Covariance and Contravariance
Hey, I posted earlier about the tensor covariant derivative, and the help was great, that makes sense to me now.
However, I am getting really really stuck on the concept of covariant vectors vs. contravariant vectors. I've looked through as many resources as I can - wikipedia, mathworld, the NASA tensor pdf (which is otherwise great), schaum's tensor outline, and I'm getting nowhere. My next step is MTW, though I am sorta intimidated by it so I haven't looked yet. I just can't understand, maybe I'm stupid.
Everyone seems to be concerned with how the components of the vector transform under a change of coordinates, but to me this feels like sophistry. The vector and manifold exist independent of coordinates - a change of coordinates is just redrawing lines over the manifold, nothing actually changes. Similarly, the change of coordinates does absolutely nothing to the vector itself, it just affects how it is written.
So then how are contravariant and contravariant vectors actually different? It seems to me like they are just vectors, forget any of this contravariant/covariant business.
Now another point a lot of these books bring up is how you can convert a covariant vector to a contravariant vector and vice versa by taking the inner product with the metric tensor. However, this seems to me like a hackish abuse of the inner product. Instead of the metric eating 2 vectors like a good inner product, it eats 1 and then chills out.
Anyways, I would appreciate any help with this!