I am looking for a literature advice about the following. I'ld like to review classical physics (basically all undergrad / grad stuff) under the aspects of a modern covariant formulation with exterior derivatives, covariant derivatives, Lie derivatives or Cartan's exterior covariant derivative.
What is the motivation? Instead of moving forward to general relativity theory as most textbooks do, I'ld like to see the machinery applied to physics I already know. I am well aware that all the different derivatives I have mentioned have different requirements on the manifold, however in classical physics the manifold in question is often $\mathrm R^3$ and fulfills all of them at once (except maybe in thermodynamics). I'ld like to see which formulation of the very same equation the author choose and why, basically what is his stand on which formulation gives the "best" physical interpretation of the equation at hand.
E.g. I have read parts of Theodor Frankel's Geometry of Physics in which he touches electromagnetism and elasticity. In his opinion in classical physics forces should be covecvors (1-forms) and not vectors because they fit much more natural to the concept of work.
Another example would be: for a scalar quantity $f$ we can write $$\Delta f=\star d\star d f=(d\delta +\delta d) f=\nabla^\alpha \nabla_\alpha f$$ while the lhs appears in standard textbooks, it might be favorable to use the second, third or fourth variant, because it will show a connection to similar types of equations easily or will stress properties of $f$ when written with $df$.
I hope the question is not unclear or too broad. I have not attached more than the basic tag to the question because in a sense it would apply too all classical fields.
