Divergence of Product of Scalar Field with Gradient of Scalar Field

Theorem
Let $R$ be a region of space.

Let $U$ and $W$ be scalar fields over $R$.

Then:
 * $\map {\operatorname {div} } {U \grad W} = U \nabla^2 W + \paren {\grad U} \cdot \paren {\grad W}$

where:
 * $\operatorname {div}$ denotes the divergence operator
 * $\grad$ denotes the gradient operator
 * $\nabla^2$ denotes the Laplacian.

Also presented as
This result can also be presented as:


 * $\map {\operatorname {div} } {U \grad W} = U \nabla^2 W + \nabla U \cdot \nabla W$

presupposing the implementation of $\grad$ as an operation using the del operator.