第九章 正交曲线坐标系中的分离变量
合集下载
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
1 1 1 2 2 2 3 3 3 1 2 3 2 3 2 3
∆V →0
∆V
A · dS = [(A1 h2 h3 )q1 +∆q1 − (A1 h2 h3 )q1 ]∆q2 ∆q3 + · · · =
∂ (∆V ) 0 1 1 1
∂ (A1 h2 h3 ) ∂q1
∆q1 ∆q2 ∆q3 + · · · .
X¨¤'8ÜÿÝ£±oÑGnANȤ "©ù´5N'© ·òª (1a) ¤{9'/ª
1 2 3 2
r = r (q1 , q2 , q3 ),
(1b)
¿Qe¡'í¥¦^aq'PÒ§ù±~¨Öþ'槱¦k9@tw& ß©XtÖöwØÙªf'¹Â§±QúvþÑ['©þ/ª\±éì© ÙGaq'PÒ§éucÚÆS Y'§´~k|'© yQ'¯K´§o$'IXâ´¢'º QIX¥§üX r Ú r + dr m'ålP ds§u
§1
¢ IX IX (x, y, z)§¥IX (r, θ, φ)§ÎIX (ρ, φ, z) Ñ´¢IX©§ 'nxI??p¢©yÄIX§ÙIP (q , q , q )§§ I m'C9 X
1 2 3
¢ IX¥ ©Î
(ds)2 = dr · dr = (dx)2 + (dy )2 + (dz )2 . (2)
ùål'½Â´Öö¤ÙG'©¢&ѧù$'½Â¿Ø´U²G½n¤¨,'©¢Sþ ·±½ÂÙ§/ª'ål©ål'½ÂØÓ§m'5ÒØÓ©¦^þã½Â§L²·¤ï Ä'´ Euclid m©u Euclid m'½Â¥'Ù§[!§·ÒØ?Ø
Xt)Ò¥'SN)Ò©'±§Ò& c¡ß'@t©du©Î' Q§'ö¨,´Ø5N'©Øv§ù$''"kÏu·rºùüED'úª© |^IX¥' Laplace ÎLªÚIC§Ò±& ±þüúª©ù´ {ü'g´§¢O%'"¡§cÙ´¥IX'¹©e¡Ñ'{'"{ü§ ·^uÙ§IX§k,'Öö±ë©
dd±á&¢5§ k
hr = 1, hθ = r, hφ = r sin θ. (12)
ù
ÝSXê'Aۿ´²w'©´&¥IX'ü ¥þ
er = sin θ cos φex + sin θ sin φey + cos θez , eθ = cos θ cos φex + cos θ sin φey − sin θez , eφ = − sin φex + cos φey . (13a) (13b) (13c)
1ÊÙ ¢IX¥©lCþ
8 ¹
∗
§1
¢IX ¥©Î ¢ IX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ¢ IX¥ ©Î . . . . . . . . . . . . . . . . . . . . . . . . . . § 2 ¥ IX¥© lCþ Helmholtz § . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helmholtz §3¥IX¥©lCþ . . . . . . . . . . . . . . . . . . . . § 3 Î IX¥© lCþ
(14)
IX'A Lª
∇u =
'"§N´wѧØÓ?QuIX'@t¥Ñy
ÝSXê© d@tÚÝSXê (9)§´&ÎIX¥'FÝÎ
∇u = 1 ∂u ∂u ∂u eρ + eφ + ez . ∂ρ ρ ∂φ ∂z (15)
qdÝSXê (12)§´&¥IX¥'FÝÎ
¢ IX¥ ©Î ©ÎÌkFÝ!ÑÝ!^ÝÚ Laplace Χ ö´FÝÚÑÝ'@Ü©Äk ÄIþ¼ê u(r) 'FÝ ∇u(r)© du ∇u ´¥þ§§½±Ðm ∇u = f e §5¿Ù¥ f ´ r '¼ê§e ' ´X r Cz'§ùIX'ü ¥þØÓ©d9X (7)§´& f = e · ∇u§ ,¡§k ∂u ∆u 1 ∆u 1 ∂u
Q J = 0 ?§½ (x, y, z)§UÃ{(½ (q , q , q )§ù
XÒ´IX'ÛX©'X ¥IX§J = r sin θ§Q r = 0 ?½ θ = 0!π ? J = 0©d½ (x, y, z)§Ã{( ½ (r, θ, φ)©¯¢þ§Q θ = 0!π ?§φ vk½Â§ Q r = 0 ?§θ!φ þvk½Â©¤± z ¶Ñ´¥IX'ÛX©Øv§ù
x y z
ei · ej = δij .
