the Laplacian operator
/ lah-PLAH-see-an, after Laplace (lah-PLAHSS) /
Of all the bundles of derivatives in PDEs, one shows up so relentlessly that it deserves its own entry: the Laplacian. In words, it answers a single question at each point — is u here bigger or smaller than the average of u just around here? If u at a point is below the local average, the Laplacian is positive; if above, negative; if u exactly equals its surrounding average, the Laplacian is zero. That comparison-to-the-average idea is the secret behind why the Laplacian governs heat, electricity, gravity, and steady states of almost everything.
Symbolically the Laplacian of u is the sum of the unmixed second derivatives. In two dimensions, Laplacian u = u_xx + u_yy; in three, u_xx + u_yy + u_zz. It is written several ways — Laplacian u, nabla^2 u, or Delta u — all the same operator, the divergence of the gradient (div grad u). A clean way to see the 'average' meaning: in one dimension, u_xx is positive exactly where the graph of u is concave up, i.e. where the point sits below the chord joining its neighbours. The Laplacian is the multi-dimensional version of that curvature, summed over every coordinate direction.
The Laplacian is the workhorse of the whole subject. Laplace's equation Laplacian u = 0 describes equilibria — a soap film, a steady temperature, an electrostatic potential in empty space; its solutions are called harmonic functions. Add a source and you get Poisson's equation Laplacian u = f. Add time and the Laplacian drives diffusion (u_t = k Laplacian u) and waves (u_tt = c^2 Laplacian u). It is rotationally symmetric — it looks the same no matter how you turn your coordinate axes — which is exactly why it models physics that has no preferred direction in space.
For u(x, y) = x^2 - y^2 we get u_xx = 2 and u_yy = -2, so Laplacian u = 2 + (-2) = 0. This u is therefore harmonic: it solves Laplace's equation. The two curvatures exactly cancel, which is the signature of equilibrium.
When the Laplacian vanishes everywhere, u is in balance — neither piling up nor draining at any point.
The Laplacian sums only the pure (unmixed) second derivatives u_xx, u_yy, u_zz; the mixed term u_xy is not part of it. In curved coordinates (polar, spherical) the Laplacian gains extra terms — it is not simply 'second derivative in r plus second derivative in theta', a trap worth remembering.