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Results tagged with laplacian
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user 13915
The Laplacian matrix is the representation of a graph in matrix form.
8
votes
Accepted
Difference between the Laplacian and the sub-Laplacian of a Lie group
As Sebastian Goette explained in his comment, the sub-Laplacian $\Delta_{sub}$ depends in general from an additional structure. … This gives you a left-invariant sub-Laplacian. …
- 3,223
8
votes
Accepted
Laplace-Beltrami and averaging
In particular (here $\dim M = n$ and the Laplacian is $\Delta = \mathrm{div}\circ \mathrm{grad}$):
$$ (\Delta u)(x) = \lim_{h\to 0} \frac{2n}{h^2}\frac{1}{|S(x,h)|}\int_{S(x,h)} [u(y)-u(x)] dy .$$
The …
- 3,223
11
votes
Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order deriv...
In terms of this frame, the Laplacian at the point $q$ is just a "sum of squares".
Locally, the construction of a local divergence-free, orthonormal frame leads to a system of first order PDEs. … partial_\phi \\ W_2 &=& \cos\theta\sin\phi \partial_\theta + \frac{\cos\phi}{\sin\theta}\partial_\phi \\
W_3 &=& -\sin\theta\partial_\theta
\end{eqnarray*}
and you can check that the standard spherical Laplacian …
- 3,223