Understanding Vector Calculus: Practical Development and Solved Problems (Dover Books on Mathematics)

113 Pages·2020·4.8 MB·English

by  Jerrold Franklin

MATHEMATICS
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his concise text was created as a workbook for learning to
use vector calculus in practical calculations and derivations.
Its only prerequisite is a familiarity with one-dimensional
differential and integral calculus. Though it often makes
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mathematics, physics, and other areas of science. D
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The two-part treatment opens with a brief text that develops T
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vector calculus from the very beginning and then addresses some N
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more detailed applications. Topics include vector differential N
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operators, vector identities, integral theorems, Dirac delta function, V
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Green’s functions, general coordinate systems, and dyadics. The T
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second part consists of answered problems, all closely related to
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the development of vector calculus in the text. Those who study L
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memorizing long equations or consulting references, they will be S
able to work out calculations as they go.
Dover original publication.
$12.95 USA PRINTED IN THE USA
ISBN-13: 978-0-486-83590-7
ISBN-10: 0-486-83590-1
51295
www.doverpublications.com 9 780486 835907
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83590-1_VectorCalc_FM.indd 2 9/4/19 9:48 AM
understanding
VECtor
calculus
83590-1_VectorCalc_FM.indd 3 10/14/19 11:09 AM
83590-1_VectorCalc_FM.indd 4 9/4/19 9:48 AM
understanding
VECtor
calculus
Practical Development and Solved Problems
Jerrold Franklin
Temple University
Dover publications, inc.
Mineola, New york
83590-1_VectorCalc_FM.indd 5 10/14/19 11:09 AM
Contents
Preface iv
1 Vector Differential Operators 1
1.1 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Divergence Theorem . . . . . . . . . . . . . . . . . . . . 6
1.3 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Summary of Operations by . . . . . . . . . . . . . . . . . . . 12
∇
1.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Vector Identities 15
2.1 Algebraic Identities . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Properties of . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
∇
Copyright 2.3 Use of bac cab . . . . . . . . . . . . . . . . . . . . . . . . . . 16
−
Copyright © 2020 by Jerrold Franklin 2.4 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
All rights reserved.
3 Integral Theorems 21

3.1 Green’s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Bibliographical Note
3.2 Gradient Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 22
Understanding Vector Calculus: Practical Development and Solved Problems
is a new work, first published by Dover Publications, Inc., in 2020. 3.3 Curl Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Integral Definition of Del . . . . . . . . . . . . . . . . . . . . . 23
3.5 Stokes’ Theorem for the Gradient . . . . . . . . . . . . . . . . . 23
International Standard Book Number
ISBN-13: 978-0-486-83590-7 3.6 Gauss’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
ISBN-10: 0-486-83590-1 3.7 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Dirac Delta Function 29
Manufactured in the United States by LSC Communications
83590101 4.1 Definition of Dirac Delta Function . . . . . . . . . . . . . . . . . 29
www.doverpublications.com 4.2 Applications of the Dirac Delta Function . . . . . . . . . . . . . 31
2 4 6 8 10 9 7 5 3 1 4.3 Singularities of Dipole Fields . . . . . . . . . . . . . . . . . . . . 31
2020 4.4 One-dimensional Delta Function . . . . . . . . . . . . . . . . . . 35
i
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Contents
Preface iixv
1 Vector Differential Operators 1
1.1 Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Divergence Theorem . . . . . . . . . . . . . . . . . . . . 6
1.3 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Summary of Operations by . . . . . . . . . . . . . . . . . . . 12
∇
1.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Vector Identities 15
2.1 Algebraic Identities . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Properties of . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
∇
2.3 Use of bac cab . . . . . . . . . . . . . . . . . . . . . . . . . . 16
−
2.4 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Integral Theorems 21
3.1 Green’s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Gradient Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Curl Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Integral Definition of Del . . . . . . . . . . . . . . . . . . . . . 23
3.5 Stokes’ Theorem for the Gradient . . . . . . . . . . . . . . . . . 23
3.6 Gauss’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.7 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Dirac Delta Function 29
4.1 Definition of Dirac Delta Function . . . . . . . . . . . . . . . . . 29
4.2 Applications of the Dirac Delta Function . . . . . . . . . . . . . 31
4.3 Singularities of Dipole Fields . . . . . . . . . . . . . . . . . . . . 31
4.4 One-dimensional Delta Function . . . . . . . . . . . . . . . . . . 35
viii
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viiiii CCOONNTTEENNTTSS
4.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Green’s Functions 39
5.1 Application of Green’s Second Theorem. . . . . . . . . . . . . . 39
5.2 Green’s Function Solution of Poisson’s Equation . . . . . . . . . 40
5.2.1 Dirichlet Boundary Condition . . . . . . . . . . . . . . . 40 Preface
5.2.2 Surface Green’s Function . . . . . . . . . . . . . . . . . . 41
5.2.3 Neumann Boundary Condition. . . . . . . . . . . . . . . 42
5.3 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
My purpose in this development of vector calculus is to present it in a way that
6 General Coordinate Systems 45
will prove useful to you for the rest of your career in science or mathematics.