(7)
dd±á&¢5§ k
φ
∂r = cos φex + sin φey , ∂ρ ∂r = −ρ sin φex + ρ cos φey , ∂φ ∂r = ez . ∂z hρ = 1, hφ = ρ, hz = 1.
(8a) (8b) (8c)
ù
ÝSXê'Aۿ´²w'§'X÷X φ Id φ φ + dφ 'ål£l¤Ø´ dφ§ ´ h dφ = ρdφ§ù´·¤Ù'©´&ÎIX'ü ¥þ
ÄkíIX¥'ÑÝLª©¥þ| A(r) 'ÑݽÂ
∇ · A = lim
∂ (∆V ) A
· dS , (19)
Ù¥ ∆V ´¹X r £¦ÙÑÝ'X§Q±þ½Â¥±w'½§¢TX'À´?¿'¤' «§ ∂(∆V ) ´ ∆V '>.¡§dS ´Ù¡È£©8 ∆V dI¡ q !q + ∆q !q !q + ∆q !q !q + ∆q ¤¤'8¡N§Oþª©f¥'È©§=¥þ|Q ∆V '>.¡'Ïþ©I 5¿'´§8¡NQ q ?'¡È´ h h ∆q ∆q § Ø´ ∆q ∆q ©u´
∇u = ∂u 1 ∂u 1 ∂u er + eθ + eφ . ∂r r ∂θ r sin θ ∂φ (16)
§1
¢IX ¥©Î
5
ÖÌ^ Laplace Χe¡ÒÑü«~^'IX!=ÎIXÚ¥I X¥' Laplace Î/ª©duIX¥' Laplace Î
r0
Ù¥ r ´ ∆V SX§þªÑ
q ?Ú q + ∆q ?ü¡' z§ /· · · 0L«Ù§o¡' z§N´ÑÙA'Lª©du ∆V = h h h ∆q ∆q ∆q § ¨ ∆q → 0!∆q → 0!∆q → 0 §r → r§%&
§%
(11a) (11b) (11c)
∂r = sin θ cos φex + sin θ sin φey + cos θez , ∂r ∂r = r(cos θ cos φex + cos θ sin φey − sin θez ), ∂θ ∂r = r sin θ(− sin φex + cos φey ). ∂φ
∇2 u =
âc¡ïÄFÝΤ¼&'²¨§·N´ß§ÎIX¥' Laplace ÎU´
∇2 u =
∂2u ∂2u ∂2u + + , ∂x2 ∂y 2 ∂z 2
¥IX¥' Laplace ÎU´
∇2 u =
∂2u 1 ∂2u ∂2u + + , ∂ρ2 ρ2 ∂φ2 ∂z 2
¢ù´Ø('©ÎIX¥'(@tAT´
x = x(q1 , q2 , q3 ),
1 2 3
y = y (q1 , q2 , q3 ),
z = z (q1 , q2 , q3 ).
