数学物理方程(谷超豪)-第三章-调和方程习题解答

§11)(),,,(21rfxxxun=L)(221nxxr++=Ln022212=∂∂++∂∂nxuxuL221)(-+=nrccrf)2(≠nrInccrf1)(21+=)2(=n21,cc)(rfu=,rxrfxrrfxuiii⋅=∂∂⋅=∂∂)()(''32''2222)(1)()(rxrfrrfrxrfxuiii⋅-⋅+⋅=∂∂312''212122)()()(rxrfrnrfrxrfxuniiniinii∑∑∑===⋅-⋅+⋅=∂∂)(1)('rfrnrf-+=0=Δu0)(1)('=-+rfrnrfrnrfrf1)()('--=)1(1')(--=nrArf2≠n1212)(crnArfn++-=+-2≠n221)(-+=nrccrf2=nrArf1')(=InrAcrf11)(+=2=nrInccrf1)(21+=2),,(φθr0sin1)(sinsin1)(12222222=∂∂⋅+∂∂∂∂⋅+∂∂∂∂⋅=Δφθθθθθururrurrru),,(φθr),,(zyxφθcossinrx=φθsinsinry=θcosrz=1222222zuyuxuu∂∂+∂∂+∂∂=Δθρsinr=1φρcos=xφρsin=y2θρsinr=θcosrz=32⎪⎪⎭⎪⎪⎬⎫∂∂+-∂∂=∂∂∂∂+∂∂=∂∂)cos()sin(sincosφρφρφφφρyuxuuyuxuu⎪⎪⎭⎪⎪⎬⎫⋅∂∂+∂∂=∂∂⋅∂∂-∂∂=∂∂ρφφφρρφφφρcossinsincosuuyuuuxu4)sincos(22ρφφφρ⋅∂∂-∂∂∂∂=∂∂uuxxu)sincos(sin)sincos(cosρφφφρφρφρφφφρρφ⋅∂∂-∂∂∂∂⋅-⋅∂∂-∂∂∂∂=uuuu+∂∂⋅+∂∂∂⋅-∂∂=22222222sincossin2cosφρφφρρφφρφuuuρρφφρφφ∂∂⋅+∂∂⋅+uu22sincossin2)cossin(22ρφφφρ⋅∂∂+∂∂∂∂=∂∂uuyyu)cossin(cos)cossin(sinρφφφρφρφρφφφρρφ⋅∂∂+∂∂∂∂++⋅∂∂+∂∂∂∂=uuuuρρφφρφφφρφφρρφφρ∂∂⋅+∂∂⋅--∂∂⋅+∂∂∂+∂∂=uuuuu2222222222coscossin2coscossin2sinρρφρρ∂∂⋅+∂∂⋅+∂∂=∂∂+∂∂uuuyuxu112222222225ρρφρρ∂∂⋅+∂∂⋅+∂∂+∂∂=∂∂+∂∂+∂∂uuzuuzuyuxu11222222222222232222zuu∂∂+∂∂ρ5rururruzuu∂∂⋅+∂∂⋅+∂∂=∂∂+∂∂11222222222θρ4ruruuθθθρcossin⋅∂∂+∂∂=∂∂)cossin(sin1sin111222222222222222rurururrururruzuyuxuθθθθφθθ⋅∂∂+∂∂+∂∂⋅++∂∂⋅+∂∂⋅+∂∂=∂∂+∂∂+∂∂θθφθθ∂∂+∂∂⋅+∂∂⋅+∂∂⋅+∂∂=uctgrrurururru222222222212sin110sin1)(sinsin1)(12222222=∂∂⋅+∂∂∂∂⋅+∂∂∂∂⋅=Δφθθθθθururrurrru3),,(zrθ222221)(1zuurrurrru∂∂+∂∂⋅+∂∂∂∂⋅=Δθ),,(zrθ),,(zyxθcosrx=θsinry=zz=rururruyuxu∂∂+∂∂+∂∂=∂∂+∂∂11222222222θ2221)(1θ∂∂+∂∂∂∂=urrurrr222221)(1zuurrurrru∂∂+∂∂+∂∂∂∂=Δθ4.1cbyax++(a,b,c)cbyaxu++=,,022=∂∂xu.022=∂∂yu0=Δuu(2)xyyx222-,222=∂∂xu,222=∂∂yu0=Δuuxyv2=,022=∂∂xv022=∂∂yv0=Δvv(3)322333