多元函数偏导，全微分，高阶偏导
一、回顾一元函数导数、微分形式
导数 $$f'(x)=\dfrac{dy}{dx}$$
微分形式$$\Delta y = f'(x) \Delta x + O(\Delta x)$$
二、多元函数一阶偏导
$$f_x'(x,y)=\dfrac{\partial z}{\partial x},f_y'(x,y)=\dfrac{\partial z}{\partial y}$$
把另一变量当作常量来求。
三、多元函数一阶全微分
$$\Delta z = f_x' \Delta x + f_y' \Delta y +O(\rho)\ ,\quad \rho=\sqrt{\Delta x^2+\Delta y^2}$$
四、多元函数二阶偏导
$$f_{xx}''(x,y)=\dfrac{\partial ^2 z}{\partial x^2}=\dfrac{\partial( \frac{\partial z}{\partial x})}{\partial x}$$
$$f_{xy}''(x,y)=\dfrac{\partial ^2 z}{\partial x \partial y}=\dfrac{\partial( \frac{\partial z}{\partial x})}{\partial y}$$
$$f_{yx}''(x,y)=\dfrac{\partial ^2 z}{\partial y \partial x}=\dfrac{\partial( \frac{\partial z}{\partial y})}{\partial x}$$
$$f_{yy}''(x,y)=\dfrac{\partial ^2 z}{\partial y^2}=\dfrac{\partial( \frac{\partial z}{\partial y})}{\partial y}$$
当$$f_{yx}'',f_{xy}''$$皆连续时，$$f_{yx}'' = f_{xy}''$$
五、连续、可偏导、可微关系
