Eܼê óª{K EÜN ©¥½n p ê ©/ªØC/ §9.4 Eܼê ê 9.4.1 Eܼê êóª{K ±¼ê~, u = f(ξ, ζ) ½Â3« D þ¼ê, ξ = φ(x, y), ζ = ψ(x, y) ½Â3« ∆ þ. ¦Eܼê u = f(φ(x, y), ψ(x, y)) k¿Â, Kb (φ(x, y), ψ(x, y)) ∈ D é?Û (x, y) ∈ ∆ ¤á. ±eo´ùb. 1/17 kJ Ik J I £ ¶ '4 òÑ
Eܼê óª{K EÜN ©¥½n p ê ©/ªØC/ Ö{²Úí25, ·kæ^þ¼ê{. P g = (g1, g2), ùp g1, g2 Ñ´½Â3 R2 ¥« ∆ þ¼ê. ù g : ∆ → R 2 ´þ¼ê. D ´ R2 ¥«, g(∆) ⊂ D. e¼ê f 3 D þk½Â, K f ◦ g Ò´½Â3 ∆ þ¼ê. ∆ D R g f f ◦ g 2/17 kJ Ik J I £ ¶ '4 òÑ
Eܼê óª{K EÜN ©¥½n p ê ©/ªØC/ 3?Ø5c, Äk)ûEܼêëY5¯K. ½n 1 ∆ Ú D Ñ´ R2 ¥«. e g : ∆ → D 3 x0 ∈ ∆ ëY, f : D → R 3 y0 = g(x0) ëY, KEܼê f ◦ g : ∆ → R 3 x0 ëY. y² Ï f 3 y0 ëY, ¤±é?¿ ε > 0 3 δ1 > 0, ¦ y ∈ B(y0, δ1) ∩ D , k |f(y) − f(y0)| 0 ¦ x ∈ B(x0, δ) ∩ ∆ , k |g(x) − g(x0)| < δ1, =, y = g(x) ∈ B(y0, δ1) ∩ D. dd x ∈ B(x0, δ) ∩ ∆ , k |f ◦ g(x) − f ◦ g(x0)| < ε. ùÒ`² f ◦ g 3 x0 ëY. 3/17 kJ Ik J I £ ¶ '4 òÑ
Eܼê óª{K EÜN ©¥½n p ê ©/ªØC/ ½n 2 ∆ Ú D Ñ´ R2 ¥«. e g : ∆ → D 3 x ∈ ∆ , f : D → R 3 y = g(x) , KEܼê f ◦ g : ∆ → R 3 x , J(f ◦ g)(x) = Jf(y)Jg(x). (9.1) . úª (9.1) L« Eܼê f ◦ g ê f Ú g êm 'X, ¡Eܼê êóª{K. x = (x1, x2), y = g(x), =, y1 = g1(x1, x2), y2 = g2(x1, x2), z = f(y1, y2) = f ◦ g, K (9.1) Ò´ ∂z ∂x1 , ∂z ∂x2 = ∂z ∂y1 , ∂z ∂y2 ∂y1 ∂x1 ∂y1 ∂x2 ∂y2 ∂x1 ∂y2 ∂x2 ! (9.2) ÐmÒ´ ∂z ∂x1 = ∂z ∂y1 ∂y1 ∂x1 + ∂z ∂y2 ∂y2 ∂x1 (9.3) ∂z ∂x2 = ∂z ∂y1 ∂y1 ∂x2 + ∂z ∂y2 ∂y2 ∂x2 (9.4) 4/17 kJ Ik J I £ ¶ '4 òÑ
Eܼê óª{K EÜN ©¥½n p ê ©/ªØC/ y² éu x ∈ ∆ h ∈ R2 ¿©, ¦ x + h ∈ ∆. Ï g 3 x , ¤± g1(x + h) − g1(x) = Jg1(x) · h + o(h) (9.5) g2(x + h) − g2(x) = Jg2(x) · h + o(h). (9.6) g 5y §ëY5, Ïd h ªu", k = g(x + h) − g(x) ªu", ùpr k wÑþ. Ï f 3 y = g(x) , ¤± f(y + k) − f(y) = Jf(y) · k + o(k). (9.7) (9.5) Ú (9.6) Ü¿å5±¤ k = Jg(x)h + o(h), ò\ (9.7) ¿5¿ o(k) = o(h), Ò f(g(x+h))−f(g(x)) = Jf(y)· Jg(x)·h+o(h) +o(k) = Jf(y)·Jg(x)·h+o(h). ùÒL²Eܼê f ◦ g 3 x , (9.1) ¤á. 5/17 kJ Ik J I £ ¶ '4 òÑ
Eܼê óª{K EÜN ©¥½n p ê ©/ªØC/ ½n 3 ∆ ⊂ Rn Ú D ⊂ Rm Ñ´«. e g : ∆ → D 3 x ∈ ∆ , f : D → Rl 3 y = g(x) , KEÜN f ◦ g : ∆ → R 3 x , J(f ◦ g)(x) = Jf(y)Jg(x). (9.8) . ^ãL«Ò´, (x1, · · · xn) g −→ (y1, · · · , ym) f −→ (z1, · · · , zl) Ñ, f ◦ g . (9.8) Ò´ ∂z1 ∂x1 · · · ∂z1 ∂xn . . . . . . . . . ∂zl ∂x1 · · · ∂zl ∂xn = ∂z1 ∂y1 · · · ∂z1 ∂ym . . . . . . . . . ∂zl ∂y1 · · · ∂zl ∂ym ∂y1 ∂x1 · · · ∂y1 ∂xn . . . . . . . . . ∂ym ∂x1 · · · ∂ym ∂xn (9.9) 6/17 kJ Ik J I £ ¶ '4 òÑ
