Laplace operator - Wikipedia

Abstraction: Second-order differential operator divergence of gradient

Key points:

• The Laplacian Δf is defined as the divergence of the gradient; in Cartesian coordinates it equals the sum of all unmixed second partial derivatives.
• Δf(p) measures how much the average value of f over small spheres centered at p deviates from f(p); solutions to Δf = 0 are called harmonic functions.
• Appears in Poisson's equation (electrostatics/gravity), diffusion equation (heat/fluid), wave equation, and Schrödinger equation.
• Fourier transform diagonalizes the Laplacian: in frequency domain it acts as multiplication by −|ξ|², making it a Fourier multiplier.
• Generalizes to the Laplace–Beltrami operator on Riemannian manifolds and to the fractional Laplacian (−Δ)^s for nonlocal analysis.
• Generates the heat semigroup e^{tΔ}; eigenfunctions on bounded domains form an orthonormal basis (Helmholtz equation).

Connections: Pierre Simon Laplace · Laplace Operator · Partial Differential Equations · Harmonic Functions · Spectral Theory

Source: http://en.wikipedia.org/wiki/Laplace_operator