Spherical coordinates + Laplacian

Question

If $f(x, y,z)=f(r\sin(\phi)\cos(\theta), r\sin(\phi)\sin(\theta),r\cos(\phi))$, what is the value of $\Delta f$, in terms of $r$, $\phi$ and $\theta$?

Answer 1

You may want to look at the gradient operator in spherical coordinates: $$\nabla f={\partial f \over \partial r}\boldsymbol{\hat r} + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta} + {1 \over r\sin\theta}{\partial f \over \partial \varphi}\boldsymbol{\hat \varphi}$$ And then: $$\Delta f=\nabla f \cdot (\Delta r,\Delta \theta, \Delta \phi)$$