Poisson bracket and curl - Second equation of MHD In the study of Magnetohydrodynamics we have:
 $$\frac{\partial \textbf{B}}{\partial t} = \nabla \times (\textbf{v} \times B)$$
In Arnold's book "Topological Method in Hydrodynamics" , equation 10.1 , p.49 shows:
$$ \frac{\partial \textbf{B}}{\partial t} = -\{\textbf{v},B\}$$
Where $\{ , \}$ is the Poisson bracket. Thus, we have:
$$ -\{\textbf{v},B\} = \nabla \times (\textbf{v} \times B) $$
But i can't see how this works, Firt because the definition of poisson bracket to functions in $\mathbb{R}^{3}$, where M is a manifold, and $v$ e $B$ are functions in $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$. Well, if someone could help...
 A: Let us recall the fact that a vector field $V$ acts on a function $f$ as follow
\begin{align}
V f = \sum_{i=1}^n V_i \frac{\partial f}{\partial x_i}.
\end{align}
Then we observe
\begin{align}
\{\boldsymbol{v}, B\}f =&\ \boldsymbol{v}(Bf)-B(\boldsymbol{v}f)=\boldsymbol{v}\left(\sum^3_{i=1}B_i\frac{\partial f}{\partial x_i}\right) -B\left(\sum^3_{i=1}v_i\frac{\partial f}{\partial x_i}\right)\\
=&\ \sum^3_{i=1}\left\{B_i \boldsymbol{v}\left(\frac{\partial f}{\partial x_i} \right)+\boldsymbol{v}(B_i)\frac{\partial f}{\partial x_i}-B(v_i)\frac{\partial f}{\partial x_i}-v_i B\left(\frac{\partial f}{\partial x_i} \right) \right\}\\
=&\ \sum^3_{i\neq j}\left(v_j\frac{\partial B_i}{\partial x_j}-B_j \frac{\partial v_i}{\partial x_j}\right)\frac{\partial f}{\partial x_i}+\sum^3_{i\neq j} (B_iv_j-v_iB_j) \frac{\partial^2 f}{\partial x_i\partial x_j}\\
=&\ \sum^3_{i\neq j}\left(v_j\frac{\partial B_i}{\partial x_j}-B_j \frac{\partial v_i}{\partial x_j}\right)\frac{\partial f}{\partial x_i}.
\end{align}
On the other hand we have
\begin{align}
-\nabla\times(\boldsymbol{v}\times B)f=&\ 
-\begin{vmatrix}
\partial_1 f & \partial_2 f & \partial_3 f\\
\partial x_1 & \partial x_2 & \partial x_3\\
B_3v_2-v_3B_2 & -B_3v_1+v_3B_1 & B_2v_1-v_2B_1
\end{vmatrix}\\
=&\ -\left(\frac{\partial }{\partial x_2}(B_2v_1-v_2B_1)+\frac{\partial }{\partial x_3}(B_3v_1-v_3B_1)\right)\frac{\partial f}{\partial x_1}\\
&\ +\left(\frac{\partial }{\partial x_1}(B_2v_1-v_2B_1)-\frac{\partial}{\partial x_3}(B_3v_2-v_3B_2)\right)\frac{\partial f}{\partial x_2}\\
&\ - \left(\frac{\partial }{\partial x_1}(-B_3v_1+v_3B_1)-\frac{\partial }{\partial x_2}(B_3v_2-v_3B_2)\right)\frac{\partial f}{\partial x_3}\\
\end{align}
which coincide with the above calculation. 
