (2) divergence of a tensor of rank 2 div2T

F (1) dyadic function -VF (1) divergence of a dyadic -"F (1) curl of a dyadic -

Appendix F

F-1 Typesetting the New Notations for Divergenceand Curl

183

To facilitate the new notations for curl and divergence, it helps to know how totypeset them. Dr. Leland Pierce of the University of Michigan has created TEXsymbols (usable also in MIEX) for the S-vector V, divergence operator V , and curloperator V . The following MIEX document:

\documentstyle[12ptl{article}\ begin{document}\font\tttt=cmsy5\def\taisvec{\nabla\!\!\!\ !\raiseO.31ex\hbox{- -}\,}\def\taidivgj vnabla\!\ !\kem-2.5pt\raiseO.5ex\hbox toopt]$\cdot$}}\def\taicurl{\nabla\!\!\!\kem-l.5pt\raiseO.8ex\hbox t07pt{\tttt\char'002}}

\begin{eqnarray*}\taicurl\bf{E} &=&-\frac{\partial\bf{B}}{vpartial t}\\\taicurl\bf{H} &=&\bf{J} + \frac{\partial \bf{D} }{vpartial t}\\\taidivg\bf{D} &=& \rho\\\taidivg\bf{B} &=&0\end{eqnarray*}\ end{document}

will produce

VE = _ aDat

aDVH=J+atVD=p

VB=O

For Macintosh'[' or PC users of Microsoft Word 6.0© or higher, the followingsteps can be used to typeset the new divergence and curl operators.

1. Pull down the "Insert" menu and select "Field."

2. Select "Equation" to insert a blank equation field in the document.

3. For Macintosh users, hold down the control key and click on the blankequation. For PC users, right click the blank equation field. Select"Change" to show the equation field.

184 Appendix F

4. Edit the equation field as follows:(a) For V vector: {EQ\O(V,\S\UP5(- »}.(b) For V operator: {EQ\O(V,\S\UP5(.»}.(c) For V operator: {EQ\O(V,\S\UP5(x»}.

Note that "V," "" and ",' are characters of Symbol Font and enteredby using "Insert Symbol" command. The size of "V" is 12 point. The sizeof "" and"." is 4 point. Note that"." is the big dot character of SymbolFont, not period, and "," is the cross character of Symbol Font, not "x,"

5. To save typing time, the above can be saved and recalled by using the"Autotext" command. For instance, select the operators V, pull down the"Edit" menu, and select "Autotext" to store the operator, say, by the name"dot." For subsequent use of the operator, type the name "dot" followedby the function key F3, then the V operator will appear automatically.

For more information, see the help menu of Microsoft Word 6.0© on thefollowing topics: Insert Field, Insert Symbol, EQ, and Autotext.

References

1. Brand, Louis, Vector and TensorAnalysis. New York: John Wiley, 1947.

2. Borisenko, A. I., and Tarapov, I. E., Vector and Tensor Analysis. NewYork: Dover Publications (English translation of 1966 Russian originalby Richard A. Silverman), 1979.

3. Tai, C. T., Dyadic Green Functions in Electromagnetic Theory, 2nd ed.,Piscataway, NJ: IEEE Press, 1994.

4. Gibbs, J. Willard, Elements of Vector Analysis, privately printed, NewHaven (first part in 1881, and second part in 1884). Reproduced in TheScientific Papers ofJ. Willard Gibbs, Vol. II. London: Longmans, Green,and Company, 1906; New York: Dover Publications, 1961, p. 26.

5. Stratton, Julius Adams, Electromagnetic Theory. New York: McGrawHill, 1941, p. 39.

6. Tai, C. T., and Fang, N. H., "A systematic treatment of vector analysis,"IEEE Trans. Educ., vol. 34, no. 2, pp. 167-174, 1991.

