北大数学物理方法(B)教案18分离变量法4:正交曲面坐标系

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WuChong-shi()Lapalce!(r,φ)#$!(x,y)%&’(x=rcosφ,y=rsinφ.)*+,-.dr=cosφdx+sinφdy,dφ=−sinφrdx+cosφrdy,/∂r∂x=cosφ,∂φ∂x=−sinφr,∂r∂y=sinφ,∂φ∂y=cosφr.012345%-6789∂∂x=∂r∂x∂∂r+∂φ∂x∂∂φ=cosφ∂∂r−sinφr∂∂φ,∂∂y=∂r∂y∂∂r+∂φ∂y∂∂φ=sinφ∂∂r+cosφr∂∂φ.:;=?@∂2∂x2=cosφ∂∂r−sinφr∂∂φcosφ∂∂r−sinφr∂∂φ=cos2φ∂2∂r2−2sinφcosφr∂2∂r∂φ+sin2φr2∂2∂φ2+sin2φr∂∂r+2sinφcosφr2∂∂φ,∂2∂y2=sinφ∂∂r+cosφr∂∂φsinφ∂∂r+cosφr∂∂φ=sin2φ∂2∂r2+2sinφcosφr∂2∂r∂φ+cos2φr2∂2∂φ2+cos2φr∂∂r−2sinφcosφr2∂∂φ.AB=?@!’C%LaplaceDE∇2≡∂2∂r2+1r∂∂r+1r2∂2∂φ2≡1r∂∂rr∂∂r+1r2∂2∂φ2.F*GHI9JKL?@M!’C%LaplaceDE∇2≡∂2∂r2+1r∂∂r+1r2∂2∂φ2+∂2∂z2≡1r∂∂rr∂∂r+1r2∂2∂φ2+∂2∂z2.WuChong-shi§18.NOPQRSLapalceTUV2WXLapalceY!(r,θ,φ)#$!(x,y,z)%&’(x=rsinθcosφ,y=rsinθsinφ,z=rcosθ.)*KLZ.dr=sinθcosφdx+sinθsinφdy+cosθdz,dθ=cosθcosφrdx+cosθsinφrdy−sinθrdz,dφ=−sinφrsinθdx+cosφrsinθdy.[*∂∂x=∂r∂x∂∂r+∂θ∂x∂∂θ+∂φ∂x∂∂φ=sinθcosφ∂∂r+cosθcosφr∂∂θ−sinφrsinθ∂∂φ,∂∂y=∂r∂y∂∂r+∂θ∂y∂∂θ+∂φ∂y∂∂φ=sinθsinφ∂∂r+cosθsinφr∂∂θ+cosφrsinθ∂∂φ,∂∂z=∂r∂z∂∂r+∂θ∂z∂∂θ=cosθ∂∂r−sinθr∂∂θ.F*GHI=KL-.∂2∂x2=sinθcosφ∂∂r+cosθcosφr∂∂θ−sinφrsinθ∂∂φsinθcosφ∂∂r+cosθcosφr∂∂θ−sinφrsinθ∂∂φ=sin2θcos2φ∂2∂r2+cos2θcos2φr2∂2∂θ2+sin2φr2sin2θ∂2∂φ2+2sinθcosθcos2φr∂2∂r∂θ−2sinφcosφr∂2∂r∂φ−2cosθsinφcosφr2sinθ∂2∂θ∂φ+cos2θcos2φ+sin2φr∂∂r+−2sin2θcosθcos2φ+cosθsin2φr2sinθ∂∂θ+2sinφcosφr2sin2θ∂∂φ,∂2∂y2=sinθsinφ∂∂r+cosθsinφr∂∂θ+cosφrsinθ∂∂φsinθsinφ∂∂r+cosθsinφr∂∂θ+cosφrsinθ∂∂φ=sin2θsin2φ∂2∂r2+cos2θsin2φr2∂2∂θ2+cos2φr2sin2θ∂2∂φ2+2sinθcosθsin2φr∂2∂