6.1 Vector Differential Operators. . . . . . . . . . . . . . . . . . . . 46
Youcanthinkofthispresentationasa‘workbook’forlearninghowtousevector
6.2 Spherical Coordinates. . . . . . . . . . . . . . . . . . . . . . . . 48
calculus in practical calculations and derivations. After studying the text and
6.3 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . 50
doing the problems, you should not have to memorize long equations or need to
6.4 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
look anything up while you work in your field. I want you to get to the point
7 Dyadics 55 where you can treat the use of vector calculus on the same level as you would
7.1 Dyad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 simple algebraic calculations, working them out as you go.
7.2 Dyadic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 The necessary background for using this book is just a knowledge of one-
7.3 Three-dimensional Taylor Expansion . . . . . . . . . . . . . . . 56 dimensional differential and integral calculus. Having had a multivariable cal-
7.4 Multipole Expansion . . . . . . . . . . . . . . . . . . . . . . . . 57 culus course is unnecessary because the multivariable aspect of vector calculus
7.5 Dipole Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 will be developed from scratch. In fact, you might have to unlearn some of the
7.6 Quadrupole Dyadic . . . . . . . . . . . . . . . . . . . . . . . . . 58 things covered in a multivariable calculus course if it relied too much on using
7.7 Tensor (Dyadic) of Inertia . . . . . . . . . . . . . . . . . . . . . 61 coordinate systems, rather than the vector methods we will introduce.
7.8 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Althoughtheexamplesinthebookareusuallyrelatedtophysicalexamples,
particularly electromagnetism, it is not necessary to have any background in
8 Answered Problems 65 physics. If any of the physics is new to you, you can simply disregard it and
8.1 Vector Differential Operators. . . . . . . . . . . . . . . . . . . . 65 concentrate on the mathematics. On the other hand, if you have a background
8.2 Vector Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . 67 in (relatively simple) physics, I think you will find the references to physics
8.3 Integral Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . 71 of interest. In treating electromagnetism, we use what are sometimes called
8.4 Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . 76 ‘natural units’ in which the potential of a point chargeq is given by q/r. In this
8.5 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 80 way, we avoid the introduction of artificial constants that might complicate the
8.6 General Coordinate Systems . . . . . . . . . . . . . . . . . . . . 82 mathematics.
8.7 Dyadics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 The book has two parts. Part one is a brief text developing vector calculus
from the very beginning, and then including some more detailed applications.
