Overview
Vector Potential
The magnetic force cannot be expressed as the gradient of a potential function. This is due to the fact that the magnetic force does not work, and therefore given that the change in work done on a particle is equal to the change in the potential, this implies that the potential would have to be a constant function.
On the other hand, it is known that
{% \nabla \cdot B = 0 %}
This means that we could possibly introduce a vector field {% A %} such that
{% \vec{B} = \nabla \times \vec{A} %}