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This is my first post here, so please be gentle. I am interested in finding the laplacian $\nabla^2$ in spherical coordinates, where $\nabla^2 = \frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$

I understand that we can find {$\frac{d}{dx}, \frac{d}{dy}, \frac{d}{dz}$} using trigonometric relations, but the amount of algebra required is frankly hazardous, and not all that enlightening.

In section 4.1.1 of Introduction to Quantum Mechanics (Griffiths, 2nd ed.), he says that there's a simpler way of doing this by change of variables. I've searched around for the last week and haven't been able to find anything like this!

I know that {$dx, dy, dz$} can be found in terms of spherical differentials with the following tensor product:

$$ \begin{pmatrix} dx\\dy\\dz \end{pmatrix} = \begin{pmatrix} \sin{\theta}\cos{\phi} & r\cos{\theta}cos{\phi} & -r\sin{\theta}\sin{\phi}\\ \sin{\theta}\sin{\phi} & r\cos{\theta}sin{\phi} & r\sin{\theta}\cos{\phi}\\ \cos{\theta} & -r\sin{\theta} & 0\end{pmatrix} \begin{pmatrix} dr\\d\theta\\d\phi\end{pmatrix}$$

And I suspect that the partial derivatives can be found in a similar way, but I don't see what it is. Any help or pointers in the right direction would be much appreciated.

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