多元函数的偏导数

1.偏导数的基本概念

（1）定义

\[设函数z=f(x,y)在点(x_0,y_0)的某邻域内有定义，如果\\ \lim_{\Delta x\rightarrow 0}\frac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}\\ 存在，则称此极限为函数z=f(x,y)在点(x_0,y_0)对x的偏导数，记为\\ \frac{\partial z}{\partial x}\Big|_{\overset{x=x_0}{y=y_0}},\frac{\partial f}{\partial x}\Big|_{\overset{x=x_0}{y=y_0}},z_x\Big|_{\overset{x=x_0}{y=y_0}}或f_x(x_0,y_0)也或f'_1(x_0,y_0),\\ 即f_x(x_0,y_0)=\lim_{\Delta\rightarrow 0}\frac{f(x_0+\Delta x,y_0)-f(x_0),y_0}{\Delta x} \]

2.高阶偏导数

\[一般情况下，函数z=f(x,y)的两个偏导数f_x(x,y)和f_y(x,y)仍然是x,y的函数,因此\\ 可以考虑f_x(x,y)和f_y(x,y)的偏导数即二阶偏导数，依次为\\ \frac{\partial}{\partial x}\Big(\frac{\partial z}{\partial x}\Big)=\frac{\partial^2z}{\partial x^2}=f_{xx}(x,y),\frac{\partial }{\partial y}\Big(\frac{\partial z}{\partial x}\Big)=\frac{\partial^2 z}{\partial y\partial x},\\ \frac{\partial }{\partial x}\Big(\frac{\partial z}{\partial x}\Big)=\frac{\partial^2 z}{\partial y\partial x}=f_{yx}(x,y),\frac{\partial }{\partial y}\Big(\frac{\partial z}{\partial y}\Big)=\frac{\partial^2 z}{\partial y^2}=f_{yy}(x,y) \]

定理：

\[若函数z=f(x,y)的两个二阶混合偏导数\frac{\partial^2 z}{\partial x \partial y},\frac{\partial ^2 z}{\partial y\partial x}在区间D内均连续，\\ 那么在该区域内这两个二阶混合偏导数必然相等\\ 也就是，二阶混合偏导数在连续的条件下与求导的次序无关 \]

3，多元复合函数的偏导数

(1)一元函数与多元复合函数复合的情形

\[定理：如果函数u=\varphi(t),v=\psi(t)都在点t可导，函数z=f(u,v)在对应点(u,v)具有\\ 连续偏导数,那么复合函数z=f[\varphi(t),\psi(t)]在点t可导,且有\\ \frac{dz}{dt}=\frac{\partial z}{\partial u}\frac{du}{dt}+\frac{\partial z}{\partial v}\frac{dv}{dt} \]

\[定理：如果函数u=u(x,y),v=v(x,y)都在点(x,y)具有对x及对y的偏导数，函数\\ z=f(u,v)在对应点(u,v)在对应点(u,v)具有连续偏导数，那么复合函数z=f[u(x,y),v(x,y)]在点\\(x,y)的两个偏导数都存在，且有\\ \frac{\partial z}{\partial x}=f_u'(u,v)\frac{\partial u}{\partial x}+f_v'(u,v)\frac{\partial v}{\partial x},\\ \frac{\partial z}{\partial y}=f_u'(u,v)\frac{\partial u}{\partial y}+f_v'(u,v)\frac{\partial v}{\partial y}, \]