Problems in Electrodynamics

Citation preview

V.V. Batygin, I.N. Toptygin

PROBLEMS in ELECTRODYNAMICS

T

PROBLEMS in ELECTRODYNAMICS

V. V. Batygin and I. N. Toptygin

PROBLEMS in ELECTRODYNAMICS Translated by

S. Chomet

Edited by

P. I. Dean

Translated from Russian V.V. BATYGIN and I.N. T0PTYG1N S bo rn ik Z a d a c h p o E le k tro d in a m ik e

Gosudarstvennoe Izdatel’stvo Fiziko Matematicheskoi Literatury 480 pp., Moscow, 1962

Copyright (c) 1964 by Academic Press Inc. (London) Ltd. and Infosearch Ltd.

All Rights Reserved

No part of this book may be produced in any form, by photostat, microfilm, or any other means, without written permission from the publishers

Library of Congress Catalog Card, Number: 64-17461 Academic Press Inc. (London) Ltd. Berkeley Square House, London, W. 1. Infosearch Ltd. London, N.W. 2.

U.S. Edition published by Academic Press Inc. I l l Fifth Avenue, New York 3, New York

P rin te d

in Great Britain by Pion Ltd., London, N.W. 2.

Contents

PAGE

P refac e

iii Problem s

C h a p te r I. V e c to r and te n s o r c a lc u lu s 1. V e c to r and te n s o r a lg e b r a . T ran sfo rm a ­ tio n s o f v e c to r s and te n s o r s 2. V e c to r a n a ly s is C h a p te r II.

E l e c t r o s t a t i c s in v a c u u m

1

185

1 7

185 191

15

194

27 27 39 42

207 207 223 225

49

240

56

247

68 68 72 77

266 266 272 281

82

286

82 88

287 299

C h a p te r III. 1. 2. 3.

E l e c t r o s t a t i c s of c o n d u c to rs an d d ie le c tr ic s B a s ic c o n c e p ts an d m e th o d s of e l e c t r o s t a t i c s C o e ffic ie n ts of p o te n tia l an d c a p a c ita n c e S p e c ia l m e th o d s of e l e c t r o s t a t i c s

S o lu t io n s

C h a p te r IV. C h a p te r V.

S te a d y c u r r e n t s M a g n e to s ta tic s

C h a p te r VI. 1. 2. 3.

E l e c t r i c a l a n d m a g n e tic p r o p e r tie s of m a tte r P o l a r i s a t i o n o f m a t t e r in a c o n s ta n t fie ld P o l a r i s a t i o n o f m a t t e r in a v a r ia b le fie ld F e r r o m a g n e tic re s o n a n c e

C h a p te r V II. 1. 2.

Q u a s i - s t a t i o n a r y e le c tr o m a g n e tic fie ld s Q u a s i- s ta tio n a r y p h e n o m e n a in l i n e a r c o n ­ d u c to r s E ddv c u r r e n t s a n d s k in e ffe c t

CO N TEN TS

11

P ro b le m s

C h a p te r VIII. 1. 2. 3.

P ro p a g a tio n of e le c tro m a g n e tic w av es P la n e w a v es in a h o m o g e n eo u s m e d iu m . R e fle c tio n and r e f r a c tio n . W ave p a c k e ts S c a tte rin g of e le c tro m a g n e tic w a v e s by m a c ro s c o p ic b o d ie s. D iffra c tio n P la n e w a v es in a n is o tro p ic and g y ro tro p ic m e d ia

Solutions

C h a p te r IX.

E le c tro m a g n e tic o s c illa tio n s in bounded b o d ie s

93

313

93

313

101

328

108

347

113

358

C h a p te r X. S p e c ial th e o ry of r e la tiv ity 1. L o re n tz tr a n s f o r m a tio n 2. F o u r -d im e n s io n a l v e c to r s and te n s o r s 3. R e la tiv is tic e le c tro d y n a m ic s

120 120 127 130

375 375 383 385

C h a p te r XI. R e la tiv is tic m e c h a n ic s 1. E n e rg y and m o m en tu m 2. T he m o tio n of c h a rg e d p a r tic le s in an e le c tro m a g n e tic fie ld

135 135

391 391

143

401

153

420

153

420

160

429

167

443

172

451

177

462

C h a p te r X II. E m is s io n of e le c tro m a g n e tic w a v es 1. T h e H e rtz v e c to r and th e m u ltip o le e x p a n ­ sio n 2. T h e e le c tro m a g n e tic fie ld of a m o v in g p o in t c h a rg e 3. In te ra c tio n of c h a rg e d p a r tic le s w ith r a d i a ­ tio n 4. E x p a n sio n of a n e le c tro m a g n e tic fie ld in t e r m s of p la n e w a v es C h a p te r XIII.

