Definition:Laplacian/Scalar Field

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

Let $U$ be a scalar field over $\R^n$.

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$.

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

The Laplacian of $U$ is defined as:

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

Also see

 * Definition:Laplacian on Vector Field