LaplacianLaplacian2002-06-04 11:32:322008-12-16 11:25:26DefinitionLaplace operator% this is the default PlanetMath preamble. as your knowledge
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\newcommand*{\norm}[1]{\lVert #1 \rVert}
\newcommand*{\abs}[1]{| #1 |}Let $(x_1, \ldots, x_n)$ be Cartesian coordinates for some open set $\Omega$
in $\sR^n$.
Then the \emph{Laplacian} differential operator $\Delta$ is defined as
$$
\Delta = \frac{\partial^2 }{\partial x_1^2} + \cdots + \frac{\partial^2 }{\partial x_n^2}.
$$
In other words, if $f$ is a twice differentiable function $f:\Omega\to \sC$, then
$$
\Delta f = \frac{\partial^2 f}{\partial x_1^2} + \cdots + \frac{\partial^2 f}{\partial x_n^2}.
$$
A coordinate independent definition of the Laplacian
is $\Delta = \nabla \cdot \nabla$, i.e., $\Delta$ is the composition of
gradient and codifferential.
A harmonic function is one for which the Laplacian vanishes.
\subsubsection*{Notes}
An older symbol for the Laplacian is $\nabla^2$ -- conceptually the scalar product of $\nabla$ with itself. This form is more favoured by physicists.
\subsubsection*{Derivation}
\htmladdnormallink{Click here}{<http://planetmath.org/?method=l2h&from=collab&id=76&op=getobj">} to see an article that derives the Laplacian in spherical coordinates.