Vector and Tensor analysis with applications, Study notes of Electronics

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VE·CTORAND

TENSORANALYSIS

WITH APPLICATIONS

by

A. I. BORISENKO

and

I. E. TARAPOV

Revised English Edition

Translated and Edited by

Richard A. Silverman

Dover Publications, Inc.

New York

Copyright© 1968 by Richard A. Silverman. All rights reserved under Pan American and Inter national Copyright Conventions.

Published in Canada by General Publishing <:;ompany, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd.

This Dover edition, first published in 1979, is an unabridged and corrected republication of the work originally published in 1968 by Prentice-Hall, Inc.

International Standard Book Number: 0-486-63833- Library of Congress Catalog Card Number: 79- Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N.Y. 10014

CON"fENTS

1 VECTOR ALGEBRA, Page 1.

1.1. Vectors and Scalars, 1. 1.1.1. Free, sliding and bound vectors, 2. 1.2. Operations on Vectors, 3. 1.2.1. Addition of vectors, 3. 1.2.2. Subtraction of vectors, 5. 1.2.3. Projection of a vector onto an axis, 6. 1.2.4. Multiplication of a vector by a scalar, 7. 1.3. (^) Bases and Transformations, 7. 1.3.1. Linear dependence and linear independence of vectors, 7. 1.3.2. Expansion of a vector with respect to other vectors, 8. 1.3.3. Bases and basis vectors, 9. 1.3.4. Direct and inverse transformations of basis vectors, 13. 1.4. Products of Two Vectors, 14. 1.4.1. The scalar product, 14. 1.4.2. The vector product, 16. 1.4.3. Physical examples, 19. 1.5. Products of Three Vectors, 20. 1.5.1. The scalar triple product, 20. 1.5.2. The vector triple product, 21. 1.5.3. "Division" of vectors, 23. J .6. Reciprocal Bases and Related Topics, 23. 1.6.1. Reciprocal bases, 23. 1.6.2. The summation convention, 26. vii

3 TENSOR ALGEBRA, Page 103.

3.1. Addition of Tensors, 103. 3.2. Multiplication of Tensors, 104. 3.3. Contraction of Tensors, 104.

CONTENTS IX

3.4. Symmetry Properties of Tensors, 105. 3.4.1. Symmetric and antisymmetric tensors, 105. 3.4.2. Equivalence of an antisymmetric second order tensor to an axial vector, 107. 3.5. Reduction of Tensors to Principal Axes, 1Q9. 3.5.1. Statement of the problem, 109. 3.5.2. The two-dimensional case, 110. 3.5.3. The three-dimensional case, 113. 3.5.4. The tensor ellipsoid, 118. 3.6. Invariants of a Tensor, 121. 3.6.1. A test for tensor character, 122. 3.7. Pseudotensors, 122. 3.7.1. Proper and improper transformations, 122. 3.7.2. Definition of a pseudotensor, 124. 3.7.3. The pseudotensor &;ki. 125. Solved Problems, 126. Exercises, 131.

4 VECTOR AND TENSOR ANALYSIS: RUDIMENTS, Page 134.

4.1. The Field Concept, 134. 4.1.1. Tensor functions of a scalar argument, 134. 4.1.2. Tensor fields, 135. 4.1.3. Line integrals. Circulation, 135. 4.2. The Theorems of Gauss, Green and Stokes, 137. 4.2.1. Gauss' theorem, 137. 4.2.2. Green's theorem, 139. 4.2.3. Stokes' theorem, 141. 4.2.4. Simply and multiply connected regions, 144. 4.3. Scalar Fields, 145. 4.3.1. Level surfaces, 145. 4.3.2. The gradient and the directional derivative,

4.3.3. Properties of the gradient. The operator 'V,

4.3.4. Another definition of grad cp, 150. 4.4. Vector Fields, 151. 4.4.1. Trajectories of a vector field, 151. 4.4.2. Flux of a vector field, 152.

X CONTENTS

4.4.3. Divergence of a vector field, 155. 4.4.4. Physical examples, 157. 4.4.5. Curl of a vector field, 161. 4.4.6. Directional derivative of a vector field, 164. 4.5. Second-Order Tensor Fields, 166. 4.6. The Operator V' and Related Differential Operators,

