Polar-coordinate Laplacian identity ($\tilde{\Delta }=-{\partial }_r^2-\frac{d}{r}{\partial }_r+r^{-2}{\Delta }_{S^d}$)

In ${\mathbb{R}}^{d+1}$, $g=d{r}^2+r^2{g}_{S^d}$ and the volume density is $r^d$. With the nonnegative Laplacian convention, $\tilde{\Delta }=-{\partial }_r^2-(d/r){\partial }_r+r^{-2}{\Delta }_{S^d}$. The identity separates radial homogeneity from the angular Laplace-Beltrami operator and derives the spectrum of the Laplacian on a sphere.