Mathematical Methods for Physicists. Answers to Miscellaneous Problems

23 Pages·2016·1.03 MB·English

by  Arfken H.J.

Answers to Miscellaneous Problems
MATHEMATICAL
METHODS FOR
PHYSICISTS
FIFTH EDITION
George B. Artken
lvlianu University
Oxford. Ohio
Hans J. Weber
Unil·ersity ofl'irginitl
Clmrloltesl•ille, Virginia
Amsterdnru Bos1oo Lmdon New York Oxford Pans San Diego
San Francisco Sing:1por(: Sydoey Tol.·yo
CHAPTER 1
1.1.1 UC=A+H. D=A- Il,thl.,,A=lC+D)/2, ll=(C-D)/2.
1.1.2 A,o, = A1 =A:= I.
Scni,,r l!dito1~ Mi!lbcm:llics B:u·b:n·al-lullanll 1.1.3 A,= Ay = ../2/2.
Senior PrnJcL1 Mmmg<!r Julm E!il<.!l'll~ 1.1.6 Fmm A= Uwt.: gctl'ortbccomponcnts A1 = B1, i = I, '2,3.
CCP<onlwmnrtipc)nr.. ~;i,l~lIoJn:1 n GOK~ianprkyl( )mn! ; hnU,nglL.:1t~,lgli:MlnmclJ. ;~ P Svot.,l uUtidon. s 1.1.7 Asloil ckglbc tP P2th4=e -(M0>Pl,u 3It i=,oOn () 2tPo4, 0 P=, - i(-22P), 0-I ,= -(12(1, ).- i-n1 ,1t0h,)0e t=)e-x((tI- .N 1Io.,t 0e- 2,th2):w) ==. l (hP2cn-,: -,o-pIPp, o,-s.2iute~)
rar.dld, as arc
Thi~ book i~· print,_.d un ocid-rru: IJafX!f' 6 Pi-P1 = (2.-1.-2) := P4- P1= (2.0, -2)-(0, 1,0);
;uxl for Ps = Pt + IPJ-1'2l = <0. -2, 2),
CCooppyyrriigghhtt r:e!n0e0w1,e dIW 1959. 51 9b8y5 G, e1o97rg0e, A19rll6\<7!,1 1E l~cvicrScicnce(USA). p~-Pt = (0, -2, 2)- (-1.0,2) = (1,-2,0) = PJ-P2.
P5- P3 = (-1, -1,2) = Pt - 1'2.
Allright~lc.<;c!'ved.
No p;~rl ol' thi~ pulllicahon may be l'<!f!ludtiCL'll or lr:.JnMmttcd ill lillY fmm or by any 1.1.8 The lliangle sille~> arc t;ivcn hy
mc:ms, ck."t.1runic or rtlCch.'lllM..~o~l, indnding phntncopy. ll."Cordmg, or :my lnfol'll'IO\Iin:n AU=U- A. nc- c-n, CA=A- C
l>lor:a,gc :tnd rttrk:v:tJ sy~ without· 1)1.,.1\JI!<>.,ion in writing from tllC pobiV..hcr.
w;th All+ llC + CA =(II- A)+ (C - II)+ (A- C)= 0.
PRLc"n(nttil!('$.,o1;sK fno;r lpk.':]:Xanl1li'st.m<:1connt ,t oH .na1n.1:0k1c.1 c1o, pIniccl~; c6f2 :mny S pCal1r tH oNflbhulr: wDnrir\k'e ,~ hOorul:lmd dho:, nF~;l~oirkX!l.:.~l lO: 1.1.11 lfv/ = v1-v., ~ = r,-rt,th..:nV; = fi,J(I'J-rJ) = flol,.
32R87 -6777.
1.2.1 (a) Using Eq. ( 1.9) gel
AAclrlnii.!hI.J:!mIiiil:lt:Por~{,E.;iv t:l'il'l'St·it•llr''' (A, cos<P+ A1,sln~)2 +(-A1-sinVJ+ Ayco~:~)2 = A~+A~.
5ht2tSp :BJ/w Swtwn.:coic. ;Suulictem itL<l.fXiC)(.)',l .S';im~l mD iego. C.•liformn 92101-4495, USA hy multiplying oul 0111d using cos ~2 + sin ~2 = I.
Acatlcm.c Pu. ". "' (b) Dcct)l}lpO.~in,g it = X L'O!i VJ + :Y sin VJ into c•'\11C.~i;m L,)lnponcuts,
84 '11\L~Ih;~kJ':;: Ro.'1ll, London WC IX 8RI{, UK wchuveAcosa' a A,• = A-Xcm;rp+A·:f!.inrp =A, CO!.fP+
hup:l/www. ." 'l:iltlemicprc.~.l'Ofll Ay,;inrp = At:osacosrp-t-AsinasinlfJ = Ac<m(a-ljJ)pnwing
thala' =a -ljJ.
A<.::Klcmic Pres.~
200 Wlk:clcr RonU, Hln lingtcm. Mwt-...chusctl~ 01803, USA 1.3.:! Front Ex. 1.2.1 we IMW the cmuponeutllccompo~!tion X' ":.,iL'lJ~ Vl +
www.ac;•dcmicprc:.:sbooli:S".com ~;~;.~-~~~·;~~~~1~~;; ~:l ~01:r~1i~1;~ .~~~~c0~:~ ~~~~~ ~e~~ !i;
c.o;-
lntcrnutiollnl Stoll!dard Book Number: 0-12...059827~2 ptuvillcd ~ -:F 0, tr /2.
PR.INTEO IN lliE UNITEI) STATES OF AMERICA 1.3.3 (a) TilC surface is :1 plane pas.o.;it'lg through the tip of u nnd pt.'l-pcn~
020)040506 98765 4 32 tliculll.rtna.
