I would like to understand Cartan's formalism of exterior covariant calculus. I think it could be useful for some calculations in physics (But If I am wrong here and it's only good for abstract considerations, please tell me in advance.) I need a reference that explains main ideas (a picture that goes with the formalism) and teaches to apply it and calculate efficiently. It is crucial that it doesn't delve into abstract algebra because I don't have necessary background. I don't mind if it omits more difficult proofs. I want to understand more or less what follows from what, but I don't mind leaving some technicalities to real mathematicians.
As for my background: I probably know most of things that are explained in introductory expositions of smooth manifolds (I do know it's a vague statement.) Also some Lie groups and algebras. I know some Riemannian and pseudo-Riemannian geometry (i.e. general relativity), however I'm not satisfied with my level of understanding. In fact I hope that this new formalism will shed some new light.
If anyone can suggest me a good source, I will be indebted. I do realize that there are no perfect books and probably none of them will give me everything that I want. But I though that giving more information will not hurt.
A very nice gentle (albeit abstract) introduction to forms and connections can be found in R.W.R Darling's Differential Forms and Connections (1), a more physics based text book would have to be Nakahara's Geometry, Topology & Physics (2) - these helped me greatly when I had a similar need to you.
Good Luck!
(1) http://www.amazon.co.uk/Differential-Forms-Connections-R-Darling/dp/0521468000
(2) http://stringworld.ru/files/Nakahara_M._Geometry_topology_and_physics_2nd_ed..pdf