(1a)
§(q , q , q ) ±L (x, y, z) '¼ê© 5 5`§·¦ Jacobi 1ª
∂x/∂q1 ∂x/∂q2 ∂x/∂q3 ∂ (x, y, z ) J= = ∂y/∂q1 ∂y/∂q2 ∂y/∂q3 = 0. ∂ (q1 , q2 , q3 ) ∂z/∂q1 ∂z/∂q2 ∂z/∂q3
1 ∂ ∇ u= ρ ∂ρ
2
1 ∂2u 1 ∂2u ∂2u + + . ∂r2 r2 ∂θ2 r2 sin2 θ ∂φ2
∂u ρ ∂ρ
1 ∂2u ∂2u + 2 2 + 2, ρ ∂φ ∂z
(17)
¥I X¥'( @tAT´
∇2 u = 1 ∂ r2 ∂r r2 ∂u ∂r + 1 ∂ 2 r sin θ ∂θ sin θ ∂u ∂θ + 1 ∂2u . r2 sin2 θ ∂φ2 (18)
©
§1
¢IX ¥©Î yQwwþªQIX¥'/ª©du dr =
3 3 i=1 (∂ r /∂qi )dqi
3
§%
(3)
(ds) = dr · dr =
2
∂r ∂r · dqi dqj . ∂qi ∂qj i,j =1
XtÑ['©þ/ª§uþª¹
18 ©yQ§·±Ñ¢IX'½ µXJ ∂r ∂r KIX (q , q , q ) ¡¢©ùk
i i i 2 1 2 1 2 3 1 2 3 1 1 1 1 1 1 1 1 1 1 −1 1 1 2 3
ei =
dª (4) ÚÝSXê'½Â§´&
1 ∂r , hi ∂qi
i = 1, 2, 3.
(6)
nxI(¢´??¢'©e¡§·^þã½Â#©Ù'¥IX ÚÎIX© ~ 1 ÎIX (ρ, φ, z)©du r = ρ cos φe + ρ sin φe + ze §%
eρ = cos φex + sin φey , eφ = − sin φex + cos φey , ez = ez . (10a) (10b) (10c)
(9)
§1
¢IX ¥©Î ~ 2 ¥IX (r, θ, φ)©du r = r sin θ cos φe
x
4 + r sin θ sin φey + r cos θez
3 i=1 i i i i i i
ei · ∇u =
u´&
1 fi = h− i ∂u/∂qi
§l
3 i=1
∂si
= lim
∆si →0
∆si
= lim
∆qi →0
hi ∆qi
=
hi ∂qi
,
∇u =
1 ∂u 1 ∂u 1 ∂u 1 ∂u ei = e1 + e2 + e3 . hi ∂qi h1 ∂q1 h2 ∂q2 h3 ∂q3 ∂u ∂u ∂u ex + ey + ez ∂x ∂y ∂z
2 2 4 6 6 7 8
c ù´¥ìÆÔnXÆ)ÆSêÆÔn{§'ë]§d c?©H?Û<E ^uÆS½Æë©H1µ©Õ^uÑÈ©
∗
ห้องสมุดไป่ตู้
c 1992–2005
1
§1
¢IX ¥©Î
2
Ùm©r©lCþ{íP '"¢S'n¯K©c®Ñ§©lCþ§ATâ >F'/Gæ^·¨'IX©Ù'8'Ò´ïÄXÛQ¥IXÚÎIX¥éa §©lCþ©·¤#'Aa§Ñ¹k Laplace Χ¤±ÄkIïÄ Laplace ÎQIX§cÙ´¥IXÚÎIX¥'/ª©
1 2 3 3
· = 0, ∂qi ∂qj
i=j
(4)
(ds) =
i=1
2
∂r ∂qi
2
3
(dqi ) ≡
i=1
2
(hi dqi )2 ,
i
(5)
Ù¥ h = |∂r/∂q | ¡ÝSXê©ùIX¥'/ª (2) q§´ dq c¡õ
ÝSXê h ©¤±§¢'9 Ò´ (ds) 'Lª¥Ø¹ dq dq ù$'¢£© yQí¢IX (q , q , q ) ¥'ü ¥þ (e , e , e )©e 'Ò´ q I '§Ïd§÷XI q §k dr = ds e = h dq e §,¡§dr = (∂r/∂q )dq § % e = h (∂r/∂q )§é e Ú e kaq@t©o@å5§Ò´
∆V →0
∆V
A · dS = [(A1 h2 h3 )q1 +∆q1 − (A1 h2 h3 )q1 ]∆q2 ∆q3 + · · · =
∂ (∆V ) 0 1 1 1
∂ (A1 h2 h3 ) ∂q1
∆q1 ∆q2 ∆q3 + · · · .