ANSWERS TO MISCELLANEOUS PROBLEMS CHAPTER 1
(b) Tite ~llrlilce is 'it spl"ll!n: lmving il a!<. a dmmcll!r; v·-:1) -1' ~ 15.8 (<~J r-Ar=A-r.
(1· - ll/2)2 - a2/4 = 0. (b) i-A, = -i·-li X (i-X A)l = 0.
1.4.1 A-n=-3(l.AXU=-2X+28f-18Z. 1.5.9 '[be triple sc.:dar proJuct A-n x Cis the volume ~umed hy the
1.4.2 A2 =A!= (U - C)2 = 82 +C!-28Cco:.;<f~.Q. VL'.Ct!.JO,.
1.4.3 0= <A +n) x {A +II)=A xU-1-n x A. 1-'i.HI A-UxC= -120,
11..44..57 {1A' .mxd lQl)!N C= :mA!tipB-.1u-~;;ainlk2l(: AR. Ii1.o:JJ :=l L-Arp2cnBdi2c(ul1:-~rlcoh~ol(.hA .1 •f .~n)l l= Q .A 1B2- cRA XXX c((IAcI XXX CAll))) === -32664i0iX+--844S80YY.-+Y +61225ii0.. i.
(A·U)2. 1.5.11 D-{ll x C)=ciA·CU x C).cu:.
1.4.9 A= U x V = 3}- 37., 1.5.1'2 (A X n). (C X DJ =\(A XU) X Cl· D= I(A-C)U- (U· C)Aj-D
A (nonn:tli;o~J) = ±<I/.J2HY+ Z). ~(A· C)(ll. D)- (A· D)(II·C).
1.4.11 ThcfiidcsarcC- A= (5. 2, 8)-(2, 1.5) = (3. 1,3) :mdn- A~
(4, H, 2)- (2, 1.5) = (2. 7. -3i).l Twic~thcarcaisgivi!nbyth~.:ircross 1.6.1 (:1) -3(14)-512CX+2.f+3i). (b) 3/196.
JWOllllcl (c) -l/f14)1f2. -2/(14)1/l, -3/04)1/l,
2cl = l32x 7 y1 -33 'Z (-24, 15, 19). 1.6.4 dFFCr=.t +F(d•·t+) -drF, (tr+,td) t=) -(dFCr-r.Vt)) =F+F hW·f+ddt. r,r+dt)-l?(l',t+dt)+
1.6.5 V{1.111) = vVu-t-uVv l'ollowt.fromthepnxluctnJicnrdillCrcnti:atiun.
Thus £1 = J24'i + 15.! + 19.!/2 = .J'i'i62J2 = 17.04. (;a) SinceV r = ~V"+*Vv = 0, VII :tnd Vv:trepm~tllclsutll:ll
1.4.14 LBAC = 45°, .trc BC = nf3 •o~dimn• • (VII) X (V11) = 0, .md vice \'Cl'S<I.
LCBA= 125,3°, arcCA=n/2radim11>,
LACB = 35.3°, ore AB = n/4nkli:ms. (b) If (VII) x (Vv) = 0, the two-dim...'f'l.~ion;al vuhmx: SJro~mxxl by
1.4.15 Cross A- n-c = UimoA IOt,>d. -A XC= A X n.orCfiiii{J ::11: V11 :md Vu. ::~lso givoo by the boobian
Bsiny.l..'iC.
1.4.16 .,=i+2Y+47~
I.S.I Volume= 1llcnr1•
LS.4 (a) A · U x C = 0. 1\ i:<; d~e pl:me of B and C. The p:amlldpipcd has vamshcs.
zero height .tlXJvl! lhe BC pl:me :u"Kl therclbrl! J:cro volume.
1.7.1 (:.t) ~~r~·~~: ~~;~;~:!~i:~~-~i~1~~;;~~·v:~-~et
{b) AxtnxC)=-X+Y+2i.
1.5.5 Tmhre2 1BwA-Ci--C-AwBi') .r ule Bivc.~ I~= /Jir x ((c) x 1') = mlr1w- I'. wrl = (b) ~~~~~i!ntiating r :~bovc we go.:t r = -c.?,·c:Kcoswt +Y sinMl =
1.5.6 T = ~~~~ [r2tv1- (l·.w>2]follow~rromfAxll).! = A2B2-(A·U)2. 1.7.5 The product rule of clifJCn:ntintitlll iu Cl11ljtlnclion with (n x h)
1.5.7 U~ing n x (b x c) = n · cb- il· he, etc. gd lu. cb- n. hcl+ ~~. (h x c), clc. give:)
lh ·nc-h · ca) + (c -1m- c -ubJ = 0. V . (11 X b) = b · (V X U)- u · {V X h).
ANSWERS TO MSCEUANEOUS PROBLEMS QW>TEA1
1.7.6 '9. (rr-1) = 3r-1- 3r--.:Jr-\7,-= :lr-1- 3r-=' = 0 1.10.1 (:~) ISk. (h) 15k. (c) 15k.
ror r > 0. Since. E ....... \?(ljr). '9 E ..... '92(1/r) = -4n"li(•·). At l.lfl.2 (:t)-T!. (b)+n.
r = 0 t~ ch:111;1: g~11Cr.tl...:t. :~ sint;!ul.1rity. 1.10.4 U o.
1.8.1 11~~.:: dl3in rulr.: gi\'C:!I -b, = *t if;, i.e. the gr:w.licnllr.tn'ifoni\S like <1 1.10.5 I.
co\'m'i:mt vector.
1.8.J lr\7 x A= O,thcn\7 ·(Axr) = r-'9 xA-A-('9 xr) =0-0=0. 1.11.1 Fur a cun.'4anl wdUI' ~•. its tliv~"gl..."f'K.:I: is h,".n.l. U~>mr, C:~t~'i.<:.' tlll:urcm
wehaVl!
