Definition:Laplacian/Vector Field/Cartesian 3-Space/Definition 2

Definition
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.

Also see

 * Equivalence of Definitions of Laplacian on Vector Field on Cartesian 3-Space