Jackson’s Laplacian in spherical Coordinates

If you took a look at one of the previous posts on how to remember the Laplacian in different forms by using a metric,  you will notice that the form of  the Laplacian that we get is:

\nabla^2 \psi = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \psi}{\partial r} \right) + \frac{1}{r^2 \sin(\theta)} \frac{\partial}{\partial \theta} \left( sin(\theta)  \frac{\partial \psi}{\partial \theta} \right)   + \frac{1}{r^2 \sin^2(\theta)} \frac{\partial^2 \psi}{\partial \phi^2}   

But in Jackson’s Classical Electrodynamics, III edition he notes the following:

20190218_135719

This is an interesting form of the Laplacian that perhaps not everyone has encountered. This can obtained from the known form by making the substitution u = r \psi and simplifying. The steps to which have been outlined below:

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