T h e ra d ia tio n e m itte d d u rin g th e in te r a c tio n of c h a rg e d p a r tic le s w ith m a tte r

A ppendix I.

T h e 6 -fu n c tio n

481

A ppendix II.

S p h e ric a l L e g e n d re fu n c tio n s

484

A ppendix III. C y lin d ric a l fu n c tio n s

487

Index

491

PREFACE

T h is b o o k c o n ta in s a b o u t 750 p r o b le m s on c l a s s i c a l e l e c t r o ­ d y n a m ic s an d its m o r e im p o r ta n t a p p lic a tio n s , in c lu d in g o v e r 150 p r o b le m s on th e s p e c ia l th e o r y of r e l a tiv ity , an d a b o u t 70 p r o b ­ le m s on v e c to r an d t e n s o r a n a ly s is . In a d d itio n to p r o b le m s i l l u s t r a t i n g fu n d a m e n ta l c o n c e p ts an d la w s of e le c tr o d y n a m ic s , w h ich c an b e s o lv e d b y p u r e ly m a th e ­ m a tic a l m e th o d s , th e c o lle c tio n in c lu d e s a l a r g e n u m b e r of m o r e c o m p lic a te d p r o b le m s (th e s e a r e in d ic a te d b y a s t e r i s k s ) . S om e of th e s o lu tio n s in v o lv e a c o n s id e r a b le a m o u n t of e ffo rt, w h ile o th e r s a r e p u r e ly th e o r e t i c a l in n a tu r e and fo llo w fr o m a le c t u r e c o u r s e on e le c tr o d y n a m ic s (p ro p a g a tio n of w a v e s in a n is o tr o p ic an d g y r o tr o p ic m e d ia , m o tio n of c h a r g e d p a r t i c l e s in th e e l e c t r o ­ m a g n e tic fie ld , r e p r e s e n t a t i o n of th e e le c tr o m a g n e tic fie ld b y a s e t of o s c i l l a t o r s , and so on). F in a lly , t h e r e a r e p r o b le m s w h ic h a r e c o n c e rn e d w ith to p ic s w hich a r e n o t w e ll c o v e r e d b y e x is tin g te x ts , f o r e x a m p le , in te r a c tio n of c h a r g e d p a r t i c l e s w ith m a tte r (C h a p te r X III), a p p lic a tio n s of c o n s e r v a tio n la w s to th e a n a ly s is of c o llis io n p r o c e s s e s and p a r t i c l e d is in te g r a tio n (C h a p te r XI), f e r r o ­ m a g n e tic r e s o n a n c e (C h a p te r VI), and so on. T h e s e c o n d p a r t of th e book g iv e s a n s w e r s an d s o lu tio n s to a la r g e n u m b e r of th e s e p ro b le m s . E a c h s e c tio n is p re fa c e d b y a s h o r t th e o r e tic a l in tro d u c tio n in w h ich th e n e c e s s a r y fo r m u la e a r e g iv en . T h e s e s h o r t in tr o ­ d u c tio n s do n o t p r e te n d to b e c o m p le te ; m o re e x te n s iv e t r e a t ­ m e n ts w ill b e found in th e books lis te d in th e b ib lio g ra p h y . T h e m a th e m a tic a l a p p e n d ic e s re v ie w th e b a s ic p r o p e r tie s of th e 6 -fu n c tio n and th e c y lin d r ic a l and s p h e r ic a l fu n c tio n s, w hich a r e n e c e s s a r y fo r th e so lu tio n of th e p ro b le m s . T h e p r e s e n t c o lle c tio n is b a s e d on l e c t u r e s g iv e n in th e D e p a rtm e n ts of E le c tr o n ic s and P h y s ic s and M e c h a n ic s of th e L e n in g ra d P o ly te c h n ic a l In s titu te . A la r g e n u m b e r of th e p r o b ­ le m s w e re s e t to th ir d and fo u rth y e a r s tu d e n ts .

iv

PREFA CE

In c o m p ilin g th is c o lle c tio n w e h av e m ade fre q u e n t u se of the w e ll-k n o w n te x ts of L. D. L an d au an d E. M. L ifs h its , I. E. T am m , Y a. I. F r e n k e l ', M. A b ra h a m an d R . B e c k e r, W .R .S m y th e , J . A. S tra tto n , an d o th e r s . A la r g e n u m b e r of m o n o g ra p h s, re v ie w p a p e r s , a n d o r ig in a l p a p e r s w e re a ls o c o n su lte d . V. Batygin and I. Toptygin

Problems

Chapter 1

V E C T O R AND T E N SO R C A L C U L U S

1.

