This may be an inadequately phrased inquiry - please allow me recognize of it - yet what is the proper means to consider the cotangent package? It appears weird to consider it as the twin of the tangent package (I am locating it weird to integrate the ideas of "maps to the ground area" with this object).
One rewarding means to think of it, if you have physics history, is as stage room. Your manifold is the arrangement room for some system of fragments, and also the cotangent package is after that the stages, so the cotangent instructions are rates. This is handy additionally with the symplectic framework on $T^*M$.
You could be curious about this MathOverflow blog post : https://mathoverflow.net/questions/17325/why-is-cotangent-more-canonical-than-tangent
(Sorry, I would certainly leave this as a comment yet I simply joined this website and also do not have adequate online reputation.)
I'm not entirely certain what you suggest by this : "It appears weird to consider it as the twin of the tangent package (I am locating it weird to integrate the ideas of "maps to the ground area" with this object)," yet possibly the adhering to will certainly aid you see why it is all-natural to take into consideration the twin room of the tangent package.
Offered a function f on our manifold, we intend to associate something like the slope of f. Well, in calculus, what identified the slope of a function? Its the vector area such that when we take its dot item with a vector v at some time p, we get the directional by-product, at p, of f along v. In a basic manifold we do not have a dot item (which is a statistics) yet we can create a covector area (something which offers a component of the cotangent package at any kind of factor) such that, when related to a vector v, we get the directional by-product of f along v. This covector area is represented df and also is called the outside by-product of f.