The Laplacian in Spherical Polar Coordinates

$f$
is spherical polar coordinates is
$\mathbf{\nabla} f = \frac{\partial f}{\partial r} \mathbf{e_r} + \frac{1}{r} \frac{\partial f}{\partial \theta} \mathbf{e_\theta} + \frac{1}{r sin \theta} \frac{\partial f}{\partial \phi} \mathbf{e_\phi}$

Now use the fact that
$\nabla^2 f = \mathbf{\nabla} \cdot (\mathbf{\nabla} f )$

With
$\mathbf{F} = F_r \mathbf{e_r} + F_\theta{e_\theta} + F_\phi \mathbf{e_\phi}$

We have
$\mathbf{\nabla} \cdot \mathbf{F}= \frac{1}{r^2} \frac{\partial (r^2 F_r)}{\partial r} + \frac{1}{r sin \theta} \frac{\partial F_\phi}{\partial \phi} + \frac{1}{r sin \theta} \frac{\partial (sin \theta F_\theta)}{\partial \theta}$

Then
\begin{aligned} \nabla^2 f &= \mathbf{\nabla} \cdot (\mathbf{\nabla} f) \\ &= \frac{1}{r^2} \frac{\partial}{\partial r}( r^2 \frac{\partial f}{\partial r}) + \frac{1}{r sin \theta} \frac{\partial}{\partial \phi}(\frac{1}{r sin \theta} \frac{\partial f}{\partial \phi}) + \frac{1}{r sin \theta} \frac{\partial f}{\partial \theta}( \frac{sin \theta}{r} \frac{\partial f}{\partial \theta}) \\ &= \frac{1}{r^2} \frac{\partial}{\partial r} ( r^2 \frac{\partial f}{\partial r}) + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2 f}{\partial \phi^2} + \frac{1}{r^2 sin \theta} \frac{\partial f}{\partial \theta}( sin \theta \frac{\partial f}{\partial \theta}) \end{aligned}