7. Tai, C. T., Generalized Vector and Dyadic Analysis, 1st ed., Piscataway,NJ: IEEE Press, 1992.

8. Hallen, E., Electromagnetic Theory (translated from Swedish edition byR. Gostrom), New York: John Wiley, 1962, p. 36.

9. Phillips, H. B., Vector Analysis. New York: John Wiley, 1933.

ISS

186 References

10. Weatherbum, C. E., Differential Geometry. Cambridge, England:Cambridge University Press, 1927, Ch. XII.

11. Van Bladel, J., Electromagnetic Fields. New York: McGraw-Hill, 1964,Appendix 2.

12. Von Helmholtz, H., "Das Princip der Kleinsten Wirkung in der Elektrodynamik," Ann. Phys. u. Chem., vol. 47, pp. 1-26, 1892.

13. Lorentz, H. A., Encyklopiidie der Mathematischen Wissenschaften,Band V. Berlin, Germany: Verlag und Druck von G. G. Teubner, 1904,part 2, p. 75.

14. Sommerfeld, A., Mechanics of Derformable Bodies. New York: Academic Press, 1950, p. 132.

15. Candel, S. M., and Poinrot, T. J., "Flame stretch"and the balance equationfor the flame area," Combust. Sci. Technol., vol. 70, pp. 1-15, 1990.

16. Truesdell, C., and Toupin, R., "The classical field theory," in Encyclopedia ofPhysics, vol. III. Berlin, Germany: Springer-Verlag, 1962, no. 1,p.346.

17. Maxwell, J. C., "A dynamical theory of electromagnetic field," in TheScience Papers ofJames Clark Maxwell, vol. 1. Cambridge, England:Cambridge University Press, 1890.

18. Tai, C.T., "On the presentation of Maxwell's theory," Proc. IEEE, vol. 60,pp. 930-945, August 1972.

19. Reynolds, A., "The general equations of motion of any entity," inScientific Papers, vol. III. Cambridge, England: Cambridge UniversityPress, 1903, pp. 9-13.

20. Penfield, Paul, Jr., and Haus, Hermann A., Electrodynamics ofMovingMedia. Cambridge, MA: The MIT Press, 1967, p. 249.

21. Crowe, Michael J., A History of Vector Analysis. New York: DoverPublications, 1985. An unabridged and corrected republication of thework first published by the University of Notre Dame Press in 1967.

22. Burali-Forti, C., and Marcolongo, R., Elements de calcul vectoriel. Paris:Librairie Scientifique, A. Hermann et Fils (French edition of the originalin Italian by S. Lattes), 1910.

23. Moon, P., and Spencer, D. E., Vectors. Princeton, NJ: Van Nostrand,1965.

24. Weatherbum, C. E., Advanced Vector Analysis. London, England: G.Bell and Sons, 1924.

25. Maxwell, J. C., A Treatise on Electricity and Magnetism. Oxford,England: Oxford University Press, 1873.

References 187

26. Heaviside, Oliver, Electromagnetic Theory. New York: vol. I, completedin 1893, vol. II in 1898, vol. III in 1912. Complete and unabridgededition of the volumes reproduced by Dover Publications with a criticaland historical introduction by Ernst Weber, 1950.

27. Wheeler, Lynde Phelps, Josiah Willard Gibbs. New Haven: YaleUniversity Press, 1952.

28. Tait, P. G., An Elementary Treatise on Quaternions. Cambridge,England: Cambridge University Press, 1890.

29. Gibbs, J. Willard, "On the role of quatemions in the algebra of vectors,"Nature, vol. XLIII, pp. 511-513, 1891. Reproduced in The ScientificPapers ofJ. Willard Gibbs, pp. 155-160.

30. Wilson, Edwin Bidwell, VectorAnalysis. New York: Charles Scribner'sSons, 1901.

31. Wilson, Edwin Bidwell, "Lectures delivered upon vector analysis and itsapplications to geometry and physics by Prof. J. Willard Gibbs, 189990." A report of 289-plus pages stored at Sterling Memorial Library,Yale University, New Haven.

32. Tai, C. T., "A survey of the improper uses of V in vector analysis," Technical Report RL 909, Radiation Laboratory, Department of ElectricalEngineering and Computer Science, the University of Michigan, AnnArbor, MI 48109, November 1994.