V e c to r an d t e n s o r a lg e b r a . T r a n s f o r m a tio n s of v e c to r s an d t e n s o r s

A v e c to r in t h r e e - d im e n s i o n a l s p a c e is d e fin e d a s a s e t o f th r e e q u a n titie s w h ich t r a n s f o r m in a c c o r d a n c e w ith th e r u le

*=i

a - 1)

w hen th e s y s te m of c o o r d in a te s is r o ta te d . H e r e A ^ a r e th e c o m p o n e n ts o f th e v e c to r a lo n g th e a x e s of th e o r ig in a l s y s te m of c o o r d in a te s , A \ a r e th e c o m p o n e n ts a lo n g th e a x e s of th e r o ta te d s y s te m , a n d a r e th e tr a n s f o r m a tio n c o e ffic ie n ts w h ich a r e e q u a l to th e c o s in e s o f th e a n g le s b e tw e e n th e & -th a x is of th e o r ig in a l s y s te m and th e z'-th a x is of th e r o ta te d s y s te m . W e s h a ll u s e th e fo llo w in g s u m m a tio n r u le : th e s u m m a tio n s ig n w ill b e o m itte d an d th e s u m m a tio n p r o c e s s w ill b e in d ic a te d by a r e p e a te d s u b s c r ip t. In a c c o rd a n c e w ith th is c o n v e n tio n E qn. (1.1) m a y be r e - w r i t t e n in th e fo rm Aj = o-ikAif A te n s o r of ra n k 2 in th r e e - d im e n s io n a l s p a c e is d e fin e d a s th e n in e -c o m p o n e n t q u a n tity (i, k = 1, 2, 3) w hich t r a n s f o r m s in a c c o rd a n c e w ith th e r u le Tlk = ^■il^knJ'lm

(L2)

w h e re th e s u m m a tio n is to b e ta k e n o v e r I an d m . S im ila r ly , a te n s o r of ra n k s in th r e e - d im e n s io n a l s p a c e is d e fin e d by th e fo llo w in g tr a n s f o r m a tio n r u le : T i k i . . . r == 3-1 i ’&kk’ • • - a rr ’ T i ‘ k ’V . . . r ’ •

In th is e x p r e s s io n th e q u a n titie s T h a v e s in d ic e s e a c h .

(1 - 3 )

2

PROBLEMS.

CHAPTER I

Q u a n titie s w hich tr a n s f o r m a s v e c to r s w hen th e c o o rd in a te s y s te m is ro ta te d m ay b eh av e in tw o d is tin c t w ays w hen th e s y s ­ te m of c o o rd in a te s is in v e rte d , i . e . it is s u b je c te d to th e t r a n s ­ fo rm a tio n x' = -x, y ' = -y, z ' = - z . V e c to rs w h o se co m p o n en ts change sig n on in v e rs io n of th e c o o rd in a te s y s te m a r e known a s p o la r v e c to r s , o r sim p ly v e c to r s . V e c to rs w hose co m p o n en ts do not change s ig n on in v e rs io n of th e c o o rd in a te s y s te m a r e c a lle d p s e u d o v e c to rs o r a x ia l v e c to r s . We s h a ll not d istin g u ish b etw een c o v a ria n t and c o n tra v a r ia n t co m p o n en ts of v e c to r s and te n s o r s b e c a u s e th is d is tin c tio n is u n im p o rta n t fo r th e p ro b le m s c o n s id e re d in th is book. T he v e c to r p ro d u c t of tw o p o la r v e c to r s is an e x am p le of an a x ia l v e c to r . S im ila rly , a te n s o r of ra n k s is r e f e r r e d to sim p ly a s a te n s o r if its co m p o n en ts tr a n s f o r m on in v e rs io n a s th e p ro d u c ts of s c o o rd in a te s , i . e . w hen th e r e s u lt of th e tr a n s f o r m a tio n is to m u ltip ly th e m by ( - l ) s . W hen th e r e ­ s u lt of th e in v e rs io n is to m u ltip ly th e c o m p o n en ts by ( - l ) s+1 th e n th e te n s o r is r e f e r r e d to a s a p s e u d o te n s o r. T he q u an tity “n

