3.1 vector calculus.pdf
TRANSCRIPT
![Page 1: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/1.jpg)
VECTORCALCULUS
Prepared byEngr. Mark Angelo C. Purio
![Page 2: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/2.jpg)
Differential Length, Area, and Volume
![Page 3: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/3.jpg)
Differential Length, Area, and Volume
Differential elements in length, area, andvolume are useful in vector calculus.
They are defined in the Cartesian,Cylindrical, and Spherical coordinate
systems
![Page 4: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/4.jpg)
Differential Length, Area, and Volume
A. Cartesian Coordinates1. Differential Displacement
2. Differential Normal Area
3. Differential Volume
![Page 5: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/5.jpg)
Differential Length, Area, and Volume
Differential Elements in the Right-handedCartesian Coordinate System
![Page 6: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/6.jpg)
Differential Length, Area, and Volume
Differential Normal Areas inCartesian Coordinates
![Page 7: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/7.jpg)
Differential Length, Area, and Volume
A. Cylindrical Coordinates1. Differential Displacement
2. Differential Normal Area
3. Differential Volume
![Page 8: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/8.jpg)
Differential Length, Area, and Volume
Differential Elements inCylindrical Coordinates
![Page 9: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/9.jpg)
Differential Length, Area, and Volume
Differential Normal Areas inCylindrical Coordinates
![Page 10: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/10.jpg)
Differential Length, Area, and Volume
A. Spherical Coordinates1. Differential Displacement
2. Differential Normal Area
3. Differential Volume
![Page 11: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/11.jpg)
Differential Length, Area, and Volume
Differential Elements inSpherical Coordinates
![Page 12: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/12.jpg)
Differential Length, Area, and Volume
Differential Normal Areas inSpherical Coordinates
![Page 13: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/13.jpg)
EXAM
PLE 3
.1 Consider the objectshown. Calculate:a) The distance BCb) The distance CDc) The surface area
ABCDd) The surface area ABOe) The surface area
AOFDf) The volume ABDCFO
a) 10b) 2.5 πc) 25 π
d) 6.25 πe) 50f) 62.5 π
![Page 14: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/14.jpg)
EXER
CISE
3.1 Disregard the differential
lengths and imagine that theobject is part of a sphericalshell. It may be describe as3 ≤ ≤ 5, 60° ≤ ≤ 90°,45° ≤ ∅ ≤ 60°where surface= 3 is the same as ,
surface = 60° is , andsurface ∅ = 45° is .
Calculatea) The distance DHb) The distance FGc) The surface area AEHDd) The surface area ABDCe) The volume of the
object
a) 0.7854b) 2.618c) 1.179
d) 4.189e) 4.276
![Page 15: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/15.jpg)
Line, Surface, and Volume Integrals
![Page 16: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/16.jpg)
Line, Surface, and Volume Integrals
By a line we mean the path along a curvein space.
Line, curve, and contour can be usedinterchangeably.
The line integral ∫ • is the of thetangential component of A along curve
L.
![Page 17: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/17.jpg)
Line, Surface, and Volume Integrals
Given a vector field A and acurve L, we define the
integral
as the line integral ofA around L.
Path of integration of avector field
![Page 18: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/18.jpg)
Line, Surface, and Volume Integrals
If the path of integration is a closed curve suchas abcba
Becomes a closed contour integral
which is called circulation of A around L
![Page 19: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/19.jpg)
Line, Surface, and Volume IntegralsGiven a vector fieldA, continuous in aregion containing
the smooth surfaceS, we define the
surface integral orthe flux of Athrough S as The flux of a vector field A
through surface S
![Page 20: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/20.jpg)
Line, Surface, and Volume Integralswhere at any pointon S , is the unit
normal to S.For a closed
surface (defining avolume):
The flux of a vector field Athrough surface S
Which is referred to a thenet outward flux of A from
S.
![Page 21: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/21.jpg)
Line, Surface, and Volume IntegralsWe define
as the volume integral of the scalar overthe volume .
The physical meaning of the line, surface, orvolume integral depends on the nature of the
physicsl quantity represented by A or .
![Page 22: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/22.jpg)
EXAM
PLE 3
.2 Given = − − ,calculate the circulation of F around a
closed path shown.