33. Lagally, Max, Vorlesungen uber Vektor-Rechnung. Leipzig, Germany:Akademische Verlagsgesellschaft, 1928.

34. Mason, M., and Weaver, W., The Electromagnetic Field. Chicago: TheUniversity of Chicago Press, 1929.

35. Schelkunoff, S. A., Applied Mathematics for Engineers and Scientists.New York: Van Nostrand, 1948.

36. Feynman, R. P., Leighton, R. B., and Sands, M., The Feynman Lectureson Physics, vol. II. Reading, MA: Addison-Wesley, 1964.

37. Gans, Richard, Einfuhrung in die Vektor-Analysis mit Anwendungen aufdie Mathematische Physik. Leipzig, Germany: B. G. Teubner, 1905.(English translation of the 6th ed. by W. M. Deans, Blakie, London,1932; 7th ed., 1950).

38. Shilov, G. E., Lectures on Vector Analysis (in Russian). Moscow,U.S.S.R.: The Government Publishing House of Technical TheoreticalLiterature, 1954. An English translation of pp. 18-21 of Ch. 2,and pp. 85-92 of Ch. 8 containing the essential formulas of Shilov'sformulation was communicated to this author by Mr. N. H. Fang of theNanjing Institute of Electronics.

188 References

39. Fang, N. H., Introduction to Electromagnetic Theory (in Chinese).Beijing, China; Science Press, 1986.

40. Morse, P. M., and Feshbach, H., Methods ofTheoretical Physics, vol. I.New York: McGraw-Hill, 1953.

41. Panofsky, W. K. H., and Phillips, M., Classical Electricity and Magnetism, 2nd ed., Reading, MA: Addison-Wesley, 1962.

42. Milne, E. A., Vectorial Mechanics. London, England: Methuen, 1948.

43. Wilson, Edwin Bidwell, "The unification of vectorial notations," Bull.Am. Math. Soc., 16, pp. 415-436, 1909-1910. Review of C. BuraliForti and R. Marcolongo, Elementi di calcolo vettoriale con numeroseapplicazioni and omografie vettoriale con applicazioni.

44. Jackson, John David, Classical Electrodynamics. New York: JohnWiley, 1962.

45. Sommerfeld, A., Electrodynamics. New York: Academic Press, 1952.

AAssociative rule, 2

BBilinear transformation, 32Bivector,135Bladel, J. Van, 99, 105Borisenko, A. J., 17, 148,155Brand, L., 17, 105Burali-Forti, C., 130

CCandel, S. M., 119Clifford, W. K., 132Closed surface theorem, 27, 92

integral form of, 93Cofactors, 11Conformal transformation, 33Convergence, 132Coordinate system

bipolar cylinders, 31, 164Cartesian or rectangular, 3cylindrical, 30, 161

Index

Dupin, 35, 100elliptical cylinder, 30, 162general curvilinear system

(GCS),23oblate spherical, 31, 163orthogonal curvilinear system

(OCS),28parabolic cylinder, 30, 162prolate spheroidal, 30, 163spherical, 30, 161

Cross-V-cross theorem, 110Cross-gradient theorem, 110Cross product or vector product, 5,

131Crowe, M. J., 127Curl

alternative definition of, 77of a dyadic function, 121in Cartesian system, 65, 132in general curvilinear system, 71Gibbs's notation for, 65, 135

189

190

Curl (cont.)linguistic notation of, 128in orthogonal curvilinear system,

64Curl theorem, 92

surface, 109Curvature

Gaussian, 39radius of, 37surface, 41

DDel operator, 62Derivatives of unit vectors, 33Differential-algebraic operators,

102, 103Differential area, 43Differential length, 43Differential volume, 44Directional cosines, 3, 10Directional radiance, 75Distributive law

for scalar products, 5for vector products, 6

Divergenceof a dyadic function, 123in Cartesian system, 65in general curvilinear system, 71Gibbs's notation for, 63, 134linguistic notation of, 128in orthogonal curvilinear system,

62Divergence theorem, 92

surface, 109Dot product, 4Dyadic algebra, 16Dyadic function, 16

antisymrnetric, 17Cartesian, 16classification of, 17scalar product of, 19symmetric, 17transpose of, 19vector product of, 21

Index

Dyadic Green function, 98Dyadic identities, 124, 166Dyadic integral theorems, 124

EEinstein, A., 174

notation, 10

FFang, N. H., 58,151Feshback, H., 154Feynman, R. P., 148Flux, 75Four-vector, 15, 178