“ l2

“ l3

“ 21

“ 22

®23

‘“ 31

“ 32

®33‘

(1.4)

is c a lle d th e tra n s fo rm a tio n m a trix . T h e d e te rm in a n t w hose e le m e n ts a r e equal to th e e le m e n ts of a giv en m a trix is c a lle d th e d e te rm in a n t of th a t m a trix . T h u s, th e d e te rm in a n t of th e m a trix g iven by Eqn. (1.4) is “ l2

“ 13

“ 21

“ 22

“ 23

“ 31

“ 32

“ 33

“n

The sum a + ft of tw o m a tr ic e s is defin ed a s th e m a trix y w hose e le m e n ts a r e equal to th e su m s of th e c o rre s p o n d in g e le m e n ts in th e com ponent m a tr ic e s . Thus T/* =

(I>6)

T he p ro d u c t a/3 of tw o m a tr ic e s is defin ed a s th e m a trix y w hose e le m e n ts a r e ob tain ed by m u ltip ly in g th e co m p o n en ts a n d /3 ^ in a c c o rd a n c e w ith th e ru le

w h e re th e su m m a tio n is to be c a r r i e d out o v e r I.

T h e m a trix y

3

VECTOR AND TENSOR CALCULUS

r e p r e s e n t s th e tr a n s f o r m a tio n w hich r e s u l t s a f te r th e s u c c e s s iv e a p p lic a tio n of th e tr a n s f o r m a tio n s /3 and « . T h e u n it m a tr ix is d efin e d by 1 0 0 1 >0 0

i =

( 1. 8 )

and d e s c r ib e s th e tr a n s f o r m a tio n y ie ld in g A £ = A i . T h e e le m e n ts of th e u n it m a tr ix m ay b e r e p r e s e n te d by th e sy m b o l 6 ;^ w h ich is su c h th a t / = k, i =A k.

1 0

(1.9)

A m a tr ix of th e fo rm ®i 0 O' 0 *2 0 ,0 0 a3‘

a

is c a lle d a d ia g o n a l m a tr ix . c o n d itio n

(1.10)

A m a tr ix w h o se e le m e n ts s a tis f y th e

(1.11)

«/*«« = 8« is c a lle d an o rth o g o n a l m a tr ix . T h e r e c i p r o c a l (or in v e r s e ) m a tr ix o'-1 is d e fin e d by aa 1= a *0 = 1,

(1.12)

an d d e s c r ib e s th e r e c i p r o c a l tr a n s f o r m a tio n , i . e . if A'i = a ^ ^ A ^ th e n Afr = F in a lly , th e tr a n s p o s e d m a tr ix 5 , is o b ta in e d fr o m a by in te rc h a n g in g th e c o lu m n s an d th e ro w s , s o th a t all “21 a12 022

a31 “32 'a13 “23 “33'

a lk — **/•

(1.13)

------ oOo-------

1. F in d th e c o s in e o f th e a n g le 9 b e tw e e n tw o d ir e c tio n s n an d n ' w h ic h a r e d e fin e d in a s p h e r ic a l s y s te m of c o o r d in a te s by th e a n g le s a , fi an d a ' , (S’ r e s p e c tiv e ly .

4

PROBLEMS.

2.

CHAPTER I

P ro v e th e id e n titie s

(A X B) • (C X D) = (A • C) (B • D) — (A • D)(B ■C); (A X B) X (C X D) = [A • (B X D)] C — [A • (B X C)J D = = [A • (C X D)] B — [B ■(C X D)J A.

