review vector analysis

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Review ofReview ofVector AnalysisVector Analysis


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Review of Vector Analysis

Vector analysis is a mathematical tool with whichelectromagnetic (EM) concepts are most conveniently

expressed and best comprehended.

A quantity is called a scalar if it has only magnitude (e.g.,

mass, temperature, electric potential, population).

A quantity is called a vector if it has both magnitude and

direction (e.g., velocity, force, electric field intensity).

The magnitude of a vector is a scalar written as A or

AA

A

Review of Vector AnalysisReview of Vector Analysis


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A unit vector along is defined as a vector whose

magnitude is unity (that is,1) and its direction is along

A

A

A

AeA !!

)(A

1!

Thus

Ae

which completely specifies in terms ofA and itsdirection Ae

A

AeAA !



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A vector in Cartesian (or rectangular) coordinates may

be represented as

or

where AX, Ay, and AZ are called the components of in the

x, y, and z directions, respectively; , , and are unitvectors in the x, y and z directions, respectively.

zzyyxx eAeAeA )A,A,A( zyx

A

A


xe

ze

ye


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Suppose a certain

vector is given by

The magnitude orabsolute value ofthe vector is

(from the Pythagorean theorem)

zyx e4e3e2V !V

385.5432V 222 !!

V



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The Radius Vector

A point P in Cartesian coordinates may be represented by

specifying (x, y, z). The radius vector (or position vector) of

point P is defined as the directed distance from the origin O

to P; that is,

The unit vector in the direction ofr is

zyx ezeyexr !

r

r

zyx

ezeyexe zyxr !

!

222



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Vector Algebra

Two vectors and can be added together to give

another vector ; that is ,

Vectors are added by adding their individual components.

Thus, if and

A B

C

BAC !

zzyyxx eAeAeA zzyyxx eBeBeBB !

zzzyyyxxx e)BA(e)BA(e)BA(C !



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Parallelogram Head torule tail rule

Vector subtraction is similarly carried out as

zzzyyyxxx e)BA(e)BA(e)BA(D

)B(ABAD

!

!!



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The three basic laws of algebra obeyed by any given vector

A, B, and C, are summarized as follows:

Law Addition Multiplication

Commutative

Associative

istributive

where k and l are scalars

ABBA !

C)BA()CB(A !

AA !

A)l()Al(!

BkAk)BA(k !



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When two vectors and are multiplied, the result is

either a scalar or a vector depending on how they are

multiplied. There are two types of vector multiplication:

1. Scalar (or dot) product:

2.Vector (or cross) product:

The dot product of the two vectors and is defined

geometrically as the product of the magnitude of and the

projection of onto (or vice versa):

where is the smaller angle between and

A

ABcosABBA

U!

BA

B

ABU

A

BA v

A B

B

B

A B



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If and then

which is obtained by multiplying and component by

component

),A,A,A(A ZYX! )B,B,B(B ZYX!

ZZYYXX BABABABA !

A B

ABBA !

CABACBA ! )(

A A ! A2

!A2

eX ex ! ey ey ! eZ ez !1

e ey ! ey ez ! e ex ! 0



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The cross product of two vectors and is defined as

where is a unit vector normal to the plane containing

and . The direction of is determined using the right-

hand rule or the right-handed screw rule.

A

A

nABesinABBA U!

B

B

e

ne

BA

irection ofand using(a) right-hand rule,(b) right-handed

screw rule

ne



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If and then

zyx

zyx

zyx

BBBAAA

eee

BA !

),A,A,A(AZYX

! )B,B,B(BZYX

!

zxyyxyzxxzxyzzy e)BABA(e)BABA(e)BABA( !



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Note that the cross product has the following basic

properties:

(i) It is not commutative:

It is anticommutative:

(ii) It is not associative:

(iii) It is distributive:

(iv)

ABB

A v{v

ABBA v!v

)BA()B(A vv{vv

CABACBA vv!v )(

0AA ! )0(sin !U



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Also note that

which are obtained in cyclic permutation and illustrated

below.

yxz

xzy

zyx

eeeeee

eee

!!

