Nabla commutation in electromagnetism 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.
 A: 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}
