40.E. If Cartesian axes are rotated in the plane by the angle $\theta$, then the new coordinates $u$, $v$ of a point are related to the original coordinates $x$, $y$ by
$x = u \cos \theta - v \sin \theta$,
$y = u \sin \theta + v \cos \theta$.
Let $f: \mathbb{R}^2 \to \mathbb{R}$ be differentiable on $\mathbb{R}^2$ and let $F(u, v) = f(x, y)$ for all $x$, $y$. Show that
$[D_1F(u, v)]^2 + [D_2F(u, v)]^2 = [D_1f(x, y)]^2 + [D_2f(x, y)]^2.$
