Laplace's equation

• Laplace equation

Laplace's equation (plural Laplace's equations)

1. (potential theory) The partial differential equation ${\displaystyle {\frac {\partial ^{2}\varphi }{\partial x_{1}^{2}}}+{\frac {\partial ^{2}\varphi }{\partial x_{2}^{2}}}+\cdots +{\frac {\partial ^{2}\varphi }{\partial x_{n}^{2}}}=0}$, commonly written ${\displaystyle \Delta \varphi =0}$ or ${\displaystyle \nabla ^{2}\varphi =0}$, where ${\displaystyle \Delta \ (=\nabla ^{2})}$ is the Laplace operator and ${\displaystyle \varphi }$ is a scalar function.
   • 1993, V. V. Sarwate, Electromagnetic Fields and Waves, New Age International Publishers, page 182:

     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.

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

     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.

The plural is rare in this form (and when used, often, although apparently not always, an error), but appears in the alternative form Laplace equations.

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• Poisson's equation on Wikipedia.Wikipedia