1Jl.4 Fmmv = w x I'WC(!:Cl '9. (w x rl--w (\1 x r) = IJ.
1.&.7 Llf:L = = - 1- (irx x# V;- ,tyhej;n ) t.h e( icnk utcnrimt-; iomfm ht) .r oentcn. of the C.1'0SS flll>t.lu<:t gives where S is the ~dU = stl.{nv f. Vtc c:· uofd tTh l=! filnli·t el {sv dolam, nc V. AI> n f. 0 is
I.R.9 la·l •. b·l.l=aJILJ,Lklh~;=iE:JI:IliJhkLJ=i(nxb) L. <~rhilmry. J~· da = 0 foiJow~.
IJLIJ Apply lhelmc-c:lh l'lllc lo g~l 1.11.2 FWill V ·I'= 3 in C.m:-.o;"tiKXll·c:m we have
A x (V x A) • ~ V(A1)- (A. V)A. ]f, , V-niT=J lfv tfT=3V=1.\• ··cla.
11tc fnc.1n1· l/2tx:e~.no;; hcc:ml'lc Vopct.1lcs only oct one A when: V is the volume cncfoSt!d by tire clo.~c:d surl~tcc S
1.8.14 V(A-n )( r) =V(A XU) ··= (cA X llh~---) =A X n. 1.11-1 Ctl\1\!1" the clc)';cll ~>Uifacc by sm:.tll (in g.cncm.l CUI'VI."ll) adj:k:cnt rect
1.8.1b FAr)= ~tJ! Tda1na"glC lle=t sS tSoE1k oi.w! SJh"yo t shAel. ud·c rec.nlul lm =gii Cv0dr.>. :n lJxc:\ec.:( :aV~rue s xcf oaArUn) " -l•Xi ntlfe a b yin= tf cogcLnr:·: 1tlh if n'cle; a."(n VcL e;>l < ecAa:tc)ch h.·
ocbc•:
1.9.2 From the BAC-CAB n1k: lttr n u·ipfc \'Ct.1.0t' product 1.11.4 G.-tuss'thcoo:mgivcsO= fv V · VlJdT = fs VlJ -cia.
v X (V )( V) = V(V. V)- (V-VJV, 1.11.5 Awlyth.!eotY~>fln•.ling ddimlion ofthcdivcrgcnt:l!lO the Vl...'t1.01' Vrp
Since V acts 0111hc vcdor fickl V. ;u'tl a·~: b = h n. c, we have to iu~.:m1junction with the mean v:lluc theorem for thl! \-olumc intt.-.grnl.
write v lim.. 1.11.7 ApplyG.-aus.-;'llw.:un:nnoV-li{JE) = Vc;o-E+c;oV-E= -Ez+£01rpp.
1.9.3 v x t~V~> = v~ x V~+lfJV x (VfPJ =0+0 =0. where fs-+0>) c;oE · da = 0.
1.9.7 Using Excrd....: 1.7.5 V · (Vu x Vu) = (Vv) (V x Vu) ).11.9 U~ingG,tw;s"ti1L•on:m we h:tve
(Vu) · (V x Vv) =0- 0=0. fn
1.9.R V2fP=V·Vrp=0,:nu.JVxV1fJ=0. Utit"= /HJ- (VxA)c/t:
1.9.9 (.~r vxl V +) r. (. rv x- \73)• ·=. vr--(V,.z ixf .r( 1=· x ,.V:~:v))z =- r2 -rI~V _., V.zr~-.V (V ·1·)) = =- V-(H xA)tfT+ / <V x H)-A
1.9.11 Since V x VlJ = O,A x A= 0, mxl p x A1Jt = -Ax p''' =-1 WxAidu+J.I Ad"C,
i(V x A)l/1 witJt V K A= R we h:wc (p-eA) x (J>- r·A)lft = .\~oo
-e(px A+A x I,)V! = -e(-A x p- iV x A+A xp)lJ = ieUlJt. \\1}CI'I! tltc Sl.nf.1cc inl,.1:\flll vmtishes.
-
ANSWERS TO MISCELLANE<XJS PROBLEMS CHAPTER 1
1.11.10 f(v:Cll-ll't:V)t!f: = j(vV · (pV)u-uV · (ftV)v)dT = 1.14.1 Gaus.s'l:lw is Mill given by E:q. 1.163.
ff (pp(Vv V·~(fvlV -u-uIVIV2vtl)) +ti T(v +V t/t( -uuVtVtu-) t·t VVpu))d. TV p=d T = 1.14.2 (:1) /11 V -Etf"C = fsE·Ju = ~ f ptfT = Q/£•• byG:.tuss"thc""
JV-(JI(ttVu - uVtJ)Jtlt: = j p(uV/1-ttVv)·da. remand Mo~xwcll-Gans.•;" C<J
(b) Totke E = E(r)i m)() wricc Gau~t.\' l:tw in r.tlli:ll fonn
1.12.5 Usmg Stnkc.-:"thcurcm f01· a loop cnclu!:ing Ih e ... -um.·nt. we gct q1i(r)j(41f£o) = V. E = ;lr~;fr) = ~£ + ~ = ~~­
J<v) ( f lntc~;r:lling E'jE = -2/r yickb; In£= -2hu· + hH:,1.c. E =
11)-tla = J.l-du =I= H-dr. cjr2• Variation of the COill~anl gives£'=;;= tJS(r)/(4:rrr2), ot·
c = q/{4neol.
