By familiar physics, I mean the physics of things I can reach out and touch. In other words, neither relativity nor analytical dynamics, etc.
After re-reading chapter 4 of MTW's Gravitation yet again, I am left wondering where the pay-off is. Similarly, reading through the introductory parts of Differential Forms with Applications to the Physical Sciences, by Harley Flanders, I have the impression that, to some extent, exterior calculus is a solution in search of a problem.
Among the purported advantages Flanders touts for exterior over tensor calculus is that tensor calculus loses the essential concepts in a "maze of indices". But, exterior calculus comes with its own, similar baggage. Ideas such as "pull-backs" render the familiar (reparameterization) intractable. Indeed, some of these complications in exterior calculus can be expressed using a comparable maze of indices. See, for example, Tensors, Differential Forms, and Variational Principles By: David Lovelock, Hanno Rund. Following the approach used in Advanced Calculus of Several Variables By: C. H. Edwards, Jr., I came up with some pretty ways of combining multi-indices and the Einstein summation convention which compellingly condense the maze of indices of exterior calculus.
But, thus far, all treatments and applications of exterior calculus I have encountered are fairly advanced. When I learned to use vectors, I didn't need an entire arsenal of theory in order to make productive use of them. I could have, and probably should have learned tensors in the same way. I can honestly say that I never really understood tensors until I applied them to such familiar topics as fluid flows and elasticity. In particular, the gradient of a velocity field opened my eyes.
Is there any treatment of "familiar" physics using differential forms which illustrates their applicability without adding the intellectual challenge of more esoteric topics such as the electromagnetic 2-form, or the sympletic geometry of phase space? It may be the case that such applications have not been made simply because it amounts to "reinventing the wheel". These topic have been effectively treated using the techniques of introductory physics, and are therefore uninteresting to the masters of differential forms. I am interested in such redundancy.