Definition:Laplacian/Scalar Field

Definition
Let $\R^n$ denote the real Cartesian space of $n$ dimensions.

Let $\map U {x_1, x_2, \ldots, x_n}$ be a scalar field over $\R^n$.

Let the partial derivatives of $U$ exist throughout $\R^n$.

The Laplacian of $U$ is defined as:


 * $\nabla^2 U := \sum_{k \mathop = 1}^n \dfrac {\partial^2 U} {\partial {x_k}^2}$

Cartesian $3$-Space
In $3$ dimensions with the standard ordered basis $\tuple {\mathbf i, \mathbf j, \mathbf k}$, this is usually rendered:

Also see

 * Laplacian on Scalar Field is Divergence of Gradient


 * Definition:Laplacian on Vector Field