Laplace's equation

Noun

 * 1)  The partial differential equation $$\frac {\partial^2 \varphi}{\partial x_1^2} + \frac{\partial^2 \varphi}{\partial x_2^2} + \cdots + \frac{\partial ^2 \varphi}{\partial x_n^2} =0$$, commonly written $$\Delta \varphi =0$$ or $$\nabla ^2\varphi =0$$, where $$\Delta \ (= \nabla^2)$$ is the Laplace operator and $$\varphi$$ is a scalar function.
 * 2) * 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.
 * 1) * 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.

Usage notes
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.