• = −
![Page 23: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/23.jpg)
EXER
CISE
3.2 Calculate the circulation of= cos ∅ + sin ∅
around the edge L of thewedge defined by 0 ≤ ≤ 2,0 ≤ ∅ ≤ 60°, = 0 and
shown.
• =
![Page 24: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/24.jpg)
DEL OPERATOR
![Page 25: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/25.jpg)
Del Operator
The del operator, written as ,is a vector differential operator.
In Cartesian coordinates,
a.k.a gradient operator
![Page 26: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/26.jpg)
Del Operator
It is not a vector in itself,Useful in defining the following:
1. The gradient of a scalar V, ( )2. The divergence of a vector A,( • )3. The curl of a vector A, ( × )4. The Laplacian of a scalar V, ( )
![Page 27: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/27.jpg)
Del Operator
In Cylindrical coordinates,
In Spherical coordinates,
![Page 28: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/28.jpg)
Gradient of a Scalar
The gradient of ascalar field V is a
vector that representsboth the magnitudeand the direction ofthe maximum spacerate of increase of V.
![Page 29: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/29.jpg)
Gradient of a ScalarIn Cartesiancoordinates,
In Cylindrical coordinates,
![Page 30: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/30.jpg)
Gradient of a ScalarIn Spherical coordinates,
![Page 31: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/31.jpg)
Gradient of a ScalarThe followingcomputationformulas on
gradient,which are
easilyproved,
should benoted:
where U and V arescalars and n is an
integer
![Page 32: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/32.jpg)
EXAM
PLE 3
.3 Find the gradient of the followingscalar fields:a) =b) =c) =
![Page 33: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/33.jpg)
EXER
CISE
3.3 Determine the gradient of the
following scalar fields:a) = +b) = + ∅ +c) = + ∅
![Page 34: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/34.jpg)
EXAM
PLE 3
.4 Given = + ,compute and the direction
derivative / in the direction+4 + 12 at (2, -1, 0).
−
![Page 35: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/35.jpg)
EXER
CISE
3.4 Given = + + . Find
gradient at point (1, 2, 3) and thedirectional derivative of at the same
point in the direction toward point(3, 4, 4).
+ + ,
![Page 36: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/36.jpg)
EXAM
PLE 3
.5 Find the angle at which line= =intersects the ellipsoid+ + 2 = 10.
= .
![Page 37: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/37.jpg)
EXER
CISE
3.5 Calculate the angle between the
normal to the surfaces+ = 3 and log − = −4at the point of intersection (-1, 2, 1)
. °
![Page 38: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/38.jpg)
Divergence of a Vector and Divergence Theorem
The divergence of A at a given point P isthe outward flux per unit volume as the
volume about P.
where ∆ is the volume enclosed by theclosed surface S in which P is located
![Page 39: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/39.jpg)
Divergence of a Vector and Divergence Theorem
a) The divergence of a vector field at point P is positivebecause the vector diverges (spreads out) at P.
b) A vector field has negative divergence (convergence)at P
c) A vector field has zero divergence at P.
![Page 40: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/40.jpg)
Divergence of a Vector and Divergence Theorem
In Cartesian coordinates,
In Cylindrical coordinates,
![Page 41: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/41.jpg)
Divergence of a Vector and Divergence Theorem
In Spherical coordinates,
Note the following properties of thedivergence of a vector field:
![Page 42: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/42.jpg)
Divergence of a Vector and Divergence Theorem
The divergence theorem states that thetotal outward flux of a vector field A
through a closed surface S is the same asthe volume integral of the divergence of
A.
Otherwise known asGauss-Ostrogradsky theorem
![Page 43: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/43.jpg)
EXAM
PLE 3
.6 Find the divergence of these vectorfields:a) = +b) = + ∅ +c) = + +∅
![Page 44: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/44.jpg)
EXER
CISE
3.6 Determine the divergence of the
following vector fields and evaluatethem at the specified points,:
![Page 45: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/45.jpg)
EXAM
PLE 3
.7 If = 10 ( + ),determine the flux of G outof the entire surface of the
cylinder= 1, 0 ≤ ≤ 1.Confirm the result usingthe divergence theorem.
Ψ =
![Page 46: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/46.jpg)
EXER
CISE
3.7 Determine the flux of= cos + sin
over the closed surface of the cylinder0 ≤ ≤ 1, = 4.Verify the divergence theorem for this case.
