Divergence of Product of Scalar Field with Curl of Vector Field

Theorem
Let $R$ be a region of space.

Let $U$ be a scalar field over $R$.

Let $\mathbf A = \curl \mathbf B$ be a vector field over $R$ whose vector potential is $\mathbf B$.

Then:
 * $\map {\operatorname {div} } {U \curl \mathbf B} = \paren {\curl \mathbf B} \cdot \paren {\grad U}$

where:
 * $\operatorname {div}$ denotes the divergence operator
 * $\grad$ denotes the gradient operator
 * $\curl$ denotes the curl operator.