Nabla calculus and conservative forces

[KayDee01]

1. The problem statement

I'm trying to show that the magnetic force is only conservative if dB/dt=0

Homework Equations

F=q[E+(v[itex]\times[/itex]B)]

Conservative if ∇[itex]\times[/itex]F=0

∇[itex]\times[/itex](A[itex]\times[/itex]B)=A(∇[itex]\cdot[/itex]B)-B(∇[itex]\cdot[/itex]A)+(B[itex]\cdot[/itex]∇)A-(A[itex]\cdot[/itex]∇)B

Maxwells equation: ∇[itex]\times[/itex]E=-∂B/∂t

The Attempt at a Solution

So the magnetic force field is conservative if
∇[itex]\times[/itex][E+(v[itex]\times[/itex]B)]=0
=∇[itex]\times[/itex]E+∇[itex]\times[/itex](v[itex]\times[/itex]B)
=-∂B/∂t+v(∇[itex]\cdot[/itex]B)-B(∇[itex]\cdot[/itex]v)+(B[itex]\cdot[/itex]∇)v-(v[itex]\cdot[/itex]∇)B

So from here I know that I need to show:
v(∇[itex]\cdot[/itex]B)-B(∇[itex]\cdot[/itex]v)+(B[itex]\cdot[/itex]∇)v-(v[itex]\cdot[/itex]∇)B=0

But when I write it out in all its components I get lost in the algebra. And then it got me thinking, if that equals zero, why doesn't ∇[itex]\times[/itex](A[itex]\times[/itex]B) always equal zero.
I know there's a simple answer to this as we did it in lectures but I can't seem to find it in my notes anywhere.

 

[mfb]

You cannot assign a scalar potential to the magnetic field. You can show that the magnetic field does not influence the speed (and therefore the energy) of the particle, however. This is related to ##\vec{v} \cdot \vec{F}##.