3. A s e t of th r e e q u a n titie s a i (i = 1, 2, 3) is giv en in a li c a r t e s i a n s y s te m s of c o o rd in a te s and it is know n th a t a ^ b i is in v a ria n t w ith r e s p e c t to ro ta tio n s and r e f le c tio n s . Show th a t if is a v e c to r (p se u d o v e c to r) th e n a ; is a ls o a v e c to r (p se u d o v e c to r ). 4. Show th a t if a i = T { k bk in e v e r y s y s te m of c o o rd in a te s , w h e re T i k is a t e n s o r of ra n k 2 w h ile bk is a v e c to r , th e n a i is a ls o a v e c to r . da 7 5. Show t h a t —— is a t e n s o r of ra n k 2. 8xk 6. Show th a t if T i k is a te n s o r of ra n k 2 and P i k is a p s e u d o te n s o r of ra n k 2, th e n T i k P i k is a p s e u d o s c a la r . 7. Show th a t th e s y m m e tr y of a te n s o r is a p r o p e r ty w hich is in v a ria n t w ith r e s p e c t to r o ta tio n s , i . e . a te n s o r w hich is s y m ­ m e tr ic ( s k e w - s y m m e tr ic ) in a g iv e n c o o rd in a te s y s te m r e m a in s s y m m e tr ic ( s k e w - s y m m e tr ic ) in a ll o th e r s y s te m s w hich a r e r o ta te d w ith r e s p e c t to th e o r ig in a l s y s te m . 8. Show th a t if th e t e n s o r S i k is s y m m e tr ic w h ile th e t e n s o r A ^ k is s k e w - s y m m e tr ic , th e n A i k S i k = 0. 9. Show th a t th e su m of th e d ia g o n a l c o m p o n e n ts of a te n s o r of r a n k 2 is an in v a ria n t q u a n tity . 10*. In s o m e c a s e s it is c o n v e n ie n t to u s e th e c y c lic c o m ­ p o n e n ts d e fin e d by a±i = * ( a x ± i a y ) / y [ 2 , a 0 = a z in s te a d of th e c a r t e s i a n c o m p o n e n ts a ^ ay, a F of a v e c to r . E x p r e s s th e s c a l a r an d v e c to r p r o d u c ts of tw o v e c to r s in t e r m s of t h e i r c y c lic c o m p o n e n ts . F in d a ls o th e c y c lic c o m p o n e n ts of th e p o s i­ tio n v e c to r in t e r m s of th e s p h e r ic a l L e g e n d re f u n c tio n s !. 11*. F in d th e c o m p o n e n ts of th e t e n s o r e ^ w h ich is r e c i p r o c a l to e ^ . C o n s id e r in p a r t i c u l a r th e c a s e w h e re is a s y m m e tr ic te n s o r d e fin e d a lo n g th e p r in c ip a l a x e s . 12. S uppose th a t in a ll c o o rd in a te s y s te m s th e c o m p o n e n ts of th e v e c to r a a r e li n e a r fu n c tio n s of th e c o m p o n e n ts of a n o th e r v e c to r b so th a t a i = e^A^AShow th a t th e q u a n tity is a

t Spherical functions are defined in Appendix II.