r∂θ+2sinφcosφr∂2∂r∂φ+2cosθsinφcosφr2sinθ∂2∂θ∂φ+cos2θsin2φ+cos2φr∂∂r+−2sin2θcosθsin2φ+cosθcos2φr2sinθ∂∂θ−2sinφcosφr2sin2θ∂∂φ,∂2∂z2=cosθ∂∂r−sinθr∂∂θcosθ∂∂r−sinθr∂∂θ=cos2θ∂2∂r2+sin2θr2∂2∂θ2−2sinθcosθr∂2∂r∂θ+2sinθcosθr2∂∂θ+sin2θr∂∂r.WuChong-shi\]^_‘abcd(e)fghiOPQV3WAB=?@Y!’C%LaplaceDE∇2≡∂2∂r2+2r∂∂r+1r2∂2∂θ2+cosθr2sinθ∂∂θ+1r2sin2θ∂2∂φ2≡1r2∂∂rr2∂∂r+1r2sinθ∂∂θsinθ∂∂θ+1r2sin2θ∂2∂φ2.WuChong-shi§18.1jklmV4W§18.1nopqrstuvwxyz{|}Z~∂2u∂x2+∂2u∂y2=0,x2+y2a2,u x2+y2=a2=f.F#$!’C9(Laplace)KL||)%(9!’|F!’9%}Z~KL1r∂∂rr∂u∂r+1r2∂2u∂φ2=0,0ra,u r=a=f(φ).¡u(r,φ)=R(r)Φ(φ)9¢£9⁄1rddrrdRdrΦ+Rr2d2Φdφ2=0,=⇒rRddrrdRdr=−1Φd2Φdφ2=λ.[*9KL9rddrrdRdr−λR=0,d2Φdφ2+λΦ=0.(R(a)Φ(φ)=f(φ)¥9[(ƒ§¤%|'“«‹›fi§¤9?@fl–⁄†}‡5%§¤·9(¶•⁄‚%§¤„”»3…‰;¿`~|´ˆ˜¯˘˙¨˚¸˝˛ˇ—9wwI.%9()Æ%ª%F%C9)#$!’Ł@!’Ø9Œº1r∂∂rr∂u∂r+1r2∂2u∂φ2=0,0ra,u r=a=f(φ).¶%}Z~æ9¶‰;%}Z~|WuChong-shi\]^_‘abcd(e)fghiOPQV5WF;9F5ıI9}Z~%Fł…Ł@!B9Fø%œßφ=0φ=2π¶ł|æ9F!9φ%([0,2π]9[u(r,φ)Fœßφ=0φ=2π%65•⁄}9A}u(r,φ)Fflœß%65|flœß()!’….%9¶ƒ%9F%}Z~9=I}‚%|=6FI%Œº•⁄.u(r,φ)Fφ=0φ=2π%|@!’%ß9(r,φ=0)(r,φ=2π)¢%(I%;ß9L9!%}Z~9I#$u(r,φ) φ=0=u(r,φ) φ=2π∂u(r,φ)∂φ φ=0=∂u(r,φ)∂φ φ=2π.9I%@%)&Laplace’#$!’(Ł@!’Ø…)*%+,9KL-.#$…?@/|F9%F!ß(x,y)=(0,0)(ł%|(9Ł@!B9Fr=0߶ł|[u(r,φ)Fr=0ß%65¶•⁄}9}u(r,φ)Fr=0ß%65|r=0ß!r%œß9()!’….%9¶(%|J0Iu(r,φ)Fr=0ß%|@%(§¤%9F(12!ß)(34%9[*9u(r,φ)F!ß(⁄%9I⁄u(r,φ) r=0⁄.56´78ˆ˜¯˘˙¨99y:z{;˚˛=1r∂∂rr∂u∂r+1r2∂2u∂φ2=0,0φ2π,0ra,u(r,φ) φ=0=u(r,φ) φ=2π,0ra,∂u(r,φ)∂φ φ=0=∂u(r,φ)∂φ φ=2π,0ra,u(r,φ) r=0⁄,0φ2π,u r=a=f(φ),0φ2π.WuChong-shi§18.1jklmV6WF9?2%@9=KLA@9BCDEF?