# Some definations

### Laplace’s equation

In n dimensions, the problem is to find twice-differentiable real-valued functions, $\varphi$ of real variables, $x_i$, $x_2$, $\cdots$, $x_n$‘, such that

${\partial^2 \varphi \over \partial x_1^2 } +{\partial^2 \varphi \over \partial x_2^2 } +\cdots+{\partial^2 \varphi \over \partial x_n^2 } = 0.$

This is often written as

$\nabla^2 \varphi = 0$

or

$\Delta \varphi = 0.$

### Harmonic function

A harmonic function is a twice continuously differentiable function $f : \mathbf{U} \rightarrow \mathbf{R}$ (where U is an open subset of Rn) which satisfies Laplace’s equation.