!

Cross product using cyclic permutation: (a) moving clockwise leads to positive results;

(b) moving counterclockwise leads to negative results



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Line Integrals

A line integral of a vector field can be calculated whenever a

path has been specified through the field.

The line integral of the field along the path P is defined asV

U!2

1

P

PPdlcosVdlV



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For segment P3,

dl!dxex (the differentiallengthdl pointsto the left)

oo

xx

x

xxo

P

xV-)edx()eV(dlVo

!! !

!03

0

4

! dlVP

field)ive(conservat00xV0xVI ooooP P PP 2 3 41

!!!



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Example. Let the vector field be given by .

Find the line integral of over the semicircular path shown

below

xoeVV !

V

V

Consider the contribution ofthe path segment located atthe angle 5

dl ! dl cos Jex

dl sin Jey

Since J ! U - 90 r

cos J ! cos( U - 90 r) ! sin U

sin J ! sin( U - 90 r) ! cos U

dl ! dl sin Uex

dl cos Uey

! adUdl

{ (sin Uex cos Uey )



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]

o

oo

yxxxo

yxxo

aV

aVdaV

deeeeaV

adeeeVI

2

)0cos180cos(sin

])(cos)([sin

)cos(sin)(

11

180

0

0

180

0 1

180

0

!

r!!

!

!

r

r

r!

!

UU

UUU

UUUU

U



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Surface Integrals

Surface integration amounts to adding up normal

components of a vector field over a given surface S.

We break the surface S into small surface elements and

assign to each element a vector

is equal to the area of the surface element

is the unit vector normal (perpendicular) to the surface

element

nedsds !

ne

ds

The flux ofa vectorfieldAthrough

surface S



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(IfS is a closed surface, is by convention directed

outward)

Then we take the dot product of the vector field at theposition of the surface element with vector . The result is

a differential scalar. The sum of these scalars over all the

surface elements is the surface integral.

is the component of in the direction of (normal

to the surface). Therefore, the surface integral can be

viewed as the flow (or flux) of the vector field through the

surface S

(the net outward flux in the case of a closed surface).

ds

ds

ds

V

UcosV

U! cosdsVdsV

V



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Example. Let be the radius vector

The surface S is defined by

The normal to the surface is directed in the +z direction

Find

V

dyddxd

cz

!

S

dsV

zyx

ezeyexV !



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V is not perpendicular to S, except at one point on the Z axis

Surface S



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5! SS cosdsVdsV

c4d(-d)]-2dc[d

dx)]d(d[cdydxcyx

ccyxdsV

cyx

ccosdxdydscyxV

2

dx

dx

dscos

222

dx

dx

dy

dy

V

222

S

222

222

!!

!

!

!U!!

!

!

U

!

!

!

!



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Introduction to ifferential Operators

An operator acts on a vector field at a point to produce

some function of the vector field. It is like a function of a

function.

If O is an operator acting on a function f(x) of the single

variable X , the result is written O[f(x)]; and means thatfirst f acts on X and then O acts on f.

Example. f(x) = x2 and the operator O is (d/dx+2)

O[f(x)]=d/dx(x2 ) + 2(x2 ) = 2x +2(x2 ) = 2x(1+x)



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An operator acting on a vector field can produce

either a scalar or a vector.

Example. (the length operator),Evaluate at the point x=1, y=2, z=-2

Thus, O is a scalar operator acting on a vector field.

Example. , ,

x=1, y=2, z=-2

Thus, O is a vector operator acting on a vector field.

)]z,y,x(V[O

O(A) ! A A yx ezey3V !)V(O

scalar32.640zy9VV)V(O 22 n!!!!

A2AAA)A(O ! yx ezey3V !

vectore65.16e49.95e4e1240)e2e(6

ez2ey6zy9)ezey3()V(O

yx

yxyx

yx22

yx

n!!

!



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Vector fields are often specified in terms of their rectangular

components:

where , , and are three scalar features functions of

position. Operators can then be specified in terms of ,

, and .

