I'm actually interested in MoonUnit's last paragraph:
I'm not too clear on the geometrical meaning myself, though I can go through the motions and manipulate tensor expressions just fine.
A contravariant vector exists in the tangent space of a specific point in the manifold being considered. In other words, if you have a basketball (the surface of which is a 2D manifold), and you glue little toothpicks tangent to it, each toothpick is a contravariant vector defined at the point it is glued to the basketball. This much makes intuitive sense to me -- I can just as easily think "tangent" whenever someone says "contravariant." If you take any point on the basketball, the set of all tangent vectors you could glue to it there forms a (2-dimensional) tangent space at that point. Contravariant vectors at that point belong to that tangent space.
Now, I know covariant vectors live in cotangent
spaces, but I'm not really clear on how to visualize a cotangent space. I have read most of John Baez' book "Gauge Fields, Knots, and Quantum Gravity," in which he makes an earnest attempt to help the reader visualize a covariant vector -- but it falls flat on me. I just can't understand what he's trying to say.
Does anyone have a clear explanation of how to "visualize" a covariant vector? Is it really even possible to visualize it?