- #1

michael879

- 698

- 7

I'm trying to derive for myself the uniqueness proof for Maxwell's equations, but I'm a little stuck at the end. I've managed to prove the following:

[tex]

\dfrac{A^\mu}{\partial{t}}\nabla{A^\mu}|_S = \dfrac{A^\mu}{\partial{t}}|_{t_0} = \nabla{A^\mu}|_{t_0} =0 \Rightarrow \forall_{\vec{r}\in V, t > t_0} F^{\mu\nu} = 0

[/tex]

Where S is the surface bounding volume V, t

_{0}is some initial time, and:

[tex]

A^\mu \equiv A_1^\mu - A_2^\mu,

F^{\mu\nu} \equiv F_1^{\mu\nu} - F_2^{\mu\nu}

[/tex]

are the difference fields between the two solutions satisfying the same boundary conditions (at surface S, time t

_{0}, and with the same charge distributions)

My problem is that I can't turn the left hand side of the first equation into any meaningful condition on the fields [itex]F^{\mu\nu}[/itex]. As far as I can tell you need to know the potentials at the boundaries in order for uniqueness to hold, rather than just the field values... What I'm looking for is some expression of [itex]F^{\mu\nu}[/itex] on the boundaries that implies the potential conditions on the left hand side of the equation