VECTOR AND TENSOR CALCULUS

5

te n s o r of r a n k 2. (M ore p r e c is e ly , is a te n s o r if a an d b a r e both p o la r v e c to r s o r p s e u d o v e c to rs , w h ile c j^ is a p s e u d o ­ te n s o r if one of th e v e c to r s is p o la r and th e o th e r is a x ia l.) 13. Show th a t th e s e t of q u a n titie s - w h e re A^k.1 is a te n s o r of ra n k 3 and a is a te n s o r of ra n k 2, is a v e c to r . 14. F in d th e tr a n s f o r m a tio n law fo r th e s e t of v o lu m e in te g r a ls TjA = f x ^ x ^ d V , w hich d e s c r ib e s s p a c e r o ta tio n s and r e f le c tio n s (x{ and xa a r e th e c a r te s ia n c o o rd in a te s ). 15. Set up th e tr a n s f o r m a tio n m a tr ic e s fo r th e b a s ic u n it v e c to r s w hich r e p r e s e n t th e tr a n s f o r m a tio n fro m c a r te s ia n c o ­ o rd in a te s to s p h e r ic a l c o o rd in a te s and v ic e v e r s a , and a ls o th e tr a n s f o r m a tio n fro m c a r te s ia n c o o rd in a te s to c y lin d r ic a l c o ­ o r d in a te s a n d v i c e v e r s a . 16. W rite down th e tr a n s f o r m a tio n m a tr ic e s f o r th e c o m ­ p o n e n ts of a v e c to r w h ich d e s c r ib e th e re f le c tio n of th e th r e e c o o rd in a te a x e s a n d th e r o ta tio n of th e c a r t e s i a n s y s te m of c o o rd in a te s abo u t th e z —a x is th ro u g h an a n g le a . 17. F in d th e tr a n s f o r m a tio n m a t­ r i x fo r th e c o m p o n e n ts of a v e c to r , w hich d e s c r ib e s th e r o ta tio n of th e c o o rd in a te a x e s d e fin e d by th e E u le r a n g le s oq, 0, oi2 (F ig . 1), by m u ltip ly in g to g e th e r th e m a tr ic e s c o rre s p o n d in g to r o ta tio n s a b o u t th e z - a x is th ro u g h a n g le o^, ab o u t th e lin e ON th ro u g h an an g le 0, an d a b o u t th e z '- a x i s th ro u g h i e . I. an a n g le oi2. 18. F in d th e m a tr ix D (arp 0, a 2) fo r th e tr a n s f o r m a tio n of th e c y c lic c o o rd in a te s of a v e c to r (c f . P r o b le m 10) w h en th e c o ­ o rd in a te s y s te m is r o ta te d th ro u g h th e E u le r a n g le s aq, 0, an d a 2 (F ig. 1). 19*. Show th a t th e m a tr ix r e p r e s e n tin g an in f in ite s im a l rAo ta tio n A of a s y s teAm of c o o rd in a te s m a y b e w r itte n in th e fo rm A o = 1 + e w h e re c is a s k e w - s y m m e tr ic m a tr ix (l^ a = ~ ^ k 0 E lu c id a te th e g e o m e tr ic a l m e a n in g of g^a* 20. Show th a t if a is a n o rth o g o n a l tr a n s f o r m a tio n m a tr ix th e n th e c o rr e s p o n d in g tr a n s p o s e d m a tr ix r e p r e s e n t s th e r e c i p r o c a l tr a n s f o r m a tio n . 21. Show th a t th e m a tr ix r e p r e s e n tin g th e r e f le c tio n o r r o ta tio n of th e th r e e b a s ic u n it v e c to r s of a c o o rd in a te s y s te m

6

PROBLEMS.

CHAPTER I

is id e n tic a l to th e m a tr ix d e s c rib in g th e tr a n s f o r m a tio n of th e c o m p o n e n ts of a v e c to r . 22*. Show th a t w hen an ev en n u m b e r of c o o rd in a te a x e s a r e r e f le c te d o r ro ta te d , th e tr a n s f o r m a tio n d e te r m in a n t is e q u al to + 1, w h ile th e c o rre s p o n d in g r e s u l t fo r an odd n u m b e r of c o ­ o rd in a te a x e s is -1 . 23. Show th a t if th e c o m p o n e n ts of tw o v e c to r s in a g iv e n s y s te m of c o o rd in a te s a r e r e s p e c tiv e ly p ro p o r tio n a l to e a c h o th e r, th e n th e y a r e a ls o p ro p o r tio n a l in any o th e r s y s te m of c o o rd in a te ? (such v e c to r s a r e c a lle d p a r a lle l) . 24*. T h e s e t of q u a n titie s e-[k i is g iv en in a ll c a r te s ia n s y s ­ te m s of c o o rd in a te s and h a s th e follo w in g p ro p e rty : tr a n s p o s itio n of any tw o s u b s c r ip ts of e ^ k i g iv e s r i s e to a ch an g e of sig n , and e 123 = Show th a t e ^ k i is a p s e u d o te n s o r of ra n k 3. 25. Show th a t th e c o m p o n e n ts of a s k e w - s y m m e tr ic te n s o r of ra n k 2 tr a n s f o r m on r o ta tio n a s th e c o m p o n e n ts of a v e c to r . 26. W rite dow n e x p r e s s io n s fo r th e c o m p o n e n ts of th e v e c to r p ro d u c t of tw o v e c to r s an d fo r th e c u r l of a v e c to r in t e r m s of th e te n s o r &ik iD e te rm in e how th e s e q u a n titie s t r a n s f o r m on r o t a ­ tio n and r e f le c tio n . 27. P r o v e th e follo w in g e q u a tio n s e l k l e lmn

