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.

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