Laplacian of a field
/ luh-PLAH-see-un /
Take the temperature in a room and ask, at each point, how the value there compares with the average of its immediate neighbors. If a point is colder than the surrounding average, heat will flow toward it; if it is exactly at the average of its surroundings, it is in balance. The Laplacian is the operator that measures this: it reports, at each point, how much the field bends away from the average of its neighborhood. It is the single most important second-order operator in mathematical physics.
For a scalar field f, the Laplacian is the scalar field nabla^2 f = div(grad f) = partial^2 f/partial x^2 + partial^2 f/partial y^2 + partial^2 f/partial z^2 — you take the gradient and then the divergence of it. It is the divergence of the field's own steepest-ascent arrows, so it is positive where the gradient is converging (a local valley) and negative where it is spreading (a local peak). For a vector field F there is also a vector Laplacian, defined component-wise in Cartesian coordinates, and tied to the other operators by the identity nabla^2 F = grad(div F) - curl(curl F). A field with nabla^2 f = 0 is called harmonic, and harmonic functions are exactly the ones equal to their own neighborhood average at every point.
The Laplacian is the heart of the three great partial differential equations: Laplace's equation nabla^2 phi = 0 (steady-state potentials, equilibrium), the heat equation partial u/partial t = k nabla^2 u (diffusion), and the wave equation partial^2 u/partial t^2 = c^2 nabla^2 u (vibrations, light, sound). In electrostatics Poisson's equation nabla^2 phi = -rho/epsilon_0 ties the potential directly to charge. Wherever something diffuses, settles to equilibrium, or oscillates, the Laplacian is what governs it.
For f(x, y) = x^2 - y^2, nabla^2 f = 2 + (-2) = 0, so f is harmonic: a saddle whose value at each point exactly equals the average of its neighbors. For f = x^2 + y^2, nabla^2 f = 4 > 0, the signature of a bowl-shaped valley.
Zero Laplacian means equilibrium (harmonic); a positive one marks a local valley, a negative one a local peak.
The neat Cartesian formula partial^2/partial x^2 + ... only looks that simple in rectangular coordinates; in polar, cylindrical, or spherical coordinates the Laplacian picks up extra terms from the scale factors, and the vector Laplacian is not just the component-wise scalar Laplacian outside Cartesian coordinates — a frequent source of error.