必要的数学知识

偏微分关系

倒易关系：$(\dfrac{\partial x}{\partial y})_z(\dfrac{\partial y}{\partial x})_z=1$
循环关系：$(\dfrac{\partial x}{\partial y})_z(\dfrac{\partial y}{\partial z})_x(\dfrac{\partial z}{\partial x})_y=-1$
复合关系1：$(\dfrac{\partial x}{\partial y})_w(\dfrac{\partial y}{\partial z})_w=(\dfrac{\partial x}{\partial z})_w$
复合关系2：$(\dfrac{\partial x}{\partial y})_w=(\dfrac{\partial x}{\partial y})_z+(\dfrac{\partial x}{\partial z})_y(\dfrac{\partial z}{\partial y})_w$

Proof:设$f(x,y,z)=0$，则

$(\dfrac{\partial f}{\partial x})_{y,z}dx+(\dfrac{\partial f}{\partial y})_{x,z}dy+(\dfrac{\partial f}{\partial z})_{x,y}dz=0$

如果x不变，则

$(\dfrac{\partial f}{\partial y})_{x,z}(\dfrac{\partial y}{\partial z})_{x}=-(\dfrac{\partial f}{\partial z})_{x,y}$

即

$(\dfrac{\partial y}{\partial z})_{x}=-(\dfrac{\partial f}{\partial z})_{x,y}/(\dfrac{\partial f}{\partial y})_{x,z}$ $(\dfrac{\partial z}{\partial x})_{y}=-(\dfrac{\partial f}{\partial x})_{y,z}/(\dfrac{\partial f}{\partial z})_{x,y}$ $(\dfrac{\partial x}{\partial y})_{z}=-(\dfrac{\partial f}{\partial y})_{x,z}/(\dfrac{\partial f}{\partial x})_{y,z}$

三者相乘可得循环关系：

$(\dfrac{\partial x}{\partial y})_z(\dfrac{\partial y}{\partial z})_x(\dfrac{\partial z}{\partial x})_y=-1$

将

$(\dfrac{\partial x}{\partial y})_{z}=-(\dfrac{\partial f}{\partial y})_{x,z}/(\dfrac{\partial f}{\partial x})_{y,z}$

中的x和y交换，即可证明倒易关系。

不难推广到n维：
$(\dfrac{\partial x_1}{\partial x_2})_{x_3}…(\dfrac{\partial x_{n-1}}{\partial x_n})_{x_1}=(-1)^n$

雅可比行列式

定义以下符号：

$\frac{\partial{(u,v)}}{\partial{(x,y)}}=\left|\begin{matrix}\dfrac{\partial u}{\partial x}&\dfrac{\partial u}{\partial y}\\\dfrac{\partial v}{\partial x}&\dfrac{\partial v}{\partial y}\end{matrix}\right|$

满足以下性质：

偏导数的表示： $(\dfrac{\partial u}{\partial x})_y=\frac{\partial{(u,y)}}{\partial{(x,y)}}$
交换律： $\frac{\partial{(u,v)}}{\partial{(x,y)}}=-\frac{\partial{(v,u)}}{\partial{(x,y)}}$ $\frac{\partial{(u,v)}}{\partial{(x,y)}}=-\frac{\partial{(u,v)}}{\partial{(y,x)}}$
倒数： $\frac{\partial{(u,v)}}{\partial{(x,y)}}=(\frac{\partial{(x,y)}}{\partial{(u,v)}})^{-1}$
结合律： $\frac{\partial{(u,v)}}{\partial{(x,y)}}=\frac{\partial{(u,v)}}{\partial{(r,s)}}\frac{\partial{(r,s)}}{\partial{(x,y)}}$

通过以上性质可以等价推导上述微分关系。

倒数性质->倒易关系
结合律->复合关系1
结合律+行列式性质->循环关系