二重积分：变量替换与 Jacobian

Consider 2 variables $u=u(x,y)$ and $v=v(x,y)$, and we want to change the variable of

$$\iint_R f(x,y) dxdy$$

into $u,v$. We define the Jacobian of variable $u,v$ with respect to $x,y$ as

$$J=\frac{\partial (u,v)}{\partial (x,y)}=\frac{\partial u}{\partial x}\frac{\partial v}{\partial y}-\frac{\partial u}{\partial y}\frac{\partial v}{\partial x}=\det\begin{pmatrix} u_x & v_x\\ u_y & v_y \end{pmatrix}$$

Then,

$$\iint_{R'} f(u,v)dudv=\iint_R f(u(x,y), v(x,y)) \textcolor{red}{|J|} dxdy$$

Inversely,

$$\iint_{S} f(x,y) dxdy = \iint_{S'} f(u,v) \textcolor{red}{\Big|\frac{1}{J}\Big|} dudv$$