stylistic:tone, structural:scholar_edition
Classical electromagnetism

In electrostatics, one can either discuss the electric field, E, or its corresponding electric potential, V. Knowledge of one makes it possible to find the other, except that potentials differing by a constant, V ↦ V + C {\displaystyle V\mapsto V+C}, correspond to the same electric field. This is because the electric field relates to changes in the potential from one point in space to another, and the constant C would cancel out when subtracting to find the change in potential. In terms of vector calculus, the electric field is the gradient of the potential, E = − ∇ V {\displaystyle \mathbf {E} =-\nabla V}. Generalizing from static electricity to electromagnetism, we have a second potential, the vector potential A, with

The general gauge transformations now become not just V ↦ V + C {\displaystyle V\mapsto V+C} but

where f is any twice continuously differentiable function that depends on position and time. The electromagnetic fields remain the same under the gauge transformation.