I don't know how to work with the 'reversed' dot product operator,
$$v\cdot \nabla$$
I arrived to expressions like this trough doing some calculus, and I don't know how to continue with the calculus because this operators that are not commutative,
$$\int \left( (E\cdot \nabla) E +c^2 (B\cdot \nabla ) B \right)d^3r \stackrel{\text{?}}{=}0$$
That needs to be zero for a set of charged particles, but I don't know how to use the operators and any properties that they have.
Hint :
\begin{align} \overbrace{ \begin{bmatrix} \mathrm a_1\dfrac{\partial \rm b_1}{\partial x_1}\boldsymbol{+}\mathrm a_2\dfrac{\partial \rm b_1}{\partial x_2}\boldsymbol{+}\mathrm a_3\dfrac{\partial \rm b_1}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm a_1\dfrac{\partial \rm b_2}{\partial x_1}\boldsymbol{+}\mathrm a_2\dfrac{\partial \rm b_2}{\partial x_2}\boldsymbol{+}\mathrm a_3\dfrac{\partial \rm b_2}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm a_1\dfrac{\partial \rm b_3}{\partial x_1}\boldsymbol{+}\mathrm a_2\dfrac{\partial \rm b_3}{\partial x_2}\boldsymbol{+}\mathrm a_3\dfrac{\partial \rm b_3}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{bmatrix}}^{\left(\mathbf{a}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\mathbf{b}} &\boldsymbol{=} \begin{bmatrix} \mathbf{a}\boldsymbol{\cdot}\boldsymbol{\nabla}\mathrm b_1\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathbf{a}\boldsymbol{\cdot}\boldsymbol{\nabla}\mathrm b_2\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathbf{a}\boldsymbol{\cdot}\boldsymbol{\nabla}\mathrm b_3\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{bmatrix} \boldsymbol{=} \overbrace{ \begin{pmatrix} \mathrm a_1\dfrac{\partial \hphantom{\rm b_1}}{\partial x_1}\boldsymbol{+}\mathrm a_2\dfrac{\partial \hphantom{\rm b_1}}{\partial x_2}\boldsymbol{+}\mathrm a_3\dfrac{\partial \hphantom{\rm b_1}}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{pmatrix}}^{\left(\mathbf{a}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)} \overbrace{ \begin{bmatrix} \mathrm b_1\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm b_2\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm b_3\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{bmatrix}}^{\mathbf{b}} \label{vecform-12}\\ \overbrace{ \begin{bmatrix} \mathrm b_1\dfrac{\partial \rm a_1}{\partial x_1}\boldsymbol{+}\mathrm b_2\dfrac{\partial \rm a_1}{\partial x_2}\boldsymbol{+}\mathrm b_3\dfrac{\partial \rm a_1}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm b_1\dfrac{\partial \rm a_2}{\partial x_1}\boldsymbol{+}\mathrm b_2\dfrac{\partial \rm a_2}{\partial x_2}\boldsymbol{+}\mathrm b_3\dfrac{\partial \rm a_2}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm b_1\dfrac{\partial \rm a_3}{\partial x_1}\boldsymbol{+}\mathrm b_2\dfrac{\partial \rm a_3}{\partial x_2}\boldsymbol{+}\mathrm b_3\dfrac{\partial \rm a_3}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{bmatrix}}^{\left(\mathbf{b}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\mathbf{a}} &\boldsymbol{=} \begin{bmatrix} \mathbf{b}\boldsymbol{\cdot}\boldsymbol{\nabla}\mathrm a_1\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathbf{b}\boldsymbol{\cdot}\boldsymbol{\nabla}\mathrm a_2\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathbf{b}\boldsymbol{\cdot}\boldsymbol{\nabla}\mathrm a_3\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{bmatrix} \boldsymbol{=} \overbrace{ \begin{pmatrix} \mathrm b_1\dfrac{\partial \hphantom{\rm b_1}}{\partial x_1}\boldsymbol{+}\mathrm b_2\dfrac{\partial \hphantom{\rm b_1}}{\partial x_2}\boldsymbol{+}\mathrm b_3\dfrac{\partial \hphantom{\rm b_1}}{\partial x_3}\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{pmatrix}}^{\left(\mathbf{b}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)} \overbrace{ \begin{bmatrix} \mathrm a_1\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm a_2\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}}\\ \mathrm a_3\vphantom{\dfrac{\tfrac{f}{g}}{\tfrac{f}{g}}} \end{bmatrix}}^{\mathbf{a}} \label{vecform-13} \end{align}