lw~hmhedc Crthtli:ei TL ,hC.cci .l.: ll.oi to'Sdt scuatnf: ht:ec cl iimniCitJe;.dI'l ltlo i sth oev cerro sths es e:mcl.:i:ot uc nocfl tohoec dw ibny! cLahrcr yloiongp 1.14.4 i-Ul~D;Ve/ 2iVlAt +·=A J/ A -==.1 f0vo,l lnXow ~(=V. /-Xl. HA, >DI ~A=- =e Ec vwvit.h A aE-/ivlt2 A= l/0t-l -in= VJ. xs o1 -1l h=at
1.12.7 The prnur i" the !':uue ''" for Ex. 1.11.3. except for A -+ V
1.12.8 Zem. 1.15.1 The mean value theorem gives lim,.-~ J f(J. )r'l11(.t") dx =
1.12.9 TillhtCis! !If';olilClodw IC"I 'lllil· mc:nm cl.n:ctclsg fmutri :o1 nc lbnys cpdm l1oso p!J 1ining V [rum u to 1.1. The -lim~u~-o~oIlIl ~J~ {:l7n~· 11 f(x)tfx = lim,._00 *.f(~11) = /(0), as
LI2.1V SUtoo;ck ct.h..;"e tlilld.•colrlcliutyl t(oJ f2 E Jx..., ,1v.1u2.. 9d,u i .=e. ..1( (;1V1V(uvv)--v(/Vl.u =). d0J... :=ll ldf s. tVpp l)y( 1.15.2 lfi(ml,;Ju-)otol. fJu fe(-x'1)1~tf(.rx =) dlxi m=, -lin00t1 1f ...(.. 0~0111)1 =,{; ~f {.Of()x. }:ets-O'u ·~tl t"l; ,= : Sl i~m-,-;.00
(11Vv- vVu) ·da = 2.{.-;CVu >< Vv) -du. 1.15.6 J::f(x)li(tl(.\"-.q))tl-r = ~ J~00f((y *+ ·Yt )/!1)8(y)t/y =
1.13.2 fll(r} = ~· tt :S r < 00, ~~(~) = ~j(xt) = f~j(x)B(.t -xl)
4S.J.
fll(r)= ~[~- 0~ r ~1.1. 1.15.9 lntL"tPling by fl"rls we fttl4.1.i-:x,C(.t")/(x)tb =
1.13.4 ·n.c gr.wit.llion:•l accdcr.Mton inlbc ::--t.lin.-ctiuu rd:Miw to the E:•rlh"s -f::, /'V).\l(J.)flx =-J'(O).
!."Urf~tce i" -,:~~)· + ~ ..... :t:f£¥ forO~:;:<< R. Tlnc-;.Lhc fO«:C 1.15.11 r'l(r- r'}= -1-8(qt-lJ;~(q2-(j~)J)(tll-t/t).
F~ = 2.1:~. lllC fot'CC F, = -x~ """ -t"~ ••m d F,, = ''~''21!J •
-y~..., -y~.lntcgraling F = -VV,yidll" thcpoccnti~11 1.15.18 (:.t)
V = ~(:.:!-!xl-!y2) = WP<-2_,.2) = ~P2(cos0).
*(ll
1.13.7 A= x 1") forL"OIU>I.mt Uimpliesn = V )(A=
~nv: r·-~n-Vr = ti-~)n. (b)
1.13.10 Fron,thc Maxwdl-F:wnday cq. V x E + Wf = 0 = V )( (E+ ~)
wcgecE+~=-Vq;. L
1.13.11 qW(i-thV tfh!e - r~~1;Su-l1+ of" E xx . (1V.1 3x .1A0) w) e= ht:t;v(c- FV f=l l-q(~E ++v V ><( All.) v=) ) l~~iug 8{x -I)=I-~f+" 1~ =1 !1111 cosmtcosm.x,
the ch:•i11 n•lc for~ = ¥, + %. VA with v = ;¥,: :uxllhc hac-c~1 c1111 =- li(t)cosmtdt = -I , 111 = 0, 1, ...
ntlc IOrv x (V x A}= V(v· A)- (v. V)A. 7t -1T 1!
ANSWERS TO MISCEl...l..Al'I:OU5 pFtOBlEMS CHAPTEA2
21.2 (a) UJlUfl ,;qtuuiOJ;. c:ach coordio:.1le diiT~.. . w.,lial
d.\= drsmHco.~¢+rco:.Hcm¢ciH-rsm01'ii11¢d¢.
cl)' = tlrsinfl:-infP +rcostl:-infPciO +rsin0cO!-i¢tltp,
clz = clrco~f-! - rsini.Jtff)
.md then fiumming tlll.!ir MJllarcs, we notice th:.1t all cross lcnns
(c) e:m~.:cl so tb:tl we uhtain
j(J)=~+fbmCOf>Jill. rrbm=L:f{l}cosl111•
- nr=l l·kncchr = f,IJo =r, It'll= rsinO.
and (hJ give (h) F'fom the 1:.1... 1. CCJU:tliOII uf (:tJ we 1-cac:l otT tfl1 = d1~ tl\";! = rt!H,
d.rJ = rsinOdfP. m·from d1· iu (:1) th;at
~~ f(t}Lll(:x - f) clx L: iiJJfr = (sinH cos fP, sin(·! ~in f/J, cos f~) = i-,
= ~ j(..t)tfx+,~,..,co.~ml }'(..l.)cosm;ttf.v ihJi j = r(co!>OCObfP.COsH hill¢, -sinl'l) = dJ-.
uo;tlf f
= + ll,h .. ,:rroo:r.mt = ((/). ~ =r(-MI"tflliinJP.sinHcosfP,O) =rsinflip.
- m=l
2.1.3 ·n\C 11-, v-, z-syl>1cm is l~.:ft-h:uuk:d.