yyxxyx--233xyxu-=,622xxu=∂∂,622xyu-=∂∂0=Δu,u323yyxv-=,622yxv=∂∂yyv622-=∂∂0=Δvv(4))(cossin,cos,sinnnxchnynxchnynxshnynxshnyshnynshnyy2)(=chnynchnyy2)(=nnxnnxxsin)(sin2-=coxnxnnxx2)(cos-=yyxxnxshnynxshny)sin()sin(-=0)sin(=Δnxshnynxshnysin511)cos(sin)cos(--++ychxyychxshx1)cos(-+=ychxshxu221)cos()cos(--+-+=∂∂ychxxshychxchxxu)cos1()cos(2ychxychx++=-)cos1()cos(2cos)cos(3222ychxshxychxyshxychxxu++-+=∂∂--)cos2()cos(23yshxchxshxyshxcoxychx--+=-2)cos(sin-+=∂∂ychxyshxyuyshxychxychxyshxyu23222sin)cos(2)cos(sin--+++=∂∂)coscoscos()cos(23yshxchxyshxyshxchxychx+++=-)sin22cos2()cos(2232222yshxshxyshxychxyuxu+-+=∂∂+∂∂-0]2)sin(cos2[)cos(223=-++=-shxyyshxychx1)cos(sin-+=ychxyv2)cos(sin-+-=∂∂ychxyshxxvyxshychxychxychxxvsin)cos(2)cos(sin23222--+++-=∂∂)cossinsinsin2()cos(223yychxxychyxshychx--+=-221)cos(sin)cos(cos--+++=∂∂ychxyychxyyv)cos1()cos(2ychxychx++=-)cos1(sin)cos(2)cos(sin3222ychxyychxychxychxyu++++-=∂∂--)sin2cossinsin2()cos(23xychychxyyychx-++=-)sin2sin2sin2()cos(2232222yxychxyshychxyvxv+-+=∂∂+∂∂-0]sin2)(sin2[)cos(223=+--+=-yxshxchyychxu,v51θrlnuru,1,ln=,0,0,0,2222=∂∂=∂∂=∂∂=θθvrvrvv01122222=∂∂+∂∂+∂∂=Δθvrrvrrvu2nnrnrnn(sincosθθθnruncos=θnnrruncos1-=∂∂θnrnnruncos)1(222--=∂∂θθnrnuncos222-=∂∂0cos])1([2222=-+-=Δ---θnrnnrrnnunnnθnrvnsin=0sin])1([2222=-+-=Δ---θnrnnrrnnvnnn(3)θθθθθcossincoslnrrrr+-θθθcossinlnrrr+.sincoslnθθθrrru-=.sincos2coslncossinsinln.cos1sincos)1(ln2222θθθθθθθθθθθθθrrrrurrrruurrurru+--=∂∂---=∂∂=∂∂-+=∂∂0sincos2cosln1sincos)1(ln1cos1=+---++=Δθθθθθθθθrrrrrrrruθθθcossinlnrrrv+=θθθθθθθθθθθθcossin)2(lnsincoscoslnsin1cossin)1(ln.2222rrrvrrrrvrrvrrv-+-=∂∂-+=∂∂=∂∂++=∂∂.0cossin)2(ln1cossin)1(ln1sin1=-+-+++=Δθθθθθθθrrrrrrrv6.)0,0byax≤≤≤≤⎪⎪⎪⎩⎪⎪⎪⎨⎧=====∂∂