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The Laplacian in Spherical Polar Coordinates

The gradient of a function  
\[f\]
  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
\[\begin{equation} \begin{aligned} \nabla^2 f &= \mathbf{\nabla} \cdot (\mathbf{\nabla} f) \\ &= \frac{1}{r^2} \frac{\partial}{\partial r}( r^2 \frac{\partial f}{\partial r}) + \frac{1}{r sin \theta} \frac{\partial}{\partial \phi}(\frac{1}{r sin \theta} \frac{\partial f}{\partial \phi}) + \frac{1}{r sin \theta} \frac{\partial f}{\partial \theta}( \frac{sin \theta}{r} \frac{\partial f}{\partial \theta}) \\ &= \frac{1}{r^2} \frac{\partial}{\partial r} ( r^2 \frac{\partial f}{\partial r}) + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2 f}{\partial \phi^2} + \frac{1}{r^2 sin \theta} \frac{\partial f}{\partial \theta}( sin \theta \frac{\partial f}{\partial \theta}) \end{aligned} \end{equation}\]