1.16.1 11V/rJ 1uiJR! =io sb atV.ln-'.lioln/1 ci dr=r ob ty:Vl hinO. tVnc:gll/l1n: n.i iltoh ·en Vn. 1 JV! 1i/.Js gifii v!e'Oni conno itlhlael .b Soiunncde: uvy fxn m{1l1 /1w =hi c1h1, 2.1.5 Vf<T~olumc (tfo•·f =th eI 3:-;c ~odmep;o;n.c nLth)c osfu lt'hlielc ccr ocsles mperondt mis. 1 . g iven hy tl\C .1bsuh1te
1.162 (IfV Vx21I'' )== V-(VV. .t hP~.l:n- PV =21' wJc; gf c~t Vt l= r-.V U·(~Vin g-I 'tHhe- Vtd exn (t'iViY > < VP)x. ( -ilithJ·xt -ilicl;2r ) 3 c!fJitft;-.,= (i-illtxJt1 -iiJJqx-.2,= -ii-Jlr.\"n"-t -ii-ll.q=\. .1 -, ) diJttiiJ.-, ·
whK:h is Ilk.: stmxt:ml Jacobian. i.e. 1~ 2 x 2llctc:nnin:ml sp.·nu\Cd by
lhi:Ve<.10f"S~,i=l,2
CHAPTEA2 2.1.6 f·romd.x2 = tlx1,tfx11 = c2t!t2-cf..:Zwchavcgoo = +l • .t:tJ =-I=
Sl2 =gJJ :u'W.Ig;1 =0, i#.i-
2.1.1 Fmrn
'2.2.1 LFn;11 1cnt,1 1t(hJ;e .ucuolm :1 psoinnteiln:ltl ' dcexfpinn.i:tsi.~oino n (fpomr jhe.c gtieotn ) a = L1 c.i1n . iJ1
we knuw that tllC vL-clon; ~ = h;i!1 m·c pointing in the dif\.:·dkm u-h=LiJ;·•)1:t·fub·c.ij=:L> c.i,h iL='L,clq1bq1
of incro!ol.'>in~ q;, i.e.~ = ij;. Sln~c I!IJ = ~ . ~· ot1hogOfWlity i} I I
<clt · i"IJ = ~1)) impliCSRij = 0 li.w/ ::#:. j. tl:iing Or11t<Jt!.on:dily. i.e. [z1 • iiJ = ~iJ-
,I
10 ANSWERS TO MISCELLANEOUS PROBLEMS CHAPTEA2 11
(:1) fmm Eq. (2.171 with Ct = iJt and cet)t = I. (Ct)2 = 0 = (it}.l ail,l : =0,
we get V •ct = _ l_ iH112h;J)-
htii"!.IJJ ,ltft *oiJio"-=f~ J .
(hi FHinl Eq. (2.221 with h'!. \12 -+ O.hJ '-':1. -+ U, WI! gt.1 V x ~~ =
.!._ [c~2...~ -t.,_!_ 01/q] =~:-inH,
"' -hJ 41tf1 112 illf.! •
~ = -hinH-Ocootl.
!.4.3 From p = (.f, v) = p(COSqJ,sinqJ) we lt<1vc p = (cos~,~;in9'l. ;J<p
OiiTcn.:Jlti;ning jj2 = I we see Llt:.ll: p. ~ = 0. l-Ienee ~ .._ ~-In (b) With V b"'VCU hy
component"~ = (-fiinqJ,COsrp) = i;. Simil,uiy we uhtain ~ =
-P. Lllld th:.ll :111 olhca·lir.~l derivatives ol' the cin:ul:tr cylindrical mlil
vcctm·s wilh rc~pccl lo the cin,,_•lar cylimlric:~l C{)(Hl.lin:llcs v:mish. the ,illcrnall! derivation of the L:ipl;tcian i" given by dolling this V
2.4.5 {ol) r= (x,_v,r.) =(.r.y)+z:Z- PP+zZ into itself. In conjum1.ion with tlu,: dcriv;ttives nflhe unit vectors
nbovc thi" gives
(b) From ~~~J. (2.32) we have V. I'= ~~ + f = 2 + 1 = 3.
FromEq.(2.:\4)with v, = p, v., =0. ll: = zwcgct Vxr =0,
2.4.13 (:1) F=c$*.
(h)
(c)
Notcllll.M, with
(~) ilVnb .l:xi n( liFfllr f i;_~!lil•ln.K l T:fu httylect fSliit•a.'t~-c11ct<lt1:1iu· optooh)t cic~no alrtii~a.g.l.i 'nte/d.1c AtJ=ico u(p/c)ni:lsiVn tnc.'ol lfttr oo,;mni•el tc r!hkmc-mvo.r-eilguniccndir.O ctliiinigO rl.siIn 2oiiaii ( :;;inH""aiii i ) =t.t7.ni-J 0iJ0 +-;I2 iHil.2f. !.'
2.4.16 lnh:gr.lling: v = iPfX<J we find fv -dr = p2w J thp = 'brwp2• l-Ienee we get Ih e staJlllanln..-.;ult usmg Ex. 2.5.18 fot: the Jow.liaiJmrt.
:fv-dl{(1fp2) =2tv= V xvb. 2.5.3 (a) v=(pwl·sinH
2.5.2 (a) From Excrcisl! 2.1.2. i.e. dilk,"L"flliating i-2 = I WI! get <hl V xv=Z<:v.
2.5.5 Resolving the u11it vccton ul" spl11..:ric~tl polar coordinates into c,utcshm
*/ihj;· = (sioNC("I!.fj!, ~illfJSUlf/J, cosf/) = i-, cm:o1mtrpi~o.n ·cnn1lcs iwnvaesr :saec ciso mLhpcl il:r-;:hmcsdp ions cE ~tm. t2lr.5ix.,1 i .ien.v olving an mthogonal
~ = r(cosHcosq;, cosHsinq;, -sinfi) = rii, X= f!.itl(-JcosVJ+ilco,.,r~cosq;-i;sinq;,
jr = i-sino,.,inq; +BcosH~>inq;+i/Jcosq;,
= r( -sintl :-:in q;, si11 (:1 cos q;, 0) = r sm HijJ. Z=fcrn;(J-6sit1H.