+∂∂.0),(,sin)0,(0),(),0(02222bxuaxxuyauyuyuxuπ)()(),(yYxXyxu=λ-=′′-=′′YYxXxX)()(0),(),0(==yauyu0)()0(==aXX⎩⎨⎧===+′′0)()0(0aXXXXλ2)(annπλλ==.sin)(xanxXnπ=),2,1(L=n0)(2=-′′nnYanYπyannyannneBeAYππ-+=∑∞=-+=1.sin)()(nyannyannxaneBeAyxuπππ⎪⎪⎭⎪⎪⎬⎫+=+=∑∑∞=-∞=11sin)(0sin)(sinnbannbannnnnxaneBeAxanBAaxπππππ⎪⎩⎪⎨⎧==+==+=+-LL,2,10,3,20111neBeAnBABAbannbannnnππbasheAbaππ--=211basheBbaππ211=L,3,20===nBAnn),(yxuxaeebashyxuybaybaππππsin)(121),()()(----⋅=xaybxshbashπππsin)(1-=7byax≤≤≤≤0,0⎪⎪⎩⎪⎪⎨⎧====∂∂+=Δ====00,00,0(00byyaxxuuuxuqpqpyu))((22gfyaxuv+-+=v,gf,v),)(22gfyaxuv+-+=,),(222222222yuyvgfyxuxu∂∂=∂∂++∂∂=∂∂)(2gfyuv++Δ=Δ∴,gpyu+=Δ,2,2qgpf-=-=v0=Δv))((2122qpyaxuv+--=v⎪⎪⎪⎩⎪⎪⎪⎨⎧-+-=--===∂∂=Δ====))((21),(20,00222200axqpbvaxqvvxvvbyyaxx),(yxv)()(),(yYxXyxv=0,00==∂∂==axxvxv⎩⎨⎧==′=+′′0)()0(0aXXXXλ0=-′′YYλ)(xXL,2,1,0,)212(2=+==nannπλλyanshByanchAYxanxXnnnnπππ212212,212cos)(+++=+=.∑∞=++++=0212cos)212212(),(nnnxanyanshByanchAyxvπππ∑∞=+=--022212cos)(2nnxanAaxqπ∑∞=++++=-+-022212cos)212212())((21nnnxanbanshBbanchAaxqpbπππ∫-+=+-=annnqaxdxanxaaqA033222.)1()12(16212cos)(ππ∫+-+=+++lnnxdxanxaaqpbbanshBbanchA022212cos)(212212πππnnqpba)1()12()(16332-++=πbanshbanchqpbnaBnnπππ212/)]2121([)12(16)1(332++-++-=.)(212[)12()1(16),(0332ybanqshnayxvnn-++-=∑∞=ππbanshxanyanshqpbπππ212/212cos]212)(+++++xanyanshqpbybanqshbshnaqpyaxyxunannπππππ212cos]212)()(212[1)12()1(16))((21),(021233222++++-+⋅⋅+-++-=∑∞=+8.),(yxu1.⎪⎪⎩⎪⎪⎨⎧=+=∂∂+∂∂=[phone]yxuyxyuxu1=u221ln1yxu++=§22.62.7),().,(yxvyxuvu..)()(dsnuvnvudxdyuvvuDC∫∫∫∂∂-∂∂=Δ-Δ0MrrrMM1ln,0=0M0MεεkεkD-rv1ln=∫∫∫∂∂-∂∂=Δ-Δ-CkDdsnurnrudxdyurru))1(ln)1(ln())1(ln)1(ln(ε∫Γ∂∂-∂∂+εdsnurnru)1ln)1(ln(εΓεk∫ΓεεΓεΓrrrnr1)1(ln)1(ln=∂∂-=∂∂∫∫ΓΓ=⋅⋅=⋅=∂∂εεππεε**2211)1(lnuu