What are some ways to derive $\left( \boldsymbol{E}\cdot \boldsymbol{E} \right) \nabla =\frac{1}{2}\nabla \boldsymbol{E}^2$?

Question

For each of the two reference books the constant equations are as follows: $$ \boldsymbol{E}\times \left( \nabla \times \boldsymbol{E} \right) =-\left( \boldsymbol{E}\cdot \nabla \right) \boldsymbol{E}+\frac{1}{2}\nabla \boldsymbol{E}^2 $$

$$ \boldsymbol{E}\times \left( \nabla \times \boldsymbol{E} \right) =-\left( \boldsymbol{E}\cdot \nabla \right) \boldsymbol{E}+\left( \boldsymbol{E}\cdot \boldsymbol{E} \right) \nabla $$ We can reason about the terms that correspond to the same in both equations. $$ \left( \boldsymbol{E}\cdot \boldsymbol{E} \right) \nabla =\frac{1}{2}\nabla \boldsymbol{E}^2 $$ What are some ways to reason about the above equation to arrive at it? (I would like to have access to a variety of reasoning options)

Answer 1

Your second equation with its $({\bf E}\cdot {\bf E})\nabla$ isnot really correct as the nabla has to act on one of the ${\bf E}$'s. The correct form is best written as $({\bf E} \cdot \nabla {\bf E})$ with the dot understood being a product between the two ${\bf E}$'s and not a scalar product with the $\nabla$. There is no standard vector notation way of saying this, but I've put the parentheses where I have to distinguish it from $({\bf E}\cdot \nabla) {\bf E}$. In components, however, the $j$-th component of the term is $$ ({\bf E} \cdot \nabla {\bf E})_j \stackrel{\rm def}{=} E_i \partial_j E_i = \frac 12 \partial_j (E_i E_i), $$
where a sum on the "$i$" index is understood.