12 ANSWERS TO MISCElLANEOUS PROBLEMS CHAPTER2 13
2.5.7 lin. vlf!...., lfr. 2.6.5 111e IOur-damcn:-;~onat fourth-rank R•em:um-Chrl:-.!olfel emv:~ture tell
2.5.t> {ll) A-Vr= Ax~ih: -1-AiyJy~ -1-A~cJ~:. =A ss-oyRmr komiflme gt,re ryne deourlac eltl !1r>ee tllla1itnei:.v-t. citm pyu:.1 1i Rrssi cktc/omm 6 1 hvd;a lpsl au4ic-r~l oe=.fI Cim2h5,1 i6i.ce ec. so6,m 2R p=tok lnm3e6 n =ctso .-m TpRho;ekn memlm al=si .
Theyc.m be thought of .1s :16 x 6 matrix. The !.}llllln<!lry under exchange
of 1)air indkcs, R;klm = Rtmik-reduc..::-.thi:-. nmLI'ix Lo 6· 7/2 = 21c om
aUyr =Y. ~=Z. ponent~. The Bianchi identity, R;ldm + Riii!Jk + R;mkl = 0, redLK:I.l\lhe
il<. intlcpeudcnl cmnponcnls to 20 bcc:msc it n::IW<.:SI'Jllls one com:tr;ainL
(h) U~ing iffi = ii. ~ = sin HfjJ ;:uul V in polar coonlinntcs from mNaokte..: ttlhmet ,ti a-u:..p1 oinnd uexsi nCg4 Ltlh:lelt op .et..rcmrou rtoaltli oown e~do yhnyun tehtel iOel~h eo1n· ien dciacne sa wlwhaicyhs
Ex. 2.5.2 fb} we: get l.lrC all difiCrcnt from each other.
A·Vr=A-i-~-t-A jj!!!:_+A-~~ 2.6.6 Zero. E:.ch compm1enl h:as at least nne repeated index ;md is therefore
ilr iJO sinfJiJf/J
= A,.i·-1-Aoii-t-AVJijJ =A. 2.7.2 The conu·action of two indices removes two in(lkcs, while the
2.5.18 From(a) -;I 'idc/; ,.-., = ;d; ;+-;2: w~.:gc~(c).;mdv•. ccvcr:a. Fro•uthem. ncr lle1ivative add~:; one, so tltat (11 + I) - 2 = 11 - I.
(;cd;;) :,r :m=d I'vJdic ;e + veIr sina . (bH)e wrec g(aL}- 1--I;I ;' -id;ddj-;r : ir[ ,=.,-d ---vI- ;-,;;c(;;/r; ;:) -+] . d d(b;-)i --1I;- Jd-I;, :22;d ;1 ;r•V h,e(rn)cle. 22..K7. 3I TTUhheetr d-socvuabl)al eri s !p't;rhtomcdsuucnc:aatu l:ooun·fi J Kt2;h _e=; A !;.' oBzu.1rr -iv~se -,c, tsoVcra2sl .: uiJ·I. "T h=at K{;~i fa,!-. .,1- scvc)o,uidJ-1r,: .m=k
IC} cf11JI(r) + ~ d1J!(r). tensor follows from the quotient theorem.
dr2 ,. dr 2.8.2 Since K;J A Jk = B1k is :1!>ccond-mnk tensor the 4UoL1entthcorcm tel1s
25.20 (.l) VxF=O. r=:;:Pf2. U!:i that Klj is ::1 SeCOJld-mnk tCilSOI~
(h) :cFo Jn?s ·c drvA: a=ti v0c.. This suggests (hut docs not pmve) lh:ltthc fnrtc is 2.8.3 S(le'iotnm,c 'r-e)v t ehfcoelro pmlh~ :n.Atls'leot te-ini'4m-1Vt =icvc ct(lo(y·r1,, ·ts-hinewcqetu iiosJ Ita'i eiLsn otart eh!n'eotou.trr s-evcme;l cls:tauoy !r:>m .: mtdhc .tal hte e-( icwi!oJ/oc ar,d s(icn,:):aitltsac ras,
(c) Pokniial = P cos (1 f r2, dipole poJcntiul. iJ11c-i<l> = -ikl1ei<l> i~o :.1 four-vcclm:
25.25 (3} (a) follows hccausc V2 is a sum of its pl;mc pol:tr pa11 .md its 2.9.2 The generalization of the tot:tlly aniJsymmctl'ic Bfjk from three to
.L-parl. illl(l i is conMalll. 11 dimcnsiuns has n indices. l·l~.:ncc the gencmli.r.ed cross product
8fj~> ... A1BJ i~o an .mtisymmct•·ic tcnsorofrankn- 2 # I for 11 # 3.
(h) This dm.'l-i not work bcc:mse angular denv;ativcs of the poku·
unit vectors do not vnnh,il, and these enter into V2, a~ l!hown 2.9.3 {:1) li;; = l(notsummcd)fore;Lclti = 1,2,3.
in Ex. 2.5.2. (b) li1JBiJk = 0 because .~/j is symmetric in i, j while Bijk is
antisymmet1ic in 1, j.
2.6.2 IIOf Arm~ja t=io nB lf~ro imn othnaet ftrr;.u:uunee o tfo r acnfe .raermhi:t:mc,rtyh conn ded: ix1;1 e= :a xe1o(oxnyl)i,n :amtc L hlm~tm s­ (c) iF o.mr ed: :1_c;h, sf.o' i1t1h ;8tl;1 i'1 18= i f•jq. t oIn lt'lei'Jr cnhoaa1n~.gt.i.ncrgo, ple .amveds q( liglilvyc lo:>n it'Jw voa ltueer mfosr.
A··_ ilx; ilx1 All _ iiY; il.\·j Bl) _ B·· hence the f.actor 2.
IJ - ilxV il.>;~ afJ - ihg il..1.~ afJ - rJ· (d) ·1h:rc arc 6 pcrmut.ttions i, j, k of I, 2, 3 in EijkBiJk = 6.