![Page 47: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/47.jpg)
Curl of a Vector and Stoke’s TheoremThe curl of A is an axial (or rotational)
vector whose magnitude is themaximum circulation of A per unit area
as the area tends to zero and whosedirection is the normal direction of thearea when the area is oriented so as to
make a circulation maximum.
![Page 48: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/48.jpg)
Curl of a Vector and Stoke’s Theorem
In Cartesiancoordinates
![Page 49: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/49.jpg)
Curl of a Vector and Stoke’s Theorem
In Cylindricalcoordinates
![Page 50: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/50.jpg)
Curl of a Vector and Stoke’s TheoremIn Spherical coordinates
![Page 51: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/51.jpg)
Curl of a Vector and Stoke’s Theorem
Note the following properties of the curl:
![Page 52: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/52.jpg)
Curl of a Vector and Stoke’s Theorem
The curl provides the maximum value ofthe circulation of the field per unit area
(or circulation density) and indicates thedirection to which this maximum value
occurs.
Measures the circulation of how muchthe field curls around P
![Page 53: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/53.jpg)
Curl of a Vector and Stoke’s Theorem
a) Curl at P points out of the pageb) Curl at P is zero.
![Page 54: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/54.jpg)
Curl of a Vector and Stoke’s TheoremStokes’s theorem statesthat the circulation of avector field A around a(closed) path L is equalto the surface integral
of the curl of A over theopen surface Sbounded by L
provided that A and× are continuouson S.
![Page 55: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/55.jpg)
EXAM
PLE 3
.8 Determine the curl of these vectorfields:a) = +b) = + ∅ +c) = + +∅
![Page 56: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/56.jpg)
EXER
CISE
3.8 Determine the curl of the following
vector fields and evaluate them at thespecified points,:
![Page 57: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/57.jpg)
EXAM
PLE 3
.9 If = + ∅, evaluate∮ • around the path shown.Confirm using Stokes’s theorem.
4.941
![Page 58: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/58.jpg)
EXER
CISE
3.9 Confirm the circulation of= cos ∅ + sin ∅
around the edge L of the wedgedefined by 0 ≤ ≤ 2, 0 ≤ ∅ ≤ 60°, =0 and shown using Stoke’s Theorem
• =
![Page 59: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/59.jpg)
EXAM
PLE 3
.10 For the vector field A,show explicitly that
× = 0;that is, the divergence of the curl of
any vector field is zero.
![Page 60: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/60.jpg)
EXER
CISE
3.10 For a scalar field V, show that× = 0;
that is, the curl of the gradient ofany scalar field vanishes.
![Page 61: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/61.jpg)
Laplacian of a ScalarThe Laplacian of a scalar field V, writtenas , is the divergence of the gradientof V.In Cartesian:
![Page 62: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/62.jpg)
Laplacian of a Scalar
In Cylindrical coordinates,
In Spherical coordinates,
![Page 63: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/63.jpg)
EXAM
PLE 3
.11Find the Laplacian of the followingscalar fields:a) =b) =c) =
![Page 64: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/64.jpg)
EXER
CISE
3.11 Determine the Laplacian of the
following scalar fields:a) = +b) = + ∅ +c) = + ∅
![Page 65: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/65.jpg)
Classification of Vector FieldsA vector field is uniquely characterized by its
divergence and curl.
Neither the divergence nor curl of a vectorfield is sufficient to completely describe a
field.
All vector fields can be classified in terms oftheir vanishing or non vanishing divergence or
curl
![Page 66: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/66.jpg)
Classification of Vector Fields
![Page 67: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/67.jpg)
Classification of Vector FieldsA vector field A is said to be solenoidal
(or divergenceless) if • =Examples:• incompressible fluids,• magnetic fields• conduction current density
under steady state conditions.
![Page 68: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/68.jpg)
Classification of Vector FieldsA vector field A is said to be irrotational
(or potential) if × =Also known as conservative field.
Examples:• electrostatic field• gravitational field
![Page 69: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/69.jpg)
SUMMARY
![Page 70: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/70.jpg)
![Page 71: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/71.jpg)
![Page 72: 3.1 VECTOR CALCULUS.pdf](https://reader030.vdocuments.us/reader030/viewer/2022033022/563db9eb550346aa9aa11eb9/html5/thumbnails/72.jpg)
Reference:Elements of Electromagnetics
by Matthew N. O. Sadiku