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The Laplacian operator for a Scalar function $\phi$ is defined by

\nabla^2 \phi = {1\over h_1h_2h_3}\left[{{\partial \over \pa...
...h_2\over h_3}{\partial \over \partial u_3}}\right)}\right]\phi
\end{displaymath} (1)

in Vector notation, where the $h_i$ are the Scale Factors of the coordinate system. In Tensor notation, the Laplacian is written
$\displaystyle \nabla^2\phi$ $\textstyle =$ $\displaystyle (g^{\lambda\kappa}\phi _{;\lambda})_{;\kappa}= g^{\lambda\kappa}{...
...\lambda\partial x^\kappa}-\Gamma^\lambda {\partial\phi\over\partial x^\lambda }$  
  $\textstyle =$ $\displaystyle {1\over\sqrt{g}}{\partial\over\partial x^j}\left({\sqrt{g}g^{ij}{\partial\phi\over\partial x^i}}\right),$ (2)

where $g_{;\kappa}$ is a Covariant Derivative and
\Gamma^\lambda \equiv {\textstyle{1\over 2}}g^{\mu\nu}g^{\la...
...l x^\mu}-{\partial g_{\mu\nu}\over \partial x^\kappa}}\right).
\end{displaymath} (3)

The finite difference form is
$\nabla^2\psi(x,y,z) = {1\over h^2}[\psi(x+h,y,z)+\psi(x-h,y,z)$
$ +\psi(x,y+h,z)+\psi(x,y-h,z)+\psi(x,y,z+h)$
$ +\psi(x,y,z-h)-6\psi(x,y,z)].\quad$ (4)
For a pure radial function $g(r)$,
$\displaystyle \nabla^2 g(r)$ $\textstyle \equiv$ $\displaystyle \nabla \cdot [\nabla g(r)]$  
  $\textstyle =$ $\displaystyle \nabla \cdot \left[{{\partial g(r)\over \partial r}\hat {\bf r}
...r\sin \theta}{\partial g(r)\over \partial \phi}\hat {\boldsymbol{\phi}}}\right]$  
  $\textstyle =$ $\displaystyle \nabla \cdot \left({\hat {\bf r}{dg\over dr}}\right).$ (5)

Using the Vector Derivative identity
\nabla\cdot (f{\bf A}) = f(\nabla\cdot{\bf A})+(\nabla f)\cdot(\bf A),
\end{displaymath} (6)

$\displaystyle \nabla^2 g(r)$ $\textstyle \equiv$ $\displaystyle \nabla \cdot [\nabla g(r)]
= {dg\over dr}\nabla \cdot \hat {\bf r}+ \nabla \left({dg\over dr}\right)\cdot \hat {\bf r}$  
  $\textstyle =$ $\displaystyle {2\over r}{dg\over dr}+ {d^2g\over dr^2}.$ (7)

Therefore, for a radial Power law,
$\displaystyle \nabla^2r^n$ $\textstyle =$ $\displaystyle {2\over r}nr^{n-1}+n(n-1)r^{n-2}= [2n+n(n-1)]r^{n-2}$  
  $\textstyle =$ $\displaystyle n(n+1)r^{n-2}.$ (8)

A Vector Laplacian can also be defined for a Vector A by

\nabla^2{\bf A}=\nabla(\nabla\cdot {\bf A})-\nabla\times(\nabla\times{\bf A})
\end{displaymath} (9)

in vector notation. In tensor notation, A is written $A_\mu$, and the identity becomes
$\displaystyle \nabla^2 A_\mu$ $\textstyle =$ $\displaystyle {A_{\mu;\lambda}}^{;\lambda} = (g^{\lambda\kappa}A_{\mu;\lambda })_{;\kappa}$  
  $\textstyle =$ $\displaystyle {g^\lambda\kappa}_{;\kappa}A_{\mu;\lambda}+g^{\lambda\kappa}A_{\mu;\lambda \kappa}.$ (10)

Similarly, a Tensor Laplacian can be given by
\nabla^2 A_{\alpha \beta}= {A_{\alpha \beta ;\lambda}}^{;\lambda}.
\end{displaymath} (11)

An identity satisfied by the Laplacian is

\nabla^2 \vert{\bf x}{\hbox{\sf A}}\vert = {{\vert{\hbox{\sf...
...sf A}}^{\rm T}\vert^2\over \vert{\bf x}{\hbox{\sf A}}\vert^3},
\end{displaymath} (12)

where $\vert{\hbox{\sf A}}\vert _2$ is the Hilbert-Schmidt Norm, ${\bf x}$ is a row Vector, and ${\hbox{\sf A}}^{\rm T}$ is the Matrix Transpose of A.

To compute the Laplacian of the inverse distance function $1/r$, where $r\equiv\vert{\bf r}-{\bf r}'\vert$, and integrate the Laplacian over a volume,

\int_V \nabla^2 \left({1\over \vert{\bf r}-{\bf r}'\vert}\right)d^3{\bf r}.
\end{displaymath} (13)

This is equal to
$\displaystyle \int_V \nabla^2 {1\over r}d^3{\bf r}$ $\textstyle =$ $\displaystyle \int_V \nabla \cdot\left({\nabla {1\over r}}\right)d^3{\bf r}
= \int_S \left({\nabla {1\over r}}\right)\cdot d{\bf a}$  
  $\textstyle =$ $\displaystyle \int_S {\partial \over \partial r}\left({1\over r}\right)\hat {\bf r}\cdot d{\bf a}
= \int_S - {1\over r^2}\hat {\bf r}\cdot d{\bf a}$  
  $\textstyle =$ $\displaystyle - 4\pi {R^2\over r^2},$ (14)

where the integration is over a small Sphere of Radius $R$. Now, for $r > 0$ and $R\to 0$, the integral becomes 0. Similarly, for $r = R$ and $R\to 0$, the integral becomes $-4\pi$. Therefore,
\nabla^2\left({1\over \vert{\bf r}-{\bf r}'\vert}\right)= -4\pi \delta^3({\bf r}-{\bf r}'),
\end{displaymath} (15)

where $\delta({\bf x})$ is the Delta Function.

See also Antilaplacian

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© 1996-9 Eric W. Weisstein