X¨¤'8ÜÿÝ£±oÑGnANȤ "©ù´5N'© ·òª (1a) ¤{9'/ª
1 2 3 2
r = r (q1 , q2 , q3 ),
(1b)
¿Qe¡'í¥¦^aq'PÒ§ù±~¨Öþ'槱¦k9@tw& ß©XtÖöwØÙªf'¹Â§±QúvþÑ['©þ/ª\±éì© ÙGaq'PÒ§éucÚÆS Y'§´~k|'© yQ'¯K´§o$'IXâ´¢'º QIX¥§üX r Ú r + dr m'ålP ds§u
§1
¢ IX IX (x, y, z)§¥IX (r, θ, φ)§ÎIX (ρ, φ, z) Ñ´¢IX©§ 'nxI??p¢©yÄIX§ÙIP (q , q , q )§§ I m'C9 X
1 2 3
¢ IX¥ ©Î
(ds)2 = dr · dr = (dx)2 + (dy )2 + (dz )2 . (2)
ùål'½Â´Öö¤ÙG'©¢&ѧù$'½Â¿Ø´U²G½n¤¨,'©¢Sþ ·±½ÂÙ§/ª'ål©ål'½ÂØÓ§m'5ÒØÓ©¦^þã½Â§L²·¤ï Ä'´ Euclid m©u Euclid m'½Â¥'Ù§[!§·ÒØ?Ø
Xt)Ò¥'SN)Ò©'±§Ò& c¡ß'@t©du©Î' Q§'ö¨,´Ø5N'©Øv§ù$''"kÏu·rºùüED'úª© |^IX¥' Laplace ÎLªÚIC§Ò±& ±þüúª©ù´ {ü'g´§¢O%'"¡§cÙ´¥IX'¹©e¡Ñ'{'"{ü§ ·^uÙ§IX§k,'Öö±ë©
dd±á&¢5§ k
hr = 1, hθ = r, hφ = r sin θ. (12)
ù
ÝSXê'Aۿ´²w'©´&¥IX'ü ¥þ
er = sin θ cos φex + sin θ sin φey + cos θez , eθ = cos θ cos φex + cos θ sin φey − sin θez , eφ = − sin φex + cos φey . (13a) (13b) (13c)
1ÊÙ ¢IX¥©lCþ
8 ¹
∗
§1
¢IX ¥©Î ¢ IX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ¢ IX¥ ©Î . . . . . . . . . . . . . . . . . . . . . . . . . . § 2 ¥ IX¥© lCþ Helmholtz § . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helmholtz §3¥IX¥©lCþ . . . . . . . . . . . . . . . . . . . . § 3 Î IX¥© lCþ
(14)
IX'A Lª
∇u =
'"§N´wѧØÓ?QuIX'@t¥Ñy
ÝSXê© d@tÚÝSXê (9)§´&ÎIX¥'FÝÎ
∇u = 1 ∂u ∂u ∂u eρ + eφ + ez . ∂ρ ρ ∂φ ∂z (15)
qdÝSXê (12)§´&¥IX¥'FÝÎ
¢ IX¥ ©Î ©ÎÌkFÝ!ÑÝ!^ÝÚ Laplace Χ ö´FÝÚÑÝ'@Ü©Äk ÄIþ¼ê u(r) 'FÝ ∇u(r)© du ∇u ´¥þ§§½±Ðm ∇u = f e §5¿Ù¥ f ´ r '¼ê§e ' ´X r Cz'§ùIX'ü ¥þØÓ©d9X (7)§´& f = e · ∇u§ ,¡§k ∂u ∆u 1 ∆u 1 ∂u
Q J = 0 ?§½ (x, y, z)§UÃ{(½ (q , q , q )§ù
XÒ´IX'ÛX©'X ¥IX§J = r sin θ§Q r = 0 ?½ θ = 0!π ? J = 0©d½ (x, y, z)§Ã{( ½ (r, θ, φ)©¯¢þ§Q θ = 0!π ?§φ vk½Â§ Q r = 0 ?§θ!φ þvk½Â©¤± z ¶Ñ´¥IX'ÛX©Øv§ù
x y z
ei · ej = δij .
