Definition:Laplacian/Vector Field

Definition
Let $\R^n \left({x_1, x_2, \ldots, x_n}\right)$ denote the real Cartesian space of $n$ dimensions.

Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\mathbf V$.

Let $\mathbf f = \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right): \mathbf V \to \mathbf V$ be a vector-valued function on $\mathbf V$.

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

The Laplacian of $\mathbf f$ is defined as:

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

Also see

 * Definition:Laplacian of Real-Valued Function