How Does Zero Divergence and Curl Determine Uniqueness in a Manifold?

[SD das]

Today when I ask a professor about maxwell eqation
He tells me " it seems that the unknowns exceed the number of equations.
What are the missing ingredients? The answer is the boundary condition .With appropriate boundary conditions, zero divergence and zero curl will nail down a unique solution of zero in the whole manifold. "
Please tell me what does " zero divergence and zero curl will nail down a unique solution of zero in the whole manifold"mean..

 

[FactChecker]

Today when I ask a professor about maxwell eqation
He tells me " it seems that the unknowns exceed the number of equations.
What are the missing ingredients? The answer is the boundary condition .With appropriate boundary conditions, zero divergence and zero curl will nail down a unique solution of zero in the whole manifold. "
Please tell me what does " zero divergence and zero curl will nail down a unique solution of zero in the whole manifold"mean..

Zero divergence and zero curl implies that anything going on inside will show up on the boundary. (There can't be an internal paired source & sink that cancel each other on the boundary and there can't be any internal swirling that does not show up on the boundary.) One consequence is that if the boundary values are all zero, the interior values must be all zero.

 

[fresh_42]

(Source: http://terpconnect.umd.edu/~petersd/246/sysplot5.gif)

The small red arrows represent a vector field, i.e. all possible tangents. A solution of the differential equation is a curve (blue lines) in this field. By fixing the initial conditions, you choose which of the lines is taken, such you get only one valid curve. The definitions of divergence (tangent vector) and curl (normal vector) can be found on Wikipedia, e.g.