(7)
dd±á&¢5§ k
φ
∂r = cos φex + sin φey , ∂ρ ∂r = −ρ sin φex + ρ cos φey , ∂φ ∂r = ez . ∂z hρ = 1, hφ = ρ, hz = 1.
(8a) (8b) (8c)
ù
ÝSXê'Aۿ´²w'§'X÷X φ Id φ φ + dφ 'ål£l¤Ø´ dφ§ ´ h dφ = ρdφ§ù´·¤Ù'©´&ÎIX'ü ¥þ
ÄkíIX¥'ÑÝLª©¥þ| A(r) 'ÑݽÂ
∇ · A = lim
∂ (∆V ) A
· dS , (19)
Ù¥ ∆V ´¹X r £¦ÙÑÝ'X§Q±þ½Â¥±w'½§¢TX'À´?¿'¤' «§ ∂(∆V ) ´ ∆V '>.¡§dS ´Ù¡È£©8 ∆V dI¡ q !q + ∆q !q !q + ∆q !q !q + ∆q ¤¤'8¡N§Oþª©f¥'È©§=¥þ|Q ∆V '>.¡'Ïþ©I 5¿'´§8¡NQ q ?'¡È´ h h ∆q ∆q § Ø´ ∆q ∆q ©u´
∇u = ∂u 1 ∂u 1 ∂u er + eθ + eφ . ∂r r ∂θ r sin θ ∂φ (16)
§1
¢IX ¥©Î
5
ÖÌ^ Laplace Χe¡ÒÑü«~^'IX!=ÎIXÚ¥I X¥' Laplace Î/ª©duIX¥' Laplace Î
r0
Ù¥ r ´ ∆V SX§þªÑ
q ?Ú q + ∆q ?ü¡' z§ /· · · 0L«Ù§o¡' z§N´ÑÙA'Lª©du ∆V = h h h ∆q ∆q ∆q § ¨ ∆q → 0!∆q → 0!∆q → 0 §r → r§%&
§%
(11a) (11b) (11c)
∂r = sin θ cos φex + sin θ sin φey + cos θez , ∂r ∂r = r(cos θ cos φex + cos θ sin φey − sin θez ), ∂θ ∂r = r sin θ(− sin φex + cos φey ). ∂φ
∇2 u =
âc¡ïÄFÝΤ¼&'²¨§·N´ß§ÎIX¥' Laplace ÎU´
∇2 u =
∂2u ∂2u ∂2u + + , ∂x2 ∂y 2 ∂z 2
¥IX¥' Laplace ÎU´
∇2 u =
∂2u 1 ∂2u ∂2u + + , ∂ρ2 ρ2 ∂φ2 ∂z 2
¢ù´Ø('©ÎIX¥'(@tAT´
x = x(q1 , q2 , q3 ),
1 2 3
y = y (q1 , q2 , q3 ),
z = z (q1 , q2 , q3 ).
