Module 1 : A Crash Course in Vectors
Lecture 6 : Laplacian
 

Example 24 :
Calculate the Laplacian of $1/r = 1/\sqrt{x^2+y^2+z^2}$.
Solution :

 

\begin{eqnarray*} \frac{\partial^2}{\partial x^2}\frac{1}{\sqrt{x^2+y^2+z^2}} &=... ...c{2x^2-y^2-z^2}{(x^2+y^2+z^2)^{5/2}}\\ &=& \frac{3x^2-r^2}{r^5} \end{eqnarray*}

Adding similar contributions from $\partial^2/\partial y^2$and $\partial^2/partial z^2$, we get

\begin{displaymath}\nabla^2\frac{1}{r} = \frac{3(x^2+y^2+z^2)-3r^2}{r^5}=\frac{3r62-3r^2}{r^5}=0\end{displaymath}

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