This question stems from physics, but it is the mathematics aspect which troubles me I believe. However, if you feel it doesn't belong here please feel free to move it to the physics forum.
It can be shown that the force on an electric dipole in a non-uniform electric field is found as: $$F_i = p_j \frac{\partial E_i}{\partial x_j} $$ which in vector-form corresponds to: $$\vec{F} = (\vec{p} \cdot \nabla) \vec{E} $$
However, using the fact that $\nabla \times \vec{E} = 0 $ the summation can also be written: $$F_i = p_j \frac{\partial E_j}{\partial x_i} $$
Now, for a magnetic dipole the force in a non-uniform magnetic field can be shown to be
$$F_i = m_j \frac{\partial B_j}{\partial x_i} (\star)$$
, which equivalent to the second expression for the electric case. However, for a magnetic field we don't in general have $\nabla \times \vec{B} = 0 $, meaning that I am not convinced that we can go back to the first form seen in the electrical case.
My question is therefore: Is there any pretty vector form of expression $\star$?
I am aware $\vec{F} = \nabla(\vec{m} \cdot\vec{B}) $ is a valid option if $\vec{m}$ is constant, but what if it is not?