二重积分的换元

二重积分的换元&&雅可比(面积变换之间的“扭曲系数”)

$$\left(\begin{array}{l}du\\ dv\end{array}\right)=\left(\begin{array}{ll}\frac{\partial u}{\partial x}& \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}& \frac{\partial v}{\partial y}\end{array}\right)\left(\begin{array}{l}dx\\ dy\end{array}\right)$$
$$\begin{gathered} du\times dv=(\frac{\partial u}{\partial x}\ast dx+\frac{\partial u}{\partial y}\ast dy)\times (\frac{\partial v}{\partial x}\ast dx+\frac{\partial v}{\partial y}\ast dy) \\ (\frac{\partial u}{\partial x}\ast dx\times \frac{\partial v}{\partial y}\ast dy)+(\frac{\partial u}{\partial y}\ast dy\times \frac{\partial v}{\partial x}\ast dx) \\ =\left[\begin{array}{ll}\frac{\partial u}{\partial x}& \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}& \frac{\partial v}{\partial y}\end{array}\right]dx\times dy \end{gathered}$$

区域限制变换后的结果

$$\begin{gathered} u,v是x,y的函数,对于区域的限制也是x,y的函数f(x,y)到f(u,v) \\ 以上为例： \end{gathered}$$

限制1	限制2	–	–	u,v满足的条件(已知f(x,y),用u,v表示x，y带入)
x=0	x=1	x	y	f(x,y)
z-y=0	z-y=1	u=z-y	v=y	f(z-y,y)