Curl product rule for a scalar field ($\nabla \times (\varphi \mathbf{F})=\varphi \nabla \times \mathbf{F}+\nabla \varphi \times \mathbf{F}$)

For a continuously differentiable scalar field $\varphi$ and vector field $\mathbf{F}$, the curl satisfiesIndeed, with fixed Cartesian basis vectors ${\mathbf{e}}_i$, the Cartesian expression for the curl and the product rule giveThe first sum is the cross product of the gradient of $\varphi$ with $\mathbf{F}$; the second is $\varphi$ times the curl of $\mathbf{F}$. The formula is useful when applying the Stokes theorem to a weighted vector field.