The divergence operator is defined as

zzyyxx e)z,y,x(Ve)x,y,x(Ve)z,y,x(V)z,y,x(V !

xV yV zV

zyx Vz

Vy

Vx

Vx

x

x

x

x

x!

xV

yV zV



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Example . Evaluate at the

point x=1, y=-1, z=2.zyx

2 e)x2(eyexV ! V

0Vz

1Vy

x2Vx

x2VyVxV

zyx

zy

2

x

!x

x!

x

x!

x

x

!!!

31x2V !!

Clearly the divergence operator is a scalar operator.



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1. - gradient, acts on a scalar to produce a vector

2. - divergence, acts on a vector to produce a scalar

3. - curl, acts on a vector to produce a vector

4. -Laplacian, acts on a scalar to produce a scalar

Each of these will be defined in detail in the subsequent

sections.

V

V

Vv

V2



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Cartesian coordinates (x,y,z)

The ranges of the coordinate variables are

A vector in Cartesian coordinates can be written as

The intersection of threeorthogonal infinite places

(x=const, y= const, and z =const)

defines point P.

gg

gg

gg

z

y

x

zzyyxxzyx eAeAeAor)A,A,A(

A

Constant x, y and z surfaces



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zyx edzedyedxdl !

Differential elements in the right handed Cartesian coordinate system

dxdydzd !R



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z

y

x

adxdy

adxdz

adydzdS !



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Cylindrical Coordinates .

- the radial distance from the z axis- the azimuthal angle, measured from the x-axis in the xy plane

- the same as in the Cartesian system.

A vector in cylindrical coordinates can be written as

Cylindrical coordinates amount to a combination of

rectangular coordinates and polar coordinates.

)z,,( JV

gg

TJe gVe

z

200

2/12

z

22

zzz

)AAA(A

eAeAeAor)AA,A(

!

JV

JJVVJV



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Positions in the x-y plane are determined by the values of

Relationship between (x,y,z) and )z,,( JV

JV and

zzx

ytanyx 122 !!J!V


R i f V A l iR i f V A l i


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JV

VJ

JV

!v

!v

!v

eee

eee

eee

z

z

z

0eeeeee

1eeeeee

z

zz

!!!

!!!

VJJJV

JJVV

Point Pand unit vectorsin the cylindricalcoordinate system


R i f V t A l iR i f V t A l i


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zand,JV

semi-infiniteplane with itsedge alongthe z - axis

Constant surfaces


R i f V t A l iR i f V t A l i


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Differential elements in cylindrical coordinates

Metric coefficient

zp adzadaddl JVV! J

dzdddv JVV!




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Planar surface( = const)

Cylindricalsurface

( =const)

dS! VdJdzaV dVdzaJ

VdJdVaz

Planar surface( z =const)J




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Spherical coordinates .

- the distance from the origin to the point P

- the angle between the z-axis and the radius

vector of P

- the same as the azimuthal angle incylindrical coordinates

),,r( JU


0 e r0 e5 T

Colatitude(polarangle)

1 24 34

0 e * 2T



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2/1222

r

rrr

)AAA(A

eAeAeAor)AA,A(

JU

JJUUJU

!

UJ

JU

JU

!v

!v

!v

eee

eee

eee

r

r

r

0eeeeee

1eeeeee

rr

rr

!!!

!!!

JJUU

JJUU

A vector A in spherical coordinates may bewritten as

Point Pand unit vectors in sphericalcoordinates




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U!

JU!

JU!

cosrz

sinsinry

cossinrx

22

11-22

1222

yx

xcos

x

yt n

z

yxt nzyxr

!!J

!U!

rzcos

zt n 11 !V!U

Rel tionships between sp ce v ri bles )z,,(

nd),,,r(),z,y,x( JVJU




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JU nd,,rConst nt surf ces




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Differential elements in the spherical coordinate system

JU JUU! adsinrardadrdl r

JUU! ddrdsinrdv 2




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J

U

J

JU

JUU!

adrdr

adrdsinr

addsinrdS r2




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e e o ecto a ys se e o ecto a ys s



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review vector analysis

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