- #1
JustinLevy
- 895
- 1
I've only learned differential equations for use in physics, and never took a rigorous math course on all their amazing features. So I'm hoping someone can teach me a bit here, in the context of this question:
Consider Maxwell's equations in vacuum, units don't matter here so I'll get rid of all constants:
[tex]\nabla \cdot \vec{E} = 0[/tex]
[tex]\nabla \cdot \vec{B} = 0[/tex]
[tex]\nabla \times \vec{E} = - \frac{\partial}{\partial t} \vec{B}[/tex]
[tex]\nabla \times \vec{B} = \frac{\partial}{\partial t} \vec{E}[/tex]
Now consider a finite region of space, with the boundary condition that the fields and their derivatives are zero on the boundary at time 0<=t<T. What solutions are there for the fields in the region during this time?
One obvious solution is: E=0, B=0 everywhere.
Is this question well posed enough to prove that this solution is unique?
If so, how? If not, what is missing?
Consider Maxwell's equations in vacuum, units don't matter here so I'll get rid of all constants:
[tex]\nabla \cdot \vec{E} = 0[/tex]
[tex]\nabla \cdot \vec{B} = 0[/tex]
[tex]\nabla \times \vec{E} = - \frac{\partial}{\partial t} \vec{B}[/tex]
[tex]\nabla \times \vec{B} = \frac{\partial}{\partial t} \vec{E}[/tex]
Now consider a finite region of space, with the boundary condition that the fields and their derivatives are zero on the boundary at time 0<=t<T. What solutions are there for the fields in the region during this time?
One obvious solution is: E=0, B=0 everywhere.
Is this question well posed enough to prove that this solution is unique?
If so, how? If not, what is missing?