LaplacianLaplacian2008-03-25 20:22:482008-03-25 20:22:48Definition% this is the default PlanetPhysics preamble. as your knowledge
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% define commands hereThe Laplacian is a vector differential operator. Like all vector operators, it is given in different forms in different coordinate systems. In general it is given by:
\[
\nabla^2 f = \Delta f = \sum_i \frac{\partial f_i}{\partial x^2_i}
\]
where the subscript $i$ refers to the different coordinate components of the vector $f$.
\subsection{Laplacian in Cartesian coordinates}
This is the simplest form. It is given by
\[
\nabla^2 = {\partial \over \partial x^2} + {\partial \over \partial y^2} + {\partial \over \partial z^2}
\]
\subsection{Laplacian in spherical coordinates}
This is given by
\[
\nabla _{cyl}^{2} = \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right) + \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2} + \frac{\partial^2}{\partial z^2}
\]
\subsection{Laplacian in cylindrical coordinates}