How does the covariant exterior derivative generalize to vector bundles?

[Matterwave]

Hi, I have some more questions about this stuff, which, as always, confuses me. I am (extremely slowly) working through Biship and Crittenden, and I'm pretty much at the point where I don't think I can understand it much at all, so, I think instead of trying to go through it all not knowing where the pay off is, I think I'll just ask my questions here (read: I've given up...lol). In the book, they define the covariant exterior derivative D (in chapter 5), but only for (as far as I can tell) forms living in a principle bundle P. From D, they obtain the curvature form from the covariant exterior derivative of the connection 1-form which also lives on P. As a physicist, I am more interested in vector bundles than principle bundles. Does this generalize trivially to the case of a vector bundle? I know that the construction of a vertical subspace, and a horizontal distribution works in the case of a vector bundle as well, correct? Even though Bishop and Crittenden seem to only ever give the definition of a connection on a vector bundle as the induced connection from an associated principle bundle...Another question is I'm used to forms on the manifold itself (existing therefore in T*M), and not forms on the bundle P (which, I assume would exist on T*P). Unless I'm just getting some terminology confused here? When people say "w is a one form on P", they mean that it lives in T*P right? They definitely don't mean that it literally lives on P right? My brain hurts...T_T

(Help!)

 

[Matterwave]

Also, knowing definitely the relationship between a "covariant exterior derivative" and simply a covariant derivative on my manifold would be my main goal in this (MTW is not very mathematically rigorous when it comes to this, and I find I cannot understand very well either Bishop and Crittenden or Cartan's books on this matter).

 

[Matterwave]

I realized my post is probably way too vague to get an answer. The book defines the covariant exterior derivative in a very similar fashion to wikipedia: http://en.wikipedia.org/wiki/Covariant_exterior_derivative

But the deal is that it's a derivative of a form on P (the bundle space) and not M (the manifold space).

Do you get the derivatives on M simply by inducement? Is it true that M is an embedded submanifold of P? If it is, it seems that it could work.