The differential forms formulation, which in many other cases (but not all!) is more useful to work with than the vector calculus formulation should probably be replaced with geometric algebra though. They are basically the same thing, but differential forms are more idiosyncratic, and some symmetries are not apparent. Neither are useful as an introduction to the aspiring physicist or engineer though.

Or open any of these and search for “Maxwell”: http://www.av8n.com/physics/maxwell-ga.htm http://arxiv.org/pdf/1101.3619.pdf http://ieeexplore.ieee.org/ielx7/5/6879517/06876131.pdf http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf http://faculty.luther.edu/~macdonal/GA&GC.pdf

I skipped the a few and went straight to the last PDF (the Survey) and coming from a stronger foundational math background, I felt right at home. (I'd imagine those who went down the physics route would be thrown off by the notation though, so I understand the author's preface). He mentions physics and CS applications, but I really think this has structural engineering applications as well -- e.g. R^4 seems like a perfect domain to explore bidirectional tensile and shear strength of materials (especially composite materials/aggregates), then perform some FEA with all of the simple operations within that come inherent to operating within the bounds of the 4-vector form. I mean that's a real trivial example (and I obviously don't know enough about structural engineering to make a good analogy) but it seems like one of those cases where you move the problem set lifted into another domain, perform your manipulation then finally lower back into your original (co-)domain. (Frequently done in algebraic topology, most famously by Perelman on Poincare)

The people who write those systems on the other hand enjoy this math very much, and tend to use Galerkin as a verb and will casually point out just how wonderful the Lax-Milgram theorem is.