Laplacian in Spherical CoordinatesLaplacianInSphericalCoordinates2005-12-30 20:39:072025-07-05 22:36:30Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe Laplacian operator in spherical coordinates is
\begin{equation}
\nabla _{sph}^{2} = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial}{\partial r}\right) + \frac{1}{r^2 sin\theta} \frac{\partial}{\partial \theta} \left( sin \theta \frac{\partial}{\partial \theta}\right) + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2}{\partial \phi^2}
\end{equation}
The derivation is fairly straight forward and begins with locating a vector {\bf r} in spherical coordinates as shown in the figure.
\begin{figure}
\includegraphics{SphericalCoordinates.eps}
\end{figure}