28. v e c to r fo rm :

e i k l e klm

^ Im•

R e - w r ite th e fo llo w in g e x p r e s s io n s in an in v a ria n t

e.in I eIrs e,Imp esip a n a r bm c t-

e,Inl

. dro!b krs e,Imp estp n kJb\c i t m

29. S h o w th a t T ^ k a i b k - T i k ak b i = 2c»>*(a x b ) w h e re is an a r b i t r a r y te n s o r of ra n k 2, a and b a r e v e c to r s , and u is a v e c to r e q u iv a le n t to th e s k e w - s y m m e tr ic p a r t of T 30. E x p r e s s th e p ro d u c t [ a * ( b x c ) ] [ a '* ( b 'x c ') ] in th e fo rm of a s u m of t e r m s c o n ta in in g only s c a l a r p ro d u c ts of v e c to r s . H int: U se th e th e o r e m on th e m u ltip lic a tio n of d e t e r ­ m in a n ts , o r th e p s e u d o te n s o r of ra n k 3 ( c f . P r o b le m 24). 31*. Show th a t: (1) th e only v e c to r w h o se c o m p o n e n ts a r e id e n tic a l in a ll s y s te m s of c o o r d in a te s is th e z e r o v e c to r , (2) any t e n s o r of ra n k 2 w h o se c o m p o n e n ts a r e id e n tic a l in a ll s y s te m s of c o o r d in a te s is p r o p o r tio n a l to 6 ^ , (3) is a te n s o r of ra n k 3, and (4) (8i k 5 i m + 8 i m5k l + 8km) is a t e n s o r of ra n k 4. 32*. S uppose th a t n is a u n it v e c to r w h ic h is su ch th a t it is e q u a lly lik e ly to lie a lo n g any d ir e c tio n in s p a c e . F in d th e

7

VECTOR AND TENSOR CALCULUS

a v e r a g e v a lu e s of its c o m p o n en ts and of th e p ro d u c ts rT[, n^ n^ , n i n ^ n i , ni?ij^ninm b y u s in g t h e i r tr a n s f o r m a tio n p r o p e r tie s r a t h e r th a n by th e d ir e c t e v a lu a tio n of th e c o rre s p o n d in g in te g r a ls . 33. F in d th e a v e r a g e s o v e r a ll d ire c tio n s of th e fo llo w in g e x p r e s s io n s : (a*n)2, (a*n)(b.n), (a-n)n, (a x n )2, (a x n ) .( b x n), (a-n) (b*n) (c-n) ( d-n) w h e re n is a u n it v e c to r w hich is su c h th a t it is e q u a lly lik e ly to lie alo n g any d ir e c tio n an d a , b, c , and d a r e c o n s ta n t v e c to r s . H int: U se th e r e s u lts of th e p re c e d in g p ro b le m . 34. O b tain a ll th e p o s s ib le in d e p en d e n t in v a ria n t fo r m s in v o lv in g th e p o la r v e c to r s n, n ', and th e p s e u d o v e c to r 1. 35. F in d th e in d e p e n d e n t p s e u d o s c a la r s w h ich c a n b e c o n ­ s tr u c te d fro m (1) tw o p o la r v e c to r s n, n ' and one p s e u d o v e c to r 1, and (2) th r e e p o la r v e c to r s n l5 n 2, n 3. 2.

V e c to r a n a ly s is

In a n a r b i t r a r y o rth o g o n a l s y s te m of c o o rd in a te s q\, q 2, cos a, y = r sin 9 sin a, z = r cos 9; hr — 1, h^ — r, /za = rsin&; eg

gradcp = er ^ - + - ^ divA = - ^ r ( / - M r)(curl A ). = — )—o Jr r sin 9 (curl A)#

_

1 _ ^ (^ s in O )-

r sin

1 dA„ r sin 9 da

(1.18)

[ ± ( A a sin f t ) - - ? ] ;

1 dAr r sin 9 da

1 d (r A ») r

.

r sin a da

dr

1 3 M e .) . r

dr dAr i r ao ’

’

i

l

a2?

. d /> 6 I - in & dr \ dr 1 * r 2 sin #a# I S1 ^ a » J ' r 2sin 29 aa 2

In th e c y lin d r ic a l s y s te m of c o o r d in a te s w e h a v e ;c = r c o s a , y — r sin a, z — z\

hr =

1. h a = r, hz = 1;

dA, i a 1 dAa div A — T dF { r A ') + T ~ d i + ~dF (curl A ), =

i» \

/

(curl A )2 =

,

dA

7

dAr

!(curl A ). = ~ r -

dA z .

(1.19)

!

1 ^