给定积分 $I=\iint_D\left[\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2\right] d x d y$, 作正则变换 $x=x(u, v), y=y(u, v)$, 区域 $D$ 变为 $\Omega$ ，如果变换满足

$$
\frac{\partial x}{\partial u}=\frac{\partial y}{\partial v}, \quad \frac{\partial x}{\partial v}=-\frac{\partial y}{\partial u}
$$


证明:

$$
I=\iint_{\Omega}\left[\left(\frac{\partial f}{\partial u}\right)^2+\left(\frac{\partial f}{\partial v}\right)^2\right] d u d v
$$