(1a)
§(q , q , q ) ±L (x, y, z) '¼ê© 5 5`§·¦ Jacobi 1ª
∂x/∂q1 ∂x/∂q2 ∂x/∂q3 ∂ (x, y, z ) J= = ∂y/∂q1 ∂y/∂q2 ∂y/∂q3 = 0. ∂ (q1 , q2 , q3 ) ∂z/∂q1 ∂z/∂q2 ∂z/∂q3
1 ∂ ∇ u= ρ ∂ρ
2
1 ∂2u 1 ∂2u ∂2u + + . ∂r2 r2 ∂θ2 r2 sin2 θ ∂φ2
∂u ρ ∂ρ
1 ∂2u ∂2u + 2 2 + 2, ρ ∂φ ∂z
(17)
¥I X¥'( @tAT´
∇2 u = 1 ∂ r2 ∂r r2 ∂u ∂r + 1 ∂ 2 r sin θ ∂θ sin θ ∂u ∂θ + 1 ∂2u . r2 sin2 θ ∂φ2 (18)
©
§1
¢IX ¥©Î yQwwþªQIX¥'/ª©du dr =
3 3 i=1 (∂ r /∂qi )dqi
3
§%
(3)
(ds) = dr · dr =
2
∂r ∂r · dqi dqj . ∂qi ∂qj i,j =1
XtÑ['©þ/ª§uþª¹
18 ©yQ§·±Ñ¢IX'½ µXJ ∂r ∂r KIX (q , q , q ) ¡¢©ùk
i i i 2 1 2 1 2 3 1 2 3 1 1 1 1 1 1 1 1 1 1 −1 1 1 2 3
ei =
dª (4) ÚÝSXê'½Â§´&
1 ∂r , hi ∂qi
i = 1, 2, 3.
(6)
nxI(¢´??¢'©e¡§·^þã½Â#©Ù'¥IX ÚÎIX© ~ 1 ÎIX (ρ, φ, z)©du r = ρ cos φe + ρ sin φe + ze §%
eρ = cos φex + sin φey , eφ = − sin φex + cos φey , ez = ez . (10a) (10b) (10c)
(9)
§1
¢IX ¥©Î ~ 2 ¥IX (r, θ, φ)©du r = r sin θ cos φe
x
4 + r sin θ sin φey + r cos θez
3 i=1 i i i i i i
ei · ∇u =
u´&
1 fi = h− i ∂u/∂qi
§l
3 i=1
∂si
= lim
∆si →0
∆si
= lim
∆qi →0
hi ∆qi
=
hi ∂qi
,
∇u =
1 ∂u 1 ∂u 1 ∂u 1 ∂u ei = e1 + e2 + e3 . hi ∂qi h1 ∂q1 h2 ∂q2 h3 ∂q3 ∂u ∂u ∂u ex + ey + ez ∂x ∂y ∂z
2 2 4 6 6 7 8
c ù´¥ìÆÔnXÆ)ÆSêÆÔn{§'ë]§d c?©H?Û<E ^uÆS½Æë©H1µ©Õ^uÑÈ©
∗
ห้องสมุดไป่ตู้
c 1992–2005
1
§1
¢IX ¥©Î
2
Ùm©r©lCþ{íP '"¢S'n¯K©c®Ñ§©lCþ§ATâ >F'/Gæ^·¨'IX©Ù'8'Ò´ïÄXÛQ¥IXÚÎIX¥éa §©lCþ©·¤#'Aa§Ñ¹k Laplace Χ¤±ÄkIïÄ Laplace ÎQIX§cÙ´¥IXÚÎIX¥'/ª©
1 2 3 3
· = 0, ∂qi ∂qj
i=j
(4)
(ds) =
i=1
2
∂r ∂qi
2
3
(dqi ) ≡
i=1
2
(hi dqi )2 ,
i
(5)
Ù¥ h = |∂r/∂q | ¡ÝSXê©ùIX¥'/ª (2) q§´ dq c¡õ
ÝSXê h ©¤±§¢'9 Ò´ (ds) 'Lª¥Ø¹ dq dq ù$'¢£© yQí¢IX (q , q , q ) ¥'ü ¥þ (e , e , e )©e 'Ò´ q I '§Ïd§÷XI q §k dr = ds e = h dq e §,¡§dr = (∂r/∂q )dq § % e = h (∂r/∂q )§é e Ú e kaq@t©o@å5§Ò´