What is $\left( \boldsymbol{e} \, \cdot \, \boldsymbol{\nabla} \right) \left( \boldsymbol{a} \times \boldsymbol{b} \right)$

Question

I am interested in determining the following: $$ \left( \boldsymbol{e} \, \cdot \, \boldsymbol{\nabla} \right) \left( \boldsymbol{a} \times \boldsymbol{b} \right) \, , $$ where $\boldsymbol{a}$, $\boldsymbol{b}$, and $\boldsymbol{e}$ are vector fields.

Can the product rule be applied here? Any help is highly appreciated.

Thank you

Answer 1

Let $\vec{x}_i$ denote the $i$th element of a right-handed orthogonal basis, so$$(e\cdot\nabla)(a\times b)=(e_i\partial_i)(\epsilon_{jkl}a_kb_l\vec{x}_j)$$ $$=e_i\epsilon_{jkl}(\partial_ia_kb_l+a_k\partial_ib_l)\vec{x}_j$$ $$=[(e\cdot\nabla)a]\times b+a\times[(e\cdot\nabla)]b.$$