We take Maxwell's equations and the fields written in terms of the potentials as input. In the left column the equations are given in the standard form while the right column gives the equivalent equation in terms of indexed components. The right column uses the totally antisymmetric tensor in 3D and assumes summation over repeated indices (Einstein notaton). So in this notation, dot products can be simply written as and any component of a cross product is written .
If the fields are written in terms of potentials, then the first two Maxwell equations are automatically satisfied.
Lets verify the first equation by plugging in the B field in terms of the potential and noticing that we can interchange
the order of differentiation.
For the second equation, we write it out in terms of the potentials and notice that the first term for the same reason as above.
Similarly we may work with the Gauss's law equation
For the fourth equation we have.
The last two equations derived are wave equations with source terms obeyed by the potentials. As discussed in the opening section of this chapter, they can be simplified with a choice of gauge.
Jim Branson 2013-04-22