14 ANSWERS TO MISCELLANEOUS PAOBLI::MS CHAPTERS 15
2.9A Given k implies p '# q fm· el''fk # o. For E!Jk -:f:. 0 rctjllirt--s either Ddimng
I~ /,=,l lJpi ,-1l11i'1,d, l~i}f ,J· = 'I· or; = r; and 1hcn j = p. lienee EtjkE,.qk = IlL"-kl = .r;~-ar;, = 4cl.l>X"11 (il1,.8fi•· + il~c,m-;J11x,. .. )
2.9.8 {UAsixn ~;,{ EU.• cx 2C.9).)4q :m=d EEppk~•;Jq (=E ;j-!,eB1;"C1"j)·A w1,e =f;t .."-t (Ji;p'~il/ -l5;q<"lJJ,)A,,BICj = 4~ll~ (i11•8/fv + ilvg1ta -lllll{1t.,)
=-A·8C,1-t-A-CBq. = (iJ1Uf~.:r + a,.J:1tk-iJ~;,~:~ .. ).
29.11 Eij = ( -01 0' ).If R= ( -sc~mlSfr;p? CslOlll\f'/'J;' ) i~arot:.tic.m.lhcn 2.10.15 .flij:k = il~o:.t:ij-r'/l.McrJ-~~~S·o
(_~;~; ~~:)(-~ ;,)(~::: -::)=(-~ ~)- = ilJ:MI}- ~~}<>J!crf! (ill.'lflk + tlkKfll-;Jil81A;)
2.9.12 If A~-= !e11,.Bu with Bu--B1;. then -~~; .. gcr/l (ilj}fJik + auw -iJfj.t:Jk)
2€mnkA~-= Em11kE/jk = (/jmllillj -lim)'~lli )Bij = B,m - B,wl = 28.,1,1. = ilt~gu-~ (ilo~1t + ilt.t:JI-iJ;~u.)
2.9.1;1 U!<:ilvlt\~ XExU.2)..9 (.C4 wXc Dh;}w l=! f:tJI•Emrri•A;BjCmD, -~ (ilJKik + iiJ:81J-UI~Jk) c 0.
2.1U.H r~2 =c t(-lirm llirn1J~- l=im -Jnli,,:11in)A2,fB.l JC111D,=A CU D-A-DU-<.:. TInh oisr dgeivr etos 0lin {=l gg~;{ m=:k ~{)m dj c+riv gc,c111ogv;t:Jtr i=an ltl-:yt1 1t1hge; ti.d eMntuitlyti gpl;y111ign1g11 Jt h=i s lbi{y.
rT2 = f~1 = l/1· r;3 = -sil)(lcosH g"i .mc.l usiug gi11 81111 = ~;:~, gives k~{ an.
r-:3 = r~, = lfr r{l = r}z =cote 2.10.16 We )1;.1\'C rrom the gcOO......OC Ct.jU.aCion tfu1 = -r::llu(l' t(xf1 .vxl frnm
2.10.9 Lei.\ he the proper time on a gOO<bic :tnd "''tsJ tlte vdociry or u (lllr.llldt.li."plucing the vekx:it)' 1i nu it: •~111-= -r'Ufl1t' tfxll.
ma.<;~; in fn.:e f.lll. "'il!n u~ sc;tL1t :!.10.17 11JCco\'arulllf derivative of a vector V1 by paro~lld trnn."rort is given
;t,l; (V·II)= 7,Iiy; ·II+ Vp dd"l.x\2t f by the limiting procLotlun: limd,1J_.u V1(t/+J•Il-VJ1,1fy l)+ 1.."; '- V"tlqi •
= (11, V0 ':~:J) 11"-y11f!:1,u"ul' = uflu11 (il11 Va- r£1, VJr) 2.11.3
involve.~ the cov;.ni:mt dcnv;tll\'e which 1':> a ft)lll'-vcctor hy the
~~~~:~\.~~~~~~~~~~~C~hat the lf~ or the gcodcsit_ L'fjll<lliOII f(w CHAPTER 3
2.1n.11 rJ2 = -p, rf2 = rit = 1/p 3.1.1 (a) -1. (h) -11. (c) 9.J2/2
2.10.13 .1.'11 =2.25 ~:12=0.60 .r;11=0.30 3.1.2 The dcll.:rmimmt or the cocrlicicuts i" equ;1l to 2. Thcrcl"m1! no
822 = 2.72 -'12~ = 0.56 ~J~ = 1.13 numrivi:1l :mlutiou exists.
2.10.14 From tim c'k:rivation of the geodcbic equ;ttion we know that 3.13 Given the pair or cqu:ttion.~
rf,.JI = 4.r:"11 (i1,,.1,'f1~ + il,.J:ua-iiH.I-:11,.). x+2y =3, 2.t +4y = 6.
ANSWERS TO MISCELLANEOUS PROBLEMS CHAPTERS 17
16
1
l<l) S~ifn et~h;e t hfien ;Ct OQI.f!ifCr idjeunstts lu.ly( tah er :s1ec.c.1oorn d2 .e qthuea tidocnt cdnifn!"icnla· .ufwl mof_ ll(ll<hll>!>C) mxl matri10. multiplic"'ion ylcl<.bo ( -'l1'1l 'l-1l11 ) ( -J:1':'!2,l/122 )
(h) u:!~xur. t•lidli~cten (1t:usw,hi.1sU7oIr.IcK2i.l"gO1d-~lC(hJ UnSuImCcI'rnalNt oornd Iclkt!~ rigllut-:I:1m:L1-1;K 1al<li.i.alk v:dain~fW\,,l.- by (h) (( ll--(tt-ttillhl1t)l2-/,1-I l-=-l lI2J"th{2h.t:l:).+- ifb'Ci'll 1l "b~22p-+ol"b2tll'bt:!t,l )s• iO tiZ+I Ji! ('b' -a/J ) .
