北京大学：数学物理方法：第十五章 正交曲面坐标系

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 x15.5 HelmholtzZ??US"/￥s ?M ?16: x15.5 HelmholtzZ??US"/￥s ?M  ?US"?
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 7u(r; ;z) = v(r; )Z(z)
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 x15.6 HelmholtzZ?oUS"/￥s ?M ?17: x15.6 HelmholtzZ?oUS"/￥s ?M  oUS"?
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 6B?DZê?('?) μ1￥è±sZ? ^ 1 sin d d ? sin d￡( )d · +?￡( ) = 0; ?1LegendreZ?
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 x15.7? 
?USMDpü ?US"??US"/￥Lapalce ???19: x15.7? 
?USMDpü ?US"??US"/￥Lapalce ?? ü ?US(r;`)?°?US(x;y)￥1" ^ x = r cos`; y = r sin`: ?N?^p dr = cos`dx+sin`dy; d` = ?sin`r dx+ cos`r dy; ' @r @x = cos`; pp`x = ? sin` r ; @r @y = sin`; @` @y = cos` r : ?vˉ?f ?￥p?E5
 @ @x = @r @x @ @r + @` @x @ @` = cos` @@r ? sin`r @@`; @ @y = @r @y @ @r + @` @y @ @` = sin` @@r + cos`r @@`: éB?ü?¤? @2 @x2 = cos` @@r ? sin`r @@` ? cos` @@r ? sin`r @@` ? = cos2` @ 2 @r2 ? 2sin` cos` r @2 @r@` + sin2` r2 @2 @`2 +sin 2 ` r @ @r + 2sin` cos` r2 @ @`; @2 @y2 = sin` @@r + cos`r @@` ? sin` @@r + cos`r @@` ? = sin2` @ 2 @r2 + 2sin` cos` r @2 @r@` + cos2` r2 @2 @`2 +cos 2 ` r @ @r ? 2sin` cos` r2 @ @`: Kaü¤?ü ?US"/￥Laplace ?? r2 · @ 2 @r2 + 1 r @ @r + 1 r2 @2 @`2 · 1r @@r r @@r ? + 1r2 @ 2 @`2: x15.7? 
?USMDpü ?US"??US"/￥Lapalce ???20: N$ 
? V[¤??US"/￥Laplace ?? r2 · @ 2 @r2 + 1 r @ @r + 1 r2 @2 @`2 + @2 @z2 · 1r @@r r @@r ? + 1r2 @ 2 @`2 + @2 @z2: x15.8? 
?USMDpoUS"/￥Lapalce ???21: x15.8? 
?USMDpoUS"/￥Lapalce ?? oUS(r; ;`)?°?US(x;y;z)￥1" ^ x = r sin cos`; y = r sin sin`; z = r cos : ?N V[3 dr = sin cos`dx+sin sin`dy +cos dz; d = cos cos`r dx+ cos sin`r dy ? sin r dz; d` = ? sin`rsin dx+ cos`rsin dy: yN @ @x = @r @x @ @r + @ @x @ @ + @` @x @ @` = sin cos` @@r + cos cos`r @@ ? sin`r sin @@`; @ @y = @r @y @ @r + @ @y @ @ + @` @y @ @` = sin sin` @@r + cos sin`r @@ + cos`r sin @@`; @ @z = @r @z @ @r + @ @z @ @ = cos @@r ? sin r @@ : x15.8? 
?USMDpoUS"/￥Lapalce ???22: N$ ü V[p @2 @x2 = sin cos` @@r + cos cos`r @@ ? sin`r sin @@` ? sin cos` @@r + cos cos`r @@ ? sin`r sin @@` ? = sin2 cos2 ` @ 2 @r2 + cos2 cos2` r2 @2 @ 2 + sin2 ` r2 sin2 @2 @`2 + 2sin cos cos 2 ` r @2 @r@ ? 2sin` cos` r @2 @r@` ? 2cos sin` cos`r2 sin @ 2 @ @` + cos2 cos2 `+sin2 ` r @ @r + ?2sin 2 cos cos2 `+cos sin2 ` r2 sin @ @ + 2sin` cos`r2 sin2 @@`; @2 @y2 = sin sin` @@r + cos sin`r @@ + cos`r sin @@` ? sin sin` @@r + cos sin`r @@ + cos`r sin @@` ? = sin2 sin2` @ 2 @r2 + cos2 sin2` r2 @2 @ 2 + cos2 ` r2 sin2 @2 @`2 + 2sin cos sin 2 ` r @2 @r@ + 2sin` cos` r @2 @r@` + 2cos sin` cos`r2 sin @ 2 @ @` + cos2 sin2 `+cos2 ` r @ @r + ?2sin 2 cos sin2 `+cos cos2 ` r2 sin @ @ ? 2sin` cos`r2 sin2 @@`; @2 @z2 = cos @@r ? sin r @@ ? cos @@r ? sin r @@ ? = cos2 @ 2 @r2 + sin2 r2 @2 @ 2 ? 2sin cos r @2 @r@ + 2sin cos r2 @@ + sin 2 r @ @r: Kaü¤?oUS"/￥Laplace ?? r2 · @ 2 @r2 + 2 r @ @r + 1 r2 @2 @ 2 + cos r2 sin @ @ + 1 r2 sin2 @2 @`2 · 1r2 @@r r2 @@r ? + 1r2 sin @@ sin @@ ? + 1r2 sin2 @ 2 @`2: