Laplace's equation

plural Laplace's equations · alternative Laplace equation

1. potential theoryThe 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.

   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.

   2 more 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.

   1993, V. V. Sarwate, Electromagnetic Fields and Waves, New Age International Publishers, page 182

   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.

   1996, Peter P. Silvester, Ronald L. Ferrari, Finite elements for electrical engineers, 3rd edition, Cambridge University Press, page 29