Index 97 Part two consists of answered problems, which are all closely related to the
developmentofvectorcalculusinthetext. Althoughthereareanswersimmedi-
ately following each problem, you should not be too quick to use the answer as
a crutch. For problems that involve working through a calculation, please try
your hardest to do it on your own. Then you can use the answer to check your
result, or possibly see another way of doing the problem. If you go immediately
to the answer, you will be learning about vector calculus, but not how to use
iii
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ii CONTENTS
4.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Green’s Functions 39
5.1 Application of Green’s Second Theorem. . . . . . . . . . . . . . 39
5.2 Green’s Function Solution of Poisson’s Equation . . . . . . . . . 40
5.2.1 Dirichlet Boundary Condition . . . . . . . . . . . . . . . 40 Preface
5.2.2 Surface Green’s Function . . . . . . . . . . . . . . . . . . 41
5.2.3 Neumann Boundary Condition. . . . . . . . . . . . . . . 42
5.3 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
My purpose in this development of vector calculus is to present it in a way that
6 General Coordinate Systems 45
will prove useful to you for the rest of your career in science or mathematics.
6.1 Vector Differential Operators. . . . . . . . . . . . . . . . . . . . 46
Youcanthinkofthispresentationasa‘workbook’forlearninghowtousevector
6.2 Spherical Coordinates. . . . . . . . . . . . . . . . . . . . . . . . 48
calculus in practical calculations and derivations. After studying the text and
6.3 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . 50
doing the problems, you should not have to memorize long equations or need to
6.4 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
look anything up while you work in your field. I want you to get to the point
7 Dyadics 55 where you can treat the use of vector calculus on the same level as you would
7.1 Dyad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 simple algebraic calculations, working them out as you go.
7.2 Dyadic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 The necessary background for using this book is just a knowledge of one-
7.3 Three-dimensional Taylor Expansion . . . . . . . . . . . . . . . 56 dimensional differential and integral calculus. Having had a multivariable cal-
7.4 Multipole Expansion . . . . . . . . . . . . . . . . . . . . . . . . 57 culus course is unnecessary because the multivariable aspect of vector calculus
7.5 Dipole Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 will be developed from scratch. In fact, you might have to unlearn some of the
7.6 Quadrupole Dyadic . . . . . . . . . . . . . . . . . . . . . . . . . 58 things covered in a multivariable calculus course if it relied too much on using
7.7 Tensor (Dyadic) of Inertia . . . . . . . . . . . . . . . . . . . . . 61 coordinate systems, rather than the vector methods we will introduce.
7.8 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Althoughtheexamplesinthebookareusuallyrelatedtophysicalexamples,
particularly electromagnetism, it is not necessary to have any background in
8 Answered Problems 65 physics. If any of the physics is new to you, you can simply disregard it and
8.1 Vector Differential Operators. . . . . . . . . . . . . . . . . . . . 65 concentrate on the mathematics. On the other hand, if you have a background
8.2 Vector Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . 67 in (relatively simple) physics, I think you will find the references to physics
8.3 Integral Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . 71 of interest. In treating electromagnetism, we use what are sometimes called
8.4 Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . 76 ‘natural units’ in which the potential of a point chargeq is given by q/r. In this
8.5 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 80 way, we avoid the introduction of artificial constants that might complicate the
8.6 General Coordinate Systems . . . . . . . . . . . . . . . . . . . . 82 mathematics.
8.7 Dyadics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 The book has two parts. Part one is a brief text developing vector calculus
from the very beginning, and then including some more detailed applications.
Index 97 Part two consists of answered problems, which are all closely related to the
developmentofvectorcalculusinthetext. Althoughthereareanswersimmedi-
ately following each problem, you should not be too quick to use the answer as
a crutch. For problems that involve working through a calculation, please try
your hardest to do it on your own. Then you can use the answer to check your
result, or possibly see another way of doing the problem. If you go immediately
to the answer, you will be learning about vector calculus, but not how to use
iixii
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xiv COPNRTEEFNACTES
vectorcalculus. Ofcourse,ifyoudon’tseehowtostart,orrunintoaroadblock,
the answer is there to help you.
Inanyevent,Ihopeyoufindmytreatmentinteresting,perhapsentertaining,
and above all, useful. I think you will see that you have nothing to fear from
vector calculus, and that it will prove to be a good friend for the rest of your
life. Feel free to use me as a resource, either by answering questions you might
have about the material in the book, or about anything else. You can contact
me at [email protected].
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