@%fl§¤·rddrrdRdr−λR=0,d2Φdφ2+λΦ=0”G9)#$JKL?@Φ(0)=Φ(2π),Φ0(0)=Φ0(2π).9H?@C;I%¿`~d2Φdφ2+λΦ=0,Φ(0)=Φ(2π),Φ0(0)=Φ0(2π).JKLMNOPQRSTUTVWXYZ[\Q]^_‘abcdefghijQ|JKLMNOPQklmnopqrQRS|λ=0Ø9·%-ZΦ0(φ)=A0φ+B0.¢£#$9⁄B0=A02π+B0,A0=A0.[*A0=0,B0st.æuλ=0(¿`9‚%¿45(Φ0(φ)=1.λ6=0Ø9%-ZΦ(φ)=Asin√λφ+Bcos√λφ.¢£#$9?@B=Asin√λ2π+Bcos√λ2π,A=Acos√λ2π−Bsin√λ2π.KLA(&’5AB%vw§¤¢5x9⁄ƒyZ%z( sin√λ2πcos√λ2π−1cos√λ2π−1−sin√λ2π =0,/2(cos√λ2π−1)=0|HKL-?¿`λm=m2,m=1,2,3,···,‚%ƒyZ(Ast,Bst.WuChong-shi\]^_‘abcd(e)fghiOPQV7W=(æ9{;¿`λm9⁄fl¿45Φm1(φ)=sinmφ,Φm2(φ)=cosmφ.lm9|}~λ0=0Qbλ06=0Q9cjΦm1(φ)=sinmφ,Φm2(φ)=cosmφ,mQN0,1,2,3,···|017%!@9-·rddrrdRdr−λR=0%Z|t·(;%’59F.%Łddt=rddr/t=lnrB9=KL·’5%·d2Rdt2−λR=0.L9λ0=0Ø9-ZR0(r)=C0+D0t=C0+D0lnr;λm=m2,m6=0Ø9-ZRm(r)=Cmemt+Dme−mt=Cmrm+Dmr−m.F9=-?C§¤§¤(#$)%Zu0(r,φ)=C0+D0lnr,um1(r,φ)=Cm1rm+Dm1r−msinmφ,um2(r,φ)=Cm2rm+Dm2r−mcosmφ.9=?@}Z~%;Zu(r,φ)=C0+D0lnr+∞Xm=1Cm1rm+Dm1r−msinmφ+∞Xm=1Cm2rm+Dm2r−mcosmφ.@⁄u r=0⁄,[lnrr−mFr=0ß(3%9L“%’5z09D0=0,Dm1=0,Dm2=0.¢£%9=?@u(r,φ) r=a=C0+∞Xm=1am(Cm1sinmφ+Cm2cosmφ)=f(φ).WuChong-shi§18.1jklmV8WC%~(}.’5C0,Cm1Cm2|«‹KL’Fourier%$-.’5C0,Cm1Cm297%!¡79J(¢¿45%£w}’5|⁄¥ƒ§¤z{d2Φdφ2+λΦ=0,Φ(0)=Φ(2π),Φ0(0)=Φ0(2π),⁄˚'“ƒ§¤wƒ§«‹›fiflw|F¿451({¿`λ0=0)¿45sinmφcosmφ({¿`λm=m2,m6=0)(£%Z2π0sinmφdφ=0,Z2π0cosmφdφ=0.F{¿`λm=m2%¿45sinmφ,cosmφ{¿`λn=n2,n6=m%¿45sinnφ,cosnφ(flfl£%Z2π0sinnφsinmφdφ=0,Z2π0sinnφcosmφdφ=0,Z2π0cosnφcosmφdφ=0.⁄˚¥“–ƒ§¤λm=m2w†–ƒ§«‹sinmφ‡cosmφ·›fiflwZ2π0sinmφcosmφdφ=0.[*9¢¿45%£wLZ2π0sin2mφdφ=π,Z2π0cos2

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