FÝ ÑÝ ^Ý Å| à | Hamilton Î §11.6 FÝ!ÑÝ!^Ý |´lþÔny¥ÄÑ5ÔnVg. ©üa: þ|Úêþ|. þ|x´@ Qkqkþ. XÚå|!6|!>|! ^| . êþ|x´kvkþ. X> |!§Ý|. 3êÆþ·^þL«Qkqkþ. Ïd, m« V ¥þ|, ±w¤´½Â3 V þnþ¼ê ~r = ~r(M) (M ∈ V ). êþ|k, 3êÆþ±^n¼ê u = u(M) (M ∈ V ) L«. À½ mIX, : M Ò±^nI (x, y, z) L«. þ|Ú êþ|Ò©OL«¤ ~r = ~r(x, y, z) Ú u = u(x, y, z). ÀJØÓIX, |L/ªØ, جUC|ÔnA5. 1/28 kJ Ik J I £ ¶ '4 òÑ
FÝ ÑÝ ^Ý Å| à | Hamilton Î 11.6.1 êþ|FÝ éu D ⊂ R3 þ½Â1wêþ| f = f(x, y, z), Ùn ê |¤þ grad f = ∂f ∂x ~i + ∂f ∂y ~j + ∂f ∂z ~k ¡ f FÝ. ~e = cos α~i + cos β~j + cos γ~k ´^{uL«ü , ¼ê f(x, y, z) ÷Tê ∂f ∂~e = ∂f ∂x cos α + ∂f ∂y cos β + ∂f ∂z cos γ = grad f · ~e. dd, ÷FݼêCz¯. FÝ¢Sþ´lêþ|þ|N grad : f 7−→ grad f, ÷v 1 ◦ grad(c1u1 + c2u2) = c1 grad u1 + c2 grad u2, Ù¥ c1, c2 ´?¿~ê. 2 ◦ grad(u1u2) = u1 grad u2 + u2 grad u1. 3 ◦ grad f(u) = f 0 (u) grad u. 2/28 kJ Ik J I £ ¶ '4 òÑ
FÝ ÑÝ ^Ý Å| à | Hamilton Î ~ 1 ¦êþ| ¡{. ) ϕ = f(x, y, z) ´1wêþ|. éu~ê c, § f(x, y, z) = c L« ¡. bù ¡ëê§ ~r(u, v) = (x(u, v), y(u, v), z(u, v)), (u, v) ∈ D, K3 ¡þk f(x(u, v), y(u, v), z(u, v)) = c. éd¦ ê, ∂f ∂xx 0 u + ∂f ∂yy 0 u + ∂f ∂z z 0 u = 0, ∂f ∂xx 0 v + ∂f ∂yy 0 v + ∂f ∂z z 0 v = 0, Ò´ grad f · ~r0 u = 0, grad f · ~r0 v = 0. ÏdFÝ grad f { ~n = ~r0 u×~r0 v |~r0 u×~r0 v | ²1. 3/28 kJ Ik J I £ ¶ '4 òÑ
FÝ ÑÝ ^Ý Å| à | Hamilton Î ~ 2 ~r = (x, y, z), r = |~r| = p x2 + y2 + z 2 . ¦ grad r. ) Ï ∂r ∂x = √ x x2+y2+z 2 = x r , ∂r ∂y = y r , ∂r ∂z = z r , ¤± grad r = ~r r . ~ 3 3:>Ö q )> |´ ϕ = q r , ùp r ´: ~r = (x, y, z) :ål, =, r = p x2 + y2 + z 2 . ¦ ϕ 3m?¿: (Ø´ :) ?FÝ9÷ ~r CzÇ. ) dFÝ$5 grad ϕ = q grad 1 r = q · (− 1 r 2 ) grad r = −q ~r r 3 . ∂ϕ ∂~r = grad ϕ · ~r r = −q 1 r 2 . 4/28 kJ Ik J I £ ¶ '4 òÑ
FÝ ÑÝ ^Ý Å| à | Hamilton Î 11.6.2 þ|ÑÝ F~ = (P, Q, R) ´m¥1wþ|, Ø@´6|, S ´| ¥µ4¡, ý, KÏþ N = ✍ S F~ · ~ndS L«36|^e, ü mS6L¡ S 6N6þ. σ(V ) L « S ¤¤« V NÈ, K 1 σ(V ) ✍ S F~ · ~ndS L«ü mSü NÈþ6LNL¡6þ. S ¤: M , e4 lim S→M 1 ∆V ✍ S F~ · ~ndS 3, Ò¡d4þ| F~ 3 M ?ÑÝ, P div F~(M). 5/28 kJ Ik J I £ ¶ '4 òÑ
