In a book I have come across the following expression
$E(\nabla \cdot E)=E\cdot(\nabla E)$, where $E=\sum_i E_i \vec{e}_i$.
Unfortunately I could not prove this, when I calculate both expression using the sum notation I end up with:
$E(\nabla \cdot E)=\sum_{i,j} E_i\partial_jE_j\vec{e}_i$
$E\cdot(\nabla E)=\sum_{i,j} E_j\partial_jE_i\vec{e}_i$
Both expressions are obviously not equivalent, have I missed something?
Kind regards, Max
Maybe it's because they are not equal, why do you expect them to be equal with your interpretation?
If for example $E = v|v|^{-3}$ (a gravitational style field) you will have $\nabla\cdot E = 0$, but $\nabla E$ is not perpendicular to $E$.
At fx $(1,0)$ we have $E=(1,0)$, $\partial_x E = (-1, 0)$, and $\partial_y E = 0$. I see no interpretation where $E\cdot\nabla E$ would be zero.
But if it's in a book it must be right, right? Anyway the book is either wrong, or they have definitions that doesn't coincide with the one you're trying to use. Check the definitions in the book.