The Laplacian and the Laplace Transform are two completely different things. The laplacian is the dot product of the gradient operator with itself, the Laplace

The Laplacian and the Laplace Transform are two completely different things. The laplacian is the dot product of the gradient operator with itself, the Laplace transform is an integral transform with the kernel e^(-s*t). The relation of the Fourier transform to the Laplacian is that FT diagonalises the Laplacian.