作者Link0763 (Miyavi雅)
看板KSU
标题[讨论] 工数偏导数及全微分
时间Sun Nov 1 17:31:47 2009
Z is a function x,y, that is z=z(x,y),x,y,z satisfy
z(z2 +3x)+3y=0. show that
ð2 z/ðx2 +ð2 z/ðy2 = 2z(x-1) / (z2 +x)3
Ans: 令 F(x,y,z) = z(z2 +3x)+3y=0
故ðz/ðx=(ðz/ðx)y = - Fx/fz = G(x,y,z),则
ð2 z/ðx2 =(ð2 z/ðx2 )y = (ð/ðx*ðz/ðx)y = (ðG/ðx)y
=ðG/ðx + ðG/ðx*ðz/ðx
=ð/ðx(-Fx/Fz) +ð/ðz(-Fx/Fz) ðz/ðx
=- FxxFz-FxFzx/(Fz)2 - FxzFz-FxFzz/(Fz)2 *(-Fx/Fz)
=- Fxx(Fz)2 +Fzz(Fz)2 -2FxzFxFz/(Fz)3 (因Fxz=Fzx)
= 2zx/(x+z2 )3
同理
ð2 z/ðy2 =- Fyy(Fz)2 +Fzz(Fy)2 -2FyzFyFz/(Fz)3
=-2z/(x+z2 ) 3
则
ð2 z/ðx2 +ð2 z/ðy2 = 2z(x-1) / (z2 +x)3
请问大大 他令G(x,y,z)是什麽意思押??
已转贴到数学板
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