Evaluate: $A \cdot \nabla R + \nabla(A \cdot R) + A \times\nabla\times R$

Question

How to evaluate: $A \cdot \nabla R + \nabla(A \cdot R) + A \times\nabla\times R$ where $A$ is a constant vector field while $R$ is a non-constant vector field.

The only simplification I can see is that $\nabla(A\cdot R) = A$.

Answer 1

If $R\colon \mathbb R^3 \to \mathbb R^3$ as suggested by $\nabla\def\sp#1{\left<#1\right\rangle}\sp{A,R} = A$ denotes the identity, we have that $\nabla R = \mathrm{Id}$, hence $A \cdot \nabla R = A$, moreover $\nabla \times R = 0$, so $A \times \nabla \times R = 0$. Hence the given term equals $2A$.