4.6.1. Differential operators in orthogonal curvi linear coordinates, 171. Solved Problems, 174. Exercises, 182.

5 VECTOR AND TENSOR ANALYSIS: RAMIFICATIONS, Page 185.

5.1. Covariant Differentiation, 185. 5.1.1. Covariant differentiation of vectors, 185. 5.1.2. Christoffel symbols, 187. 5.1.3. Covariant differentiation of tensors, 190. 5.1.4. Ricci's theorem, 191. 5.1.5. Differential operators in generalized co ordinates, 192. 5.2. Integral Theorems, 196. 5.2.1. Theorems related to Gauss' theorem, 197. 5.2.2. Theorems related to Stokes' theorem, 198. 5.2.3. Green's formulas, 201. 5.3. Applications to Fluid Dynamics, 203. 5.3.1. Equations of fluid motion, 203. 5.3.2. The momentum theorem, 208. 5.4. Potential and Irrotational Fields, 211. 5.4.1. Multiple-valued potentials, 213. 5.5. Solenoidal Fields, 216. 5.6. Laplacian Fields, 219. 5.6.1. Harmonic functions, 219. 5.6.2. The Dirichlet and Neumann problems, 422. 5.7. The Fundamental Theorem of Vector Analysis, 223. 5.8. Applications to Electromagnetic Theory, 226. 5.8. I. Maxwell's equations, 226. 5.8.2. The scalar and vector potentials, 228. 5.8.3. Energy of the electromagnetic field. Poynting's vector, 230. Solved Problems, 232. Exercises, 247. BIBLIOGRAPHY, Page 251. INDEX, Page 253.

2 VECTOR ALGEBRA CHAP.^1

elastic body leads to the concept of the deformation (or strain) tensor. We

will defer further discussion of tensors until Chapter 2, concentrating our attention for now on vectors.

A vector A is represented by a directed line segment, whose direction and

length coincide with the direction and magnitude (measured in the chosen system of units) of the quantity under consideration. Vectors are denoted by

boldface letters, A, B, ... and their magnitudes by IAI, IBI,... or by the

corresponding lightface letters A, B,... When working at the blackboard, it is customary to indicate vectors by the presence of little arrows, as in

A.,D, ....

The vector of magnitude zero is called the zero vector, denoted by 0

(ordinary lightface zero). This vector cannot be assigned a definite direction, or alternatively can be regarded as having any direction at all. Vectors can be compared only if they have the same physical or geometri

cal meaning, and hence the same dimensions. Two such vectors A and B

measured in the same system of units are said to be equal, written A = B, if

they have the same magnitude (length) and direction.

1.1.1. Free, sliding and bound vectors. A vector like the velocity of a body undergoing uniform translational motion, which can be displaced parallel

to itself and applied at any point, is called a free vector [see Fig. 1.1(a)]. In

(al (b) (cl FIG. 1.1. (a) A free vector, which can be displaced parallel to itself; (b) A sliding v�ctor, which can be displaced along its line of action; (c) A bound vector.

three dimensions a free vector is uniquely determined by three numbers, e.g., by its three projections on the axes of a rectangular coordinate system. It can also be specified by giving its magnitude (the length of the line segment representing the vector) and two independent angles a. and �(any two of the angles between the vector and the coordinates axes). A vector like the force applied to a rigid body fastened at a fixed point,

SEC. 1.2 VECTOR ALGEBRA 3

which can only be displaced along the line containing the vector, is called a

sliding vector [see Fig. I. I(b)]. In three dimensions a sliding vector is deter

mined by five numbers, e.g., by the coordinates of the point of intersection M

of one of the coordinate planes and the line containing the vector (two

numbers), by the magnitude of the vector (one number) and by two inde

pendent angles r1. and � between the vector and two of the coordinate axes

(two numbers).

A vector like the wind velocity at a given point of space, which is referred

to a fixed point, is called a bound vector [see Fig. l.l(c)]. In three dimensions

a bound vector is determined by six numbers, e.g., the coordinates of the

initial and final points of the vector (Mand Nin the figure).

Free vectors are the most general kind of quantity specified by giving a

magnitude and a direction, and the study of sliding and bound vectors can

always be reduced to that of free vectors. Therefore we shall henceforth

consider only free vectors.

1.2. Operations on Vectors

1.2.1. Addition of vectors. Given two vectors A and B, suppose we put the

initial point of B at the final point of A. Then by the sum A + B we mean

the vector joining the initial point of A to the final point of B. This is also

the diagonal of the parallelogram constructed on A and B, in the way shown

in Fig. 1.2(a). It follows that the sum A + B + C + · · · of several vectors

c

(a J (b) ( c)

FIG. 1.2. (a) The sum of two vectors A+ B = C;

(b) The sum of several vectors A+ B + C + · · · = N;

(c) Associativity of vector addition: (A+ B) + C = A+

(B + C) = A+ B + C.

B

A, B, C,... is the vector closing the polygon obtained by putting the initial

point of B at the final point of A, the initial point of C at the final point of B,

and so on, as in Fig. l.2(b). The physical meaning of vector addition is clear

if we interpret A, B, C,... as consecutive displacements of a point in space.

SEC. 1.2 VECTOR ALGEBRA 5

(o) (b)

F10. 1.3. Rotation of a sphere about its axis.

(a) The directed Jine segment a3 does not lie in the plane of the segments

a1 and a2;

(b) Two ways of carrying a circular arc from one position (A1B1) to

another (A2B2).

