Definition:Laplacian/Scalar Field/Cartesian 3-Space

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 $\R^3$.

The Laplacian of $U$ is defined as:

Also see

 * Definition:Laplacian on Vector Field on Cartesian 3-Space