设 z=f(x,y) ，u=u(x,y) ,v=v(x,y) ，求 z 关于 x,y 的二阶偏导数的展开式

泮安宁

z=f(u,v)，u=u(x,y)，v=v(x,y)
这个复合函数的二阶偏导具体要怎么写
\(\frac{\partial^2z}{\partial x\partial y}=\frac{\partial\left( \frac{\partial z}{\partial x}\right)}{\partial y}\)
这个具体要怎么展开

马奕琛

\(以下默认u_{xy}=u_{yx}，v_{xy}=v_{yx}，f_{uv}=f_{vu}\)

luyuanhong

楼上 马奕琛 的解答很好！已收藏。

elim

记 \(\dfrac{\partial w}{\partial t} = \partial_t w = w_t\) , 则 \(z_x = z_u u_x + z_v v_x,\)
\((z_x)_y=(z_u)_yu_x+z_u(u_x)_y+(z_v)_yv_x+z_v(v_x)_y\)
\(\qquad=z_{uu}u_xu_y+z_{uv}v_yu_x+z_uu_{xy}+z_{uv}u_yv_x+z_{vv}v_xv_y+z_vv_{xy}\)
\(\qquad=z_{uu}u_xu_y+z_{uv}(u_xv_y+u_yv_x)+z_{vv}v_xv_y+z_uu_{xy}+z_vv_{xy}\)