\begin{definition}[Definition:Laplacian/Vector Field/Cartesian 3-Space/Definition 2]
Let $R$ be a region of Cartesian $3$-space $\R^3$.
Let $\map {\mathbf V} {x, y, z}$ be a vector field acting over $R$.
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
Let $\mathbf V$ be expressed as vector-valued function:
:$\mathbf V := V_x \mathbf i + V_y \mathbf j + V_z \mathbf k$
The '''Laplacian''' on $\mathbf V$ is defined as:
:$\nabla^2 \mathbf V = \nabla^2 V_x \mathbf i + \nabla^2 V_x \mathbf j + \nabla^2 V_y \mathbf k$
where $\nabla^2 V_x$ and so on are the laplacians of $V_x$, $V_y$ and $V_z$ as scalar fields.
\end{definition}