Laplacian

The Laplacian is a differential operator equal to the divergence of a scalar field’s gradient. It measures local variation and, in Euclidean coordinates, sums the field’s second derivatives along each axis.

Gradient: A vector field of first derivatives that points toward the steepest increase of a scalar field. The Laplacian begins by taking the gradient of the scalar field.
Heat equation: A partial differential equation describing how temperature or another diffusing quantity changes over time. The Laplacian gives the spatial spreading term in the heat equation.
Partial derivative: A derivative measuring how a multivariable function changes along one coordinate while others are held fixed. The Euclidean Laplacian sums second partial derivatives along coordinate directions.
Directional derivative: A derivative measuring a function’s rate of change along a specified direction. Unlike a single directional rate, the Laplacian combines second-order change across directions.
Pierre-Simon Laplace: A French mathematician and astronomer whose work shaped celestial mechanics and probability theory. The operator bears Laplace’s name through its role in potential theory.
Divergence: A differential operator measuring the net outward flux of a vector field per unit volume. Applying divergence to the gradient produces the Laplacian.
Poisson's equation: A partial differential equation relating a function’s Laplacian to a prescribed source term. It uses the Laplacian to connect fields to their sources.
Scalar field: A function assigning a scalar value to each point in a space. The standard Laplacian acts on scalar fields.
Curl: A differential operator measuring the local rotation of a vector field in three dimensions. Curl acts on vector fields and measures rotation, unlike the scalar Laplacian’s net curvature.
Potential theory: The study of scalar potentials and harmonic functions arising in fields such as gravitation and electrostatics. The Laplacian became central to equations governing gravitational and electrostatic potentials.

Linked from 21 pages