Definition:Laplacian/Vector Field

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

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

Let $\mathbf f: \R^n \to \R^n$ be a vector-valued function on $\R^n$:
 * $\forall \mathbf x \in \R^n: \map {\mathbf f} {\mathbf x} := \ds \sum_{k \mathop = 0}^n \map {f_k} {\mathbf x} \mathbf e_k$

where each of $f_k: \R^n \to \R$ are real-valued functions on $\R^n$.

That is:
 * $\mathbf f := \tuple {\map {f_1} {\mathbf x}, \map {f_2} {\mathbf x}, \ldots, \map {f_n} {\mathbf x} }$

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 $\tuple {\mathbf i, \mathbf j, \mathbf k}$, this is usually rendered:

Also see

 * Definition:Laplacian of Real-Valued Function