The Laplacian-Differential-Operator, often simply referred to as the Laplacian, is a fundamental operator in mathematical analysis, particularly in the study of Partial Differential Equations and Vector Calculus. It is named after Pierre-Simon Laplace, although it was actually first introduced by Jean le Rond d'Alembert.
The Laplacian operator, denoted by Δ (the Greek letter Delta) or ∇² (nabla squared), acts on a scalar field or a vector field, depending on the context:
The Laplacian emerged from the work on Potential Theory in the 18th century. Initially, Laplace used it to solve problems related to gravitational fields, but its applications have since expanded vastly. Its use became widespread in physics and engineering, particularly in problems involving heat conduction, fluid dynamics, and electromagnetic fields.
Here are some key applications of the Laplacian:
The concept of the Laplacian has been extended to: