The Laplacian measures how a point differs from its surroundings
Take any point, draw a tiny sphere around it, and compare the average value on that sphere with the value at the centre. That difference is what the Laplacian measures. The same simple idea turns up in gravity, heat flow, waves, quantum mechanics and even software that finds edges in photographs.
Formally, the Laplace operator takes the gradient of a function and then its divergence. In ordinary Cartesian coordinates it is just the sum of the second partial derivatives along each axis, and it has handy forms in cylindrical and spherical coordinates too. It is the simplest example of what mathematicians call an elliptic operator.
Pierre-Simon de Laplace, who lived from 1749 to 1827, first used it in celestial mechanics. He noticed that applying it to the gravitational potential of some distribution of mass gives back a constant multiple of the mass density. Where the Laplacian is zero, the functions are called harmonic, and they describe the possible gravitational potentials in empty space. Electrostatics works the same way: the charge distribution is the negative Laplacian of the electric potential.
Diffusion offers perhaps the most intuitive picture. Suppose a chemical has spread out and settled into equilibrium. With no sources or sinks inside a region, the net flow through its boundary is zero, and that condition turns out to be exactly Laplace's equation. So its solutions are the possible settled states of anything that diffuses. When things are not yet settled, the Laplacian at a point tells you how much that point is acting as a source or a sink.
From there it spreads everywhere. Poisson's equation for electric and gravitational potentials, the diffusion equation for heat and fluid flow, the wave equation and the Schrödinger equation of quantum mechanics all contain it. Image-processing software uses it to detect blobs and edges, it lies at the core of Hodge theory and de Rham cohomology in pure mathematics, and it is essentially the generator of standard Brownian motion, the random jiggling of particles.
Source: Laplace operator