Eܼê óª{K EÜN ©¥½n p ê ©/ªØC/ ~ 1 Äk£eê½Â. ¤¢3: (x0, y0) êÙ ¢Ò´Eܼê u = f(x0 + t cos α, y0 + t cos β) éCþ t ¦, ùp ~e = (cos α, cos β) ´ü þ. ¤±âEÜ¼ê¦ 5K, k ∂u ∂~e = df dt = ∂f ∂x dx dt + ∂f ∂y dy dt = ∂f ∂x cos α + ∂f ∂y cos β 5`, =¦Ø´ü þ, Xéþ ~l = (l1, l2), E, k ∂u ∂~l = df(x0 + tl1, x0 + tl2) dt = ∂f ∂xl1 + ∂f ∂yl2 5¿, ·Ï~S.^/d0L«ékCþê, ^/∂0L« éõCþ ê. 7/17 kJ Ik J I £ ¶ '4 òÑ
Eܼê óª{K EÜN ©¥½n p ê ©/ªØC/ ~ 2 u = f(x, y, z), y = ϕ(x, r), z = ψ(x, y, r). ¦ ∂u ∂x, ∂u ∂r . ) ùp²LEÜ, r u ¤gCþ x, r ¼ê u = f(x, y, z) = f(x, ϕ(x, r), ψ(x, y, r)) = f(x, ϕ(x, r), ψ(x, ϕ(x, r), r)) = u(x, r). Ïd·k ∂u ∂x = f 0 x + f 0 yϕ 0 x + f 0 z (ψ 0 x + ψ 0 yϕ 0 x ), ∂u ∂r = f 0 yϕ 0 r + f 0 z (ψ 0 yϕ 0 r + ψ 0 r ). 8/17 kJ Ik J I £ ¶ '4 òÑ
Eܼê óª{K EÜN ©¥½n p ê ©/ªØC/ ~ 3 u:>Ö q )> ´ u = q r , ùp r ´: ~r = (x, y, z) :ål r = p x2 + y2 + z 2 . r 6= 0 , ¦§3m?¿:?F Ý9÷ ~r CzÇ, ¿y²¼ê u ÷v Laplace § ∂ 2u ∂x2 + ∂ 2u ∂y2 + ∂ 2u ∂z2 = 0. ) Ï ∂u ∂x = −q 1 r 2 ∂r ∂x = −q 1 r 2√ x x2+y2+z 2 = −q x r 3 , |^¼ê'ugCþ é¡5, ØJ u ©O'u y, z ê. Ïd, grad u = −q ~r r 3 . P ~r þü þ ~r0 = ~r r , K> u ÷ ~r CzÇ ∂u ∂~r0 = grad u · ~r0 = grad u · ~r r = −q ~r r 3 · ~r r = − q r 2 . UYé u ¦ ê ∂ 2u ∂x2 = q ∂ ∂x − x r 3 = q −r 3+3r 2xr0 x r 6 = −q 1 r 3 + q 3x 2 r 5 . d¼êéugCþé¡5q ∂ 2u ∂y2 = −q 1 r 3 + q 3y 2 r 5 , ∂ 2u ∂z2 = −q 1 r 3 + q 3z 2 r 5 . ¤± u = q r (r 6= 0) ÷v Laplace §. ¡ Laplace §)NÚ¼ê. 9/17 kJ Ik J I £ ¶ '4 òÑ
Eܼê óª{K EÜN ©¥½n p ê ©/ªØC/ ½n 4 (¼ê©¥½n) f(x, y) 3« D ¥. eë (x0, y0) Ú (x0 + h, y0 + k) ã L ¹3 D ¥, K3 θ ∈ (0, 1), ¦ f(x0 + h, y0 + k) = f(x0, y0) + f 0 x (x0 + θh, y0 + θk)h + f 0 y (x0 + θh, y0 + θk)k. y² ò f(x, y) 3 L þ´¼ê: ϕ(t) = f(x0 + th, y0 + tk), t ∈ [0, 1]. Ï f(x, y) Ú t ¼ê x0 + th 9 y0 + tk Ñ, ¤± ϕ(t) 3 [0, 1] . â¼ê©¥½n, 3 θ ∈ (0, 1) ¦ ϕ(1) − ϕ(0) = ϕ0 (θ). d ¼ê êóª{K, k ϕ 0 (θ) = f 0 x (x0 + θh, y0 + θk)h + f 0 y (x0 + θh, y0 + θk)k u´¤y¤á. 10/17 kJ Ik J I £ ¶ '4 òÑ