## Euclidean Laplacian in spherical polar coordinates

We can express laplace of Euclidean space in sphere polar form,

$\displaystyle \Delta f= \frac{\partial^2 f}{\partial r^2}+ \frac{N-1}{r} \frac{\partial f}{\partial r}+ \frac{1}{r^2} \Delta_{\mathbb{S}^{N-1}} f$

Suppose ${r}$ is the distance function to the origin. ${(\theta^1,\cdots,\theta^{n-1})}$ is a local coordinates of ${\mathbb{S}^{N-1}}$. And ${\phi:(\theta^1,\cdots,\theta^{n-1})\rightarrow \mathbb{S}^{n-1}}$ is the diffeomorphism. Then ${(r,\theta^1,\cdots,\theta^{n-1})}$ is a coordinates of ${\mathbb{R}^N}$. Metric under this coordiates is

$\displaystyle g=\sum_i(d(r\phi^i))^2=\sum_i(\phi^idr+rd\phi^i)^2=(dr)^2+r^2\sum_i(d\phi^i)^2$

$\displaystyle =(dr)^2+r^2h_{ij}(\theta)d\theta^i d\theta^j$

$\displaystyle \Delta f=tr(Hess(f))=Hess(f)(\frac{\partial }{\partial r},\frac{\partial }{\partial r})+\frac{1}{r^2}h^{ij}Hess(f)(\frac{\partial }{\partial \theta^i},\frac{\partial }{\partial\theta^j})$

One can prove that

$\displaystyle Hess(f)(\frac{\partial }{\partial r},\frac{\partial }{\partial r})=\frac{\partial^2 f}{\partial r^2}+ \frac{N-1}{r} \frac{\partial f}{\partial r}$

Note that ${h_{ij}(\theta)d\theta^i d\theta^j}$ is the stardand metric of ${\mathbb{S}^{N-1}}$, then

$\displaystyle \Delta_{\mathbb{S}^{N-1}}f=h^{ij}Hess(f)(\frac{\partial }{\partial \theta^i},\frac{\partial }{\partial\theta^j})$

Our formula is proved.