40.Y. If $f: \mathbb{R}^2 \to \mathbb{R}$ has continuous second partial derivatives and if $F(r, \theta) = f(r \cos \theta, r \sin \theta)$ for $r > 0$, $\theta \in \mathbb{R}$, show that \begin{align*} D_{xx}f(x, y) + D_{yy}f(x, y) &= D_{rr}F(r, \theta) + \frac{1}{r}D_rF(r, \theta) + \frac{1}{r^2}D_{\theta\theta}F(r, \theta) \\ &= \frac{1}{r}D_r(rD_rF(r, \theta)) + \frac{1}{r^2}D_{\theta\theta}F(r, \theta), \end{align*} where $x = r \cos \theta$, $y = r \sin \theta$.