2. Both of the following operations carry a circular arc A1B1 on a sphere into a new position A2B2 [see Fig. 1.3(b)]: a) A rotation through ex.� radians about the axis OA1 (the directed line segment ex�) followed by a rotation through ex.;^ radians about the axis OB2 (the directed line segment ex;); b) A rotation through ex.; radians about the axis 081 (the directed line segment ex;) followed by a rotation through ex.; radians about the axis OA2 (the directed line segment ex;). Hence ex� + ex; must equal ex; + ex; if rotations. add like vectors. But this is impossible, since ex� + ex; and ex; + ex; lie in nonparallel planes.

Remark. Unlike the case of finite rotations of a rigid body, it turns out

that infinitesimal rotations are vectors (see Prob. 10, p. 44).

1.2.2. Subtraction of vectors. By the vector -A ("minus A") we mean the vector with the same magnitude as A but with the opposite direction. Each of the vectors A and -A is called the opposite or negative of the other. Obviously, the only vector equal to its own negative is the zero vector, and the sum of a vector and its negative is the zero vector. If X+B=A,

1 Note that ex; = ex;, ex� = ex;.

6 VECTOR ALGEBRA (^) CHAP. 1

then adding -B to both sides of the

equation, we obtain

X+ B+ (-B) =A+ ( -B). ( 1.1)

But

X + B+ (-B) =X+ [B+ (-B)]

=X+O=X,

and hence ( 1.1) implies

X =A+ (-B).

FIG. 1.4. Subtraction of two vectors. The right-hand side is the result of sub-

tracting B from A and is written simply

as A - B, without any intervening parentheses. Thus subtraction of B from

A reduces to addition of A and the negative of B, as shown in Fig. 1.4.

1.2.3. Projection of a vector onto an axis. Given a vector A and an axis

u (see Fig. 1.5), by the projection of A onto u, denoted by Au, we mean the

+u

(o l (b) FIG. 1.5. (a) Projection of a vector onto an axis; (b) Projection of one vector onto the direction of another.

length of the segment cut from u by the planes drawn through the end points

of A perpendicular to u, taken with the plus sign if the direction from the

projection (onto u) of the initial point of A to the projection of the end point

of A coincides with the positive direction of u, and with the minus sign

otherwise. By the unit vector corresponding to the axis u we mean the vector

u0 of unit length whose direction coincides with that of u (thus u0 = lu01 = 1).

If <p =(A, u0) denotes the angle between A and u0, 2 then

Au = A cos <p = A cos (A, u0). (1.2)

2 Given two vectors A and B, the angle between A and B, denoted by (A, B), will always be chosen to lie between 0 and 7t.

8 VECTOR ALGEBRA CHAP. 1

In other words, n vectors A1, A2, • • • , An are said to be linearly independent

if (1.3) implies

Two linearly dependent vectors are collinear. This follows from Sec.

1.2.4 and the fact that

implies

if c1 -=ft 0 or

if C2 -=ft 0.

Three linearly dependent vectors are coplanar, i.e., lie in the same plane (or are parallel to the same plane). In fact, if

where at least one of the numbers c1, c2, c3 is nonzero, say c3, then

where

C=mA+nB

m=--C C

n= --C C

i.e., C lies in the same plane as A and B (being the sum of the vector mA collinear with A and the vector nB collinear with B).

1.3.2. Expansion of a vector with respect to other vectors. Let A and B be two linearly independent (i.e., noncollinear) vectors. Then any vector C coplanar with A and B has a unique expansion

C=mA+nB (1.6)

with respect to A and B. In fact, since A, B and C are coplanar, (I .4) holds

with at least one nonzero coefficient, say c3. Dividing (1.4) by c3, we get ( 1.6),

where m and n are the same as in (1.5). To prove the uniqueness of the

expansion (I .6), suppose there is another expansion

C=m'A+ n'B. (1.7)

Subtracting (1.7) from (1.6), we obtain

(m - m')A+(n -- n')B=0.

But then m=m', n=n' since A and B are linearly independent. In other

SEC. 1.3 VECTOR ALGEBRA 9

words, the coefficients m and n of the expansion (1.6) are uniquely deter

mined.

Let A, B and C be three linearly independent (i.e., noncoplanar) vectors.

Then any vector D has a unique expansion

D = mA + nB + pC (1.8)

with respect to A, B and C. To see this,

draw the vectors A, B, C and D from a

common origin 0 (see Fig. 1.6). Then

through the end point of D draw the three planes parallel to the plane of the vectors

A and B, A and C, B and C. These planes,

together with the planes of the vectors A

and B, A and C, B and C form a parallel

epiped with the vector Das one of its diag

onals and the vectors A, B and C (drawn

from the origin 0) along three of its edges.

