Mathematically rigorous text on classical electrodynamics.

Solution 1:

Electromagnetic Theory and Computation A Topological Approach by Paul W. Gross and P. Robert Kotiuga. Free in http://library.msri.org/books/Book48/.

Also, this thread: https://physics.stackexchange.com/questions/44973/electromagnetism-for-mathematician.

Solution 2:

Any book on differential gemoetry with gauge theory and bundles would probably do the job. Why don't you have a look at Y. Choquet-Bruhat, C. DeWitt-Morette's Analysis, Manifolds and Physics (both volumes)? In volume I, chapter Vbis ("Connections on a Principal Fibre Bundle") classical electrodynamics is treated as gauge theory on certain fibre bundles.

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