| Version 8 |
Version 7 |
| Let $(x_1, \ldots, x_n)$ be Cartesian coordinates for some open set $\Omega$ |
Let $(x_1, \ldots, x_n)$ be Cartesian coordinates for some open set $\Omega$ |
| in $\sR^n$. |
in $\sR^n$. |
| Then the \emph{Laplacian} differential operator $\Delta$ is defined as |
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}. |
\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 |
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}. |
\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 |
A coordinate independent definition of the Laplacian |
| is $\Delta = \nabla \cdot \nabla$, i.e., $\Delta$ is the composition of |
is $\Delta = \nabla \cdot \nabla$, i.e., $\Delta$ is the composition of |
| gradient and divergence. |
gradient and divergence. |
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| A harmonic function is one for which the Laplacian vanishes. |
A harmonic function is one for which the Laplacian vanishes. |
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| \subsubsection*{Notes} |
\subsubsection*{Notes} |
| An older symbol for the Laplacian is $\nabla^2$ -- conceptually the scalar product of $\nabla$ with itself. |
An older symbol for the Laplacian is $\nabla^2$ -- conceptually the scalar product of $\nabla$ with itself. |
| This form may be more favoured by physicists. |
This form may be more favoured by physicists. |
