Meaning of Laplace's equation | Babel Free

Definitions

The partial differential equation (∂²φ)/(∂x_1²)+(∂²φ)/(∂x_2²)+⋯+(∂²φ)/(∂x_n²)=0, commonly written Δφ=0 or ∇²φ=0, where Δ(=∇²) is the Laplace operator and φ is a scalar function.

Examples

“In practical problems, since the charges are confined to small regions while major part of the space is charge-free, it is obvious that Laplace's Equation has far greater utility than Poisson's equation.”

“Numerous problems in electrical engineering require a solution of Laplace's equation in two dimensions. This section outlines a classic problem that leads to Laplace's equation, then develops a finite element method for its solution.”

“2002, Gerald D. Mahan, Applied Mathematics, Kluwer Academic / Plenum, page 141, Laplace's equation appears in a variety of physics problems and several examples are provided below. The relevance of Laplace's equation to complex variables is provided by the following important theorem.”

CEFR level