\begin{definition}[Definition:Laplacian/Vector Field]
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 $\mathbf V: \R^n \to \R^n$ be a vector field on $\R^n$:
:$\forall \mathbf x \in \R^n: \map {\mathbf V} {\mathbf x} := \ds \sum_{k \mathop = 0}^n \map {V_k} {\mathbf x} \mathbf e_k$
where each of $V_k: \R^n \to \R$ are real-valued functions on $\R^n$.
That is:
:$\mathbf V := \tuple {\map {V_1} {\mathbf x}, \map {V_2} {\mathbf x}, \ldots, \map {V_n} {\mathbf x} }$
Let the partial derivative of $\mathbf V$ with respect to $x_k$ exist for all $f_k$.
The '''Laplacian of $\mathbf V$''' is defined as:
{{begin-eqn}}
{{eqn | l = \nabla^2 \mathbf V
      | o = :=
      | r = \sum_{k \mathop = 1}^n \dfrac {\partial^2 \mathbf V} {\partial {x_k}^2}
}}
{{end-eqn}}
\end{definition}