(c) lll"tfliCCS to M>lvc x + 2_v = 3. Given,, • -" = (3- x)f2. 'Tlli~> 3.2.5 A fO\Cior(-l)can hi! pulled out of each fOW giving the (-J)>~ O\!Cnlll.
b: Ute g~:l'lt!IOll Mllutinn fo•: a.·hit"'l)' V'..ll~ of x. 3.2.7 A2=B2=C2=1
3.!.7 '11te G;tt1ss climin:tlion yields AB~BA~C
10x1 -!-9XX22--l!--28-tX:'lJ -+1- 43XX44 -1-1-- SXx55 + Hhti == 510 BCcA~~cACB~~AB.
IU.r1 + 23x4 + 44x~ - 60xc, = -5 3.2.8 II= 6.
16.14 +48x5- 30x(i = 15 3.2.9 Expanding the cmmnul:aton. we lin~l [A.[B, Cl! = A[ B. Cl -
48-t) -1- 498Xfi = 215 I B. CIA~ ABC- ACB- BCA+CBA,IB.IA.CII ~BAG­
- 11316X6 = -443R BCA- ACB +CAB, IC.IA. Bll ~ CAB- CBA- ABC+ BAG,
:md ~;ubh'aL1ing Lite lo:Jstlk)uhlc conumriHtrn· from the sl!coud yields
.sxXX-.o~.4J. t====h a (5((t2 - 1.X-515I 5\- : 11=+--o .46J .w2O902;.8l '.(1-1o.(\9 1c-S/-4.5x)86s/4. 45-4~88X5 !l)==i.1/o - 1400 6..-234 :\=91.2~ 22;..t0,,:"j J .) 4j=3l U7 -,= 0 .-3(5).99.6 6, 33..22..1132 tlLBhl;'ye;l l fd~lkii1 r1r,1=\e:lgc .0-0t1 l1ml=1j!.:, , u1shh1i;minkxck f Jeo=l ltlr hJi0ieti, >pB : luAki;c ,Gtuhtlb ieO.cm ~tnldSu a mCmls 'MAios.1 B L'&" mmIJCdp Iil'tIltyllilwo "f nhocsm:rr. nti.c >=d k.
x~ = (10 -X5-4x4- 8.0-9.\2)/10 = LE8. 3.2.14 E:~p:mtling in C.lltcsi.:m compotJclllS .tnd multiplying we get
3.2.1 W(Ir:i,t i"nfg hth,ek ),prB<XCb.K=.-1< Lm,a,t'ri1i1c'-crxlk )i.n( A1~B1)1C1 =o r( ~t(l}b~:r ,.C,lll!lt,lmllh..,.,mlitSl. c,A:kB) == n(a- h-n l):!(n+·hio) =-fua ]xaibbl.J + LJ#kn)otu;ht = n·h+iemcc;btat =
~.: f/~~h,:'"C'ttk := A(BC) =<I:,. ll;m(~ ''~c~~~:)),bccau..eprOO­ 3.2.19 lf:m (ltJCr:lturP(.'Oillllllll~ wilhJ" .tndJv. the x :md yl:omponcnts
ucl~ of n~;d and c(lmpbmmll'lcrh ar.: a.-.soctall\'t! the l~rcntlll.~~alll of:m :IIJb'Uio:armomeuiUmotll!l'ittOf,thcni[P.J~I = IP. J,Jr -J"J,~I
['ll; danpp.,.:l for all m.atrix dc.mcol. .. = J,~IP, Jrl +IP, J,l..l"-J,,jP,J,I- IP.JvJJ, = 0.
3.2.2 Mulliplyial& Otrl (A+ B)lA- B)= A2 + BA- AB- B2 = A2- 3.2.21 If A = ±i~. Ll'lCil n = TA j, uniquely solve~.! by the 1"1!:11 number
B'+IB.AI. T = ±B/A,wllCJ"I!A, B:tn:lbclcngth~ufthcD.)JT\!S(londingvectors..
3.2.3 tI.nz r't:e!,nt,)u)s: :f't tl."· 1m(La;tr,itxr 1c11lrc·~u~tr)c n+tcs,1 AJL =,, ,,(,a,,u,J':!.. nw) e=: hcatvlAe r<tI):;,, t+i;t,..'(:!C(AII'rI2n) /+· Il'f=ll =AT·IAJ/ iAs 2v asloid t,h saoi iAsl-l'n =' =U O-.SI1oA, =le t TlAl = -hBAY. =A (=T A-X=h )wyAiAtwho.iotuhtr
3.2.4 (n) (tit+ j/)t)-ltr2 + ih::.) = c11 -t/2 + i(h1 - 1}2) corrc.'lponds lusl!ufgcneralily. Then(~)=(~~)(~) lead;; to B
10 ( -~: ~;:)-( -~:~!:~) = (-o~'_=-,;~~ ~;:=::~).i.e. y = BJAmulT = (~ ~) is.tsolution. 8U1110is (~ ~) withanyx.z.
St(h<hIetn c+iolar irr/lcJy,t;, }p(ioCt Inh:!od l+clnh ci~f.> :cf2 oh)l o·=mld us(lt llriolplat·l i:;c:!l(;-ludlhiOttmihl n(::X. ),o 'C+:mIItl lli .(<sttu1l! bJlhtirr2;st t-c 1t-iCo/n2/.J t) ,_2.23 <FHoLern1 ,cil ie;-, :.hf:I sJkk, u :=:m) d0= cl1O u (r ti-~:: 1f:1. -:h:cf:t.~ 1:,kk~:),. w=e g(eBl Afo)Ir kt he-= = pr<oLdu:,t.,1b 1c,tlrc,Jmke) n=ls (A(h;B~:,)ua~.:; J:=).