\begin{definition}[Definition:Laplacian/Vector Field/Cartesian 3-Space/Definition 1]
Let $R$ be a region of Cartesian $3$-space $\R^3$.
Let $\map {\mathbf V} {x, y, z}$ be a vector field acting over $R$.
The '''Laplacian''' on $\mathbf V$ is defined as:
:$\nabla^2 \mathbf V = \dfrac {\partial^2 \mathbf V} {\partial x^2} + \dfrac {\partial^2 \mathbf V} {\partial y^2} + \dfrac {\partial^2 \mathbf V} {\partial z^2}$
\end{definition}