the Laplace-Beltrami operator
/ lah-PLASS bell-TRAH-mee /
The ordinary Laplacian in the plane, u_xx + u_yy, measures how much a function at a point differs from its average over a tiny circle around it — it is the engine behind heat flow, diffusion, and equilibrium. The Laplace-Beltrami operator is exactly this idea moved onto a curved manifold: it is the right notion of 'second derivative' or 'how much above or below the local average' once you only have a metric and no flat coordinates. It governs heat, harmonic functions, and vibration on any Riemannian manifold.
On a Riemannian manifold (M, g) the Laplace-Beltrami operator is the divergence of the gradient, Delta u = div(grad u). In local coordinates this is Delta u = (1/sqrt(det g)) partial_i ( sqrt(det g) g^{ij} partial_j u ), where g^{ij} is the inverse metric and the sqrt(det g) factors are exactly what make the operator metric-correct (they encode how volume is distorted). Equivalently Delta = -d^* d on functions, the composition of the exterior derivative and its formal adjoint. Two sign conventions are in use: analysts often take Delta = -div grad so that it is a nonnegative operator with nonnegative eigenvalues (the 'geometer's Laplacian'), while classical usage takes Delta = div grad; always check which one a source means, because the same symbol can differ by a sign. On flat R^n it reduces to the familiar sum of second partials; on the round sphere its eigenfunctions are the spherical harmonics.
The Laplace-Beltrami operator is the central object of spectral geometry and the bridge between geometry and analysis on manifolds: its eigenvalues encode the shape of the manifold ('hearing the shape of a drum'), its heat kernel encodes curvature and topology through asymptotic expansions, and it appears in the harmonic-map equation, the linearized minimal-surface equation, and the wave and Schrodinger equations on curved space. The honest caveat: the Laplace-Beltrami operator on functions is only one of several Laplacians on a manifold. On differential forms there is the Hodge Laplacian, and on sections of a bundle there is the rough (connection) Laplacian; these agree on functions but differ on forms by curvature terms (the Bochner-Weitzenbock formulas). Conflating them is a common and consequential error.
On the unit circle, Delta = d^2/dtheta^2 and its eigenfunctions are sin(n theta) and cos(n theta) with eigenvalues n^2 — these are the pure tones of a vibrating ring, the simplest instance of the Laplacian hearing geometry.
On the circle the Laplacian's eigenvalues n^2 are the resonant frequencies of the loop.
Watch the sign convention: the 'geometer's Laplacian' Delta = -div grad has nonnegative spectrum, while the classical Delta = div grad has nonpositive spectrum; a stated eigenvalue inequality is meaningless without saying which is meant.