GGans, R., 149Gauge condition, 98Gauss, theorem or divergence

theorem, 92generalized, 91generalized surface, 106

Gibbs,J. VV., 15,24,63, 129, 135Gradient

alternative definition of, 75in Cartesian system, 65in general curvilinear system, 70method of, 78in orthogonal curvilinear system,

62of a vector, 87, 122

Gradient theorem, 92surface, 109

Green's theoremfirst dyadic-dyadic, 126first scalar, 94first vector, 95first vector-dyadic, 125scale-vector, 168second dyadic-dyadic, 126second scalar, 94second vector, 95second vector-dyadic, 125surface, 114

Index

HHallen's formula, 92Hamilton, W. R., 129Haus, H., 124Heaviside, 0., 129, 141, 149Helmholtz, H. L. F. von, 116Helmholtz theorem, 97

transport theorem, 116Helmholtz wave equation

scalar, 98vector, 98

IIdemfactor, 19, 25, 122Integral theorems, 167

relationship between, 170

JJackson, J. D., 130Jacobian


KKronecker afunction, 12

LLagally, M., 147Laplace, P. S., 87Laplacian

of a scalar, 86surface, 114of a vector, 87, 153

Leighton, R. B., 148Lemma 4.1,60Lemma 4.2, 82Lemma 5.1, 112Line integrals, classification of, 44Linguistic notation, 105, 128Lorentz, H. A., 119

transform, 174

MMarcolongo, R., 130Mason, M., 147Material derivative, 119Maxwell's equations, 96, 175

Maxwell's theorem, 119Metric coefficients, 28Milne, E. A., 156Moon,~, 127,130,152Morse, P. M., 154

NNomenclature and notations,

181

oOperand, 132Operator(s), 132

binary, 132cascade, 132cross, 133curl, 58, 64, 140del, 62, 127differential-algebraic surface,

102, 103divergence, 58, 63, 140dot, 133gradient, 58, 62, 139Hamilton, 62invariance of, 65nabla, 62unary,132

pPanofsky, W. K. H., 155Penfield, P., 124Phillips, H. B., 97Phillips, M., 155Poinsot, T. J., 119Poisson's equation


Position vector, 23Potential function

dynamic scalar, 97dynamic vector, 97electrostatic, 96magnetostatic vector, 97

QQuarternion, 129, 131

191

192

RRadiance, 75Reynolds transport theorem, 120Rotation of Cartesian coordinate

system, 8

SS-vector,59

typesetting of, 183, 184Sands, M., 148Scalar product, 4Schelkunoff, S. A., 147Shear, 75Shilov, G. E., 151Six-vector, 15, 179Sommerfeld, A., 177Special theory of relativity, 174Spencer, D. E., 127, 130, 152Stokes, G. G., 110Stokes theorem, 110Stratton, J. A., 73, 130, 153Surface curl, 103

theorem, 109of Weatherburn, 105

Surface gradient, 101theorem, 109of Weatherburn, 105

Surface integrals, classification of,48

Surface symbolic expression, 100with two functions, 110with two surface S-vectors, 113

Surface symbolic vector, 99partial, 111

TTai, C. T., 21, 58, 63,119,127,146Tarapov, I. E., 17, 148, 155Tensor, 17, 179Toupin, R., 119Transform of field vectors, 176

Index

Triple productof dyadics, 21of vectors, 6

Truesdell, C., 119Twisted differential operator, 146Twisted vector function, 146Typesetting of S-vector, divergence

and curl operator, 183, 184

UUnit vectors, 3

derivatives of, 33transformation of, 79, 161

VVector or vector function, 1

axial, 15components of, 3controvariant components of, 25covariant components of, 25irrotational, 95orthogonal transformation of, 8polar, 14position, 23primary, 23reciprocal, 24reciprocal unitary, 24screw, 15solenoidal, 95unitary, 24

Vector identities, 82, 165Vector Laplacian, 153Vector product,S

formal (FVP), 143Volume integrals, classification of,

56

WWeatherbum, C. E., 99,104Weaver, W., 147Weber, E., 151Wilson, E. B., 129, 141, 160

general vector and dyadic analysis 0780334132

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