How do I compute the Laplacian of a function in terms of a given (general) coordinate transformation?

Question

Consider a coordinate transformation $\boldsymbol{x} = \boldsymbol{x}(\boldsymbol{\xi})$ (with Jacobian $\partial \boldsymbol{x}/\partial \boldsymbol{\xi})$, the scalar function $f(\boldsymbol{x}) = f(\boldsymbol{x}(\boldsymbol{\xi})) = \bar{f}(\boldsymbol{\xi})$ and the definitions

$$\boldsymbol{\nabla}_\xi (\cdot) = \partial (\cdot)/\partial \boldsymbol{\xi},$$ $$\boldsymbol{\nabla}_x (\cdot) = \partial (\cdot)/\partial \boldsymbol{x}.$$

I know how to write the gradient of $f(\boldsymbol{x})$, $\partial f(\boldsymbol{x})/\partial \boldsymbol{x}$ in this case: I start with

$$\boldsymbol{\nabla}_\xi \bar{f}(\boldsymbol{\xi}) = \boldsymbol{\nabla}_x f(\boldsymbol{x})\cdot\partial \boldsymbol{x}/\partial \boldsymbol{\xi} = \boldsymbol{\nabla}_x f(\boldsymbol{x})\cdot\boldsymbol{J} = \boldsymbol{J}^{\mathrm{T}}\cdot\boldsymbol{\nabla}_x f(\boldsymbol{x}),$$

so that $$\boldsymbol{\nabla}_x f(\boldsymbol{x}) = \boldsymbol{J}^{-\mathrm{T}}\cdot\boldsymbol{\nabla}_x \bar{f}(\boldsymbol{\xi}).$$

But how do I obtain an expression for the Laplacian

$$\boldsymbol{\nabla}_x\cdot\boldsymbol{\nabla}_x f(\boldsymbol{x})?$$

This seems basic, but I can't figure out how to do it. I've searched the internet and math.stackexchange but haven't found any answers for the general case.

Answer 1

You should use the Laplace-Beltrami operator on general Riemannian manifolds, for example:

You can find more information here.