FÝ ÑÝ ^Ý Å| à | Hamilton Î þ| F~ 3:ÑÝL«´d:)6NUå. e div F~(M) > 0, K3 M :)6N, d:Ò¡/ ’0 , e div F~(M) < 0, K3 M :á Â6N, d:Ò¡/¦0½/®0. div F~ ??", F~ Ò¡Ã |. d Gauss úªÚÈ©¥½n, ✍ S F~ · ~ndS = ✝ V ∂P ∂x + ∂Q ∂y + ∂Q ∂z dV = ∂P ∂x + ∂Q ∂y + ∂Q ∂z (M0 )σ(V ), Ù¥ M0 ´ S S:, § S  M ªu M. Ïdk div F~(M) = ∂P ∂x + ∂Q ∂y + ∂Q ∂z (M). ùÒÑÝOúª: div F~ = ∂P ∂x + ∂Q ∂y + ∂Q ∂z . (11.1) Gauss úª±L« ✍ ∂V F~ · ~ndS = ✝ V div F dV. ~ (11.2) 6/28 kJ Ik J I £ ¶ '4 òÑ
FÝ ÑÝ ^Ý Å| à | Hamilton Î dc¡ùªf, ± Gauss úªÔn)º: ü mS6LÔ NL¡6þ uÔNS¤k/ 0½/®0êÚ. Ïd, Gauss úª¡ ÑÝúª. div : F~ → div F~ ´þ|êþ|N, §÷v±e${K: 1 ◦ div(c1F~1 + c2F~2) = c1 div F~1 + c2 div F~2, c1, c2 ´?¿~ê. 2 ◦ div(uF~) = u div F~ + grad u · F~, Ù¥ u ´êþ|. ~ 4 ¦>|rÝ E~ = q r 3~r ÑÝ, Ù¥ ~r = (x, y, z), r = |~r|. ) div E~ = div q r 3 ~r = q r 3 div ~r + grad q r 3 · ~r = 3q r 3 + q r 3 0 grad r · ~r = 3q r 3 − 3q r 4 ~r r · ~r = 0. 7/28 kJ Ik J I £ ¶ '4 òÑ
FÝ ÑÝ ^Ý Å| Ã | Hamilton Î 11.6.3 þ|^Ý F~ = (P, Q, R) ´m V ¥1w þ|, p0 ∈ V, ~n ´?¿ü þ. ± p0 %, ¿©ê ε »Ru ~n Dε ⊂ V, ∂Dε ´ Dε >. ~τ L« ∂Dε þ÷ (= ~n N) ü þ, KÈ© ~n ~τ Dε ☞ ∂Dε F~ · d~r = ☞ ∂Dε F~ · ~τ ds þ| F~ 3 ± ∂Dε þþ. Ïd 1 d(Dε) ☞ ∂Dε F~ · ~τ ds N þ| F~ 3 Dε þ²þ^=G, ùp d(Dε) ´ Dε ¡È. 8/28 kJ Ik J I £ ¶ '4 òÑ
FÝ ÑÝ ^Ý Å| à | Hamilton Î e ε → 0 , 4 Ω~n(p0) = lim ε→0 1 d(Dε) ☞ ∂Dε F~ · ~τ ds 3, Kd4¡þ| F~ 3 p0 ?7 ~n 6þ (µþ). §N F~ 3 p0 :^=G. F~ ´6|, 3 p0 ?/¤/µ, 3 p0 Ó, Ó3 F~ ^eò^=å5. UCÓ¶Ý, KÓ^=Ýu )Cz. 3¶ ~n, 3ùþ Ω~n(p0) . â Stokes úª, k ☞ ∂Dε F~ · d~r = ☎ Dε ~Ω · ~ndS, ùp ~Ω = ∂R ∂y − ∂Q ∂z , ∂P ∂z − ∂R ∂x , ∂Q ∂x − ∂P ∂y . dÈ©¥½n, 3: p 0 ∈ Dε ¦ ☎ Dε ~Ω · ~ndS = (~Ω · ~n)(p 0 )d(Dε). 9/28 kJ Ik J I £ ¶ '4 òÑ
FÝ ÑÝ ^Ý Å| à | Hamilton Î Ï k Ω~n(p0) = ~Ω(p0) · ~n. ddª, ~n ~Ω(p0) , Ω~n(p0) |~Ω(p0)|. u´, ¡ ~Ω(p) F~ 3 p :^Ý, P rot F~(p). =, rot F~ = ∂R ∂y − ∂Q ∂z , ∂P ∂z − ∂R ∂x , ∂Q ∂x − ∂P ∂y . (11.3) ¡ S þ Stokes úª±L ☞ ∂S F~ · d~r = ☎ S rot F~ · ~ndS. (11.4) rot F~ ??", F~ ¡Ã^|. dþ| F~ ÷µ4þ". ^ݱw¤´þ|þ|N, §÷vXe${K: 1 ◦ rot(c1F~1 + c2F~2) = c1 rot F~1 + c2 rot F~2, c1, c2 ´~ê. 2 ◦ rot(uF~) = u rot F~ + grad u × F~, u ´êþ|. 3 ◦ div(F~1 × F~2) = F~2 · rot F~1 − F~1 · rot F~2. 10/28 kJ Ik J I £ ¶ '4 òÑ