Definition:Laplacian/Scalar Field

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

Let $\map f {x_1, x_2, \ldots, x_n}$ denote a real-valued function on $\R^n$.

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

Let the partial derivative of $f$ with respect to $x_k$ exist for all $x_k$.

The Laplacian of $f$ is defined as:

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

Also see

 * Definition:Laplacian of Vector-Valued Function