If the numbers m, n and pare such that mA,

nB and pC have magnitudes equal to

pC c FIG. 1.6. An arbitrary vector D has a unique expansion with respect to three noncoplanar vectors A, B and C.

the lengths of the corresponding edges of the parallelepiped, then clearly

D = mA + (nB + pC) = mA + nB + pC

as shown in Fig. 1.6.

To prove the uniqueness of the expansion (1.8), suppose there is another

expansion

D = m'A + n'B + p'C. (1.9)

Subtracting (1.9) from (1.8), we obtain

(m - m')A + (n - n')B + ( p - p')C = 0.

But then m = m', n = n', p' = p since A, B and C are linearly independent by

hypothesis.

Remark. It follows from the above considerations that any four vectors

in three-dimensional space are linearly dependent.

1.3.3. Bases and basis vectors. By a basis for three-dimensional space we

mean any set of three linearly independent vectors e1, e2, e3• Each of the

vectors e1, e2, ea is called a basis vector. Given a basis e1, e2, ea, it follows from

the above remark that every vector A has a unique expansion of the form

SEC. 1.3 VECTOR^ ALGEBRA^11

The great merit of vectors in applied problems is that equations describing

physical phenomena can be formulated without reference to any particular

coordinate system. However, in actually carrying out the calculations needed

to solve a given problem, one must eventually cast the problem into a form

involving scalars. This is done by introducing a suitable coordinate system,

and then replacing the given vector (or tensor) equations by an equivalent

system of scalar equations involving only numbers obeying the ordinary

rules of arithmetic. The key step is to expand the vectors (or tensors) with

respect to a suitable basis, corresponding to the chosen system of co

ordinates.

Consider, for example, the case of two dimensions. The position of a

point Min the plane is uniquely determined by its radius vector r relative to

some fixed point 0 which can be chosen arbitrarily and is independent of any

coordinate system. However, before making any calculations, we must intro

duce a coordinate system. Then the position of the point M is given by two

numbers p and q (called its coordinates), which now depend both on the

coordinate system and on the units of measurement. In a rectangular system,

these coordinates are just the (signed) distances p = x1 and q = x2 between

M and two perpendicular lines going through the origin of coordinates.

Holding one coordinate fixed, say p = const, and continuously varying the

other coordinate, we obtain a coordinate curve. Thus there are two coordinate

curves passing through every point of the plane. In rectangular coordinates

these curves are simply the lines parallel to the coordinate axes. As the basis

vectors corresponding to the coordinates p and q, we choose the unit vectors

(i.e., the vectors of unit length) tangent to the coordinate curves at the point

M. In rectangular coordinates these are just the unit vectors i1 and i2 parallel

to the coordinate axes.

Clearly, the basis vectors i1 and i2 of a rectangular coordinate system are

independent of the point M and always intersect at right angles. Suppose,

however, that the position of M is specified in polar coordinates, i.e., by

giving the distance Rbetween Mand a fixed point 0 (called the pole) and the

angle cp between the line joining 0 to Mand a fixed ray (called the polar axis)

drawn from 0. The coordinate curves are then the circles of radius Rand the

rays of inclination cp, and the corresponding basis vectors are the unit vectors

eR and e<P shown in Fig. 1.9(a). Note that although eR and e<P vary from point

to point, they always intersect at right angles (compare the basis at M with

that at N). Coordinate systems whose basis vectors intersect at right angles

are called orthogonal systems, and are the systems of greatest importance in

the applications. Coordinate systems whose coordinate curves are not

straight lines are called systems of curvilinear coordinates (as opposed to

12 VECTOR ALGEBRA

(al ( b)

Fm. 1.9. Curvilinear coordinates in the plane.

(a) Polar coordinates;

(b) "Generalized" polar coordinates.

CHAP. 1

rectangular or oblique coordinates). Three-dimensional curvilinear co

ordinates will be considered in detail in Sec. 2.8.

Given a system of polar coordinates R and e with pole 0, consider a

system of rectangular coordinates x1 and x2 with origin 0, and suppose the

x1-axis coincides with the polar axis of the system of polar coordinates. Then

the relation between the rectangular coordinates x1' x2 and the polar

coordinates R, cp is given by the formula

X1 = R cos cp, x2 = R sin cp ( 1.10)

and

R = �x� + x� (0 (^) < R < oo),

tan cp = -^ X2 (0 (^) < cp < 27t). X More generally, consider the system of "generalized" polar coordinates u and

e whose coordinate curves are the ellipses

J

2 2 U =^ X1^ +^ X a^2 b^2

(a> 0, b > 0, a=:/= b) and the rays

tan e = - ax bx

where the analogue of ( 1.10) is now

X1^ =^ QU COS^ 8,

(0 < u < oo)

(o < e <^ 27t),

X2 =^ bu^ sin^ 8.