1) Laplace transformation in ODE.
2) Laplacian (del squared) of a PDE
3) Laplacian matrix in matrix-tree theorem for calculating spanning trees
And couple of other places I have encountered these things . Are they correlated ? If so then Where should I start the process of learning ? I am thinking starting from khan academy. Am I in right direction ?
The first one has nothing to do with the other two, beyond being tied to the name of http://en.wikipedia.org/wiki/Pierre-Simon_Laplace.
The second and third are related. The simplest connection is that the Laplacian matrix of a lattice graph is, up to a scale factor, an approximation of the Laplacian at the points of the lattice.
More precisely, suppose $L_{d,n}$ is the Laplacian matrix of a lattice in dimension $d$ with $n$ points in each direction (for a total of $n^d$ points). Then as $n \to \infty$, $\frac{1}{n^d} L_{d,n}$ converges (in a certain weak sense) to the Laplacian on the unit hypercube. This is nothing but the centered difference approximation for the Laplacian in $d$ dimensions.
