全微分

1.全微分的概念

（1）定义

\[如果函数z=f(x,y)在点(x,y)的某邻域内有定义,如果函数在点(x,y)的全增量\\ \Delta z=f(x+\Delta x,y+\Delta y)-f(x,y)可表示为\Delta z=A\Delta x+B\Delta y +o(\rho),其中A和B不依赖\\ 于\Delta x和\Delta y而仅与x和y有关，\rho=\sqrt{(\Delta x)^2+(\Delta y)^2},那么称函数z=f(x,y)在点(x,y)\\ 可微分，而且A\Delta x+B\Delta y称为函数z=f(x,y)在(x,y)的全微分，记作dz,及\\ dz=A\Delta x+B\Delta y.\\ 如果函数在区域D内处处都可微，那么称这函数在D内可微分 \]

可微的必要条件

\[如果函数z=f(x,y)在点(x,y)可微分，则函数z=f(x,y)在点(x,y)的偏导数\\ \frac{\partial z}{\partial x},\frac{\partial z}{\partial y}比存在，且dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy\\ 对于可微的三元函数u=f(x,y,z)，也有du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy+\frac{\partial u}{\partial z}dz\\ 可微\Rightarrow偏导存在\nRightarrow可微\\ \]

可微的等价定义

\[对于函数z=f(x,y),若\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}存在，且\\ \lim_{\rho\rightarrow 0}\frac{\Delta z-\frac{\partial z}{\partial x}\Delta x-\frac{\partial z}{\partial y}\Delta y}{\rho}=0\\ 则z=f(x,y)可微 \]

可微的充分条件

\[如果函数z=f(x,y)的偏导数\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}在点(x,y)连续，那么函数在该点可微分 \]

\[证明：\Delta z=A.\Delta x+B.\Delta y +o(\rho)\qquad A=\frac{\partial z}{\partial x}\quad B=\frac{\partial z}{\partial y}\\ \Delta z=f(x+\Delta x,y+\Delta y)-f(x,y+\Delta y)+f(x,y+\Delta y)-f(x,y)\\ g(x+\Delta x)-g(x)=g'(\xi)\Delta x\\ =g'(x+\theta\Delta x).\Delta x \qquad \xi介于x与x+\Delta x之间\quad 0<\theta <1\\ 由拉氏定理有\\ f(x+\Delta x,y+\Delta y)-f(x,y+\Delta y)=f_1'(x+\theta_1\Delta x,y+\Delta y).\Delta x \qquad (0<\theta_1<1)\\ f(x,y+\Delta y)-f(x,y)=f_2'(x,y+\theta_2\Delta y).\Delta y\qquad(0<\theta_2<1)\\ 又\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}连续\\ 故：\lim_{\overset{\Delta x\rightarrow 0}{\Delta y\rightarrow 0}}f_1'(x+\theta_1\Delta x,y+\Delta y)=f_1'(x,y)\\ \lim_{\overset{\Delta x\rightarrow 0}{\Delta y\rightarrow 0}}f_2'(x,y+\theta_2\Delta y)=f_2'(x,y)\\ 由极限与无穷小的关系，有f_1'(x+\theta_1\Delta x,y+\Delta y)=f_1'(x,y)+\xi_1(\xi_1\rightarrow 0)\\ f_2'(x,y+\theta_2\Delta y)=f_2'(x,y)+\xi_2(\xi_2\rightarrow 0)\\ 从而\Delta z=\Big[f_1'(x,y)+\xi_1\Big]\Delta x+\Big[f_2'(x,y)+\xi_2\Big]\Delta y\\ =f_1'(x,y)\Delta x+f_2'(x,y)\Delta y+\xi_1\Delta x+\xi_2\Delta y\\ 又0\leq\Big|\frac{\xi_1\Delta x+\xi_2\Delta y}{\sqrt{(\Delta x)^2+(\Delta y)^2}}\Big|\leq|\xi_1|\frac{|\Delta x|}{\sqrt{(\Delta x)^2+(\Delta y)^2}}+|\xi_2|\frac{|\Delta y|}{\sqrt{(\Delta x)^2+(\Delta y)^2}}\leq|\xi_1|+|\xi_2|\\ 故由夹逼准则\lim_{\overset{\Delta x\rightarrow 0}{\Delta y\rightarrow 0}}\frac{\xi_1\Delta x+\xi_2\Delta y}{\sqrt{(\Delta x)^2+(\Delta y)^2}}=0\\ \therefore \xi_1\Delta x+\xi_2\Delta y=o(\rho)\\ 从而\Delta z=f_1'(x,y)\Delta x+f_2'(x,y)\Delta y+o(\rho)\\ 故...可微 \]

全微分形式的不变性

\[设z=f(u,v),u=u(x,y),v=(x,y),则dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy=\frac{\partial z}{\partial u}du+\frac{\partial z}{\partial v}dv \]