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From what I've read the tangent bundle is defined as the disjoint union of the tangent spaces to each point on a manifold, i.e. [itex] TM = \bigcup_{p\in M} T_{p}M[/itex]. By disjoint union is it meant that the new set [itex]TM[/itex] is created by constructing ordered pairs [itex](p,\mathbf{v})[/itex] such that each [itex]p\in M[/itex] indexes a family of vectors in the tangent space [itex]T_{p}M[/itex] to that point. In doing so, although two tangent vectors may essentially be 'the same' (in that they have the same components), they are distinct as they are 'fixed' at different points on the manifold [itex]M[/itex] and, as such, belong to different tangent spaces?! Is it then correct to write [itex]TM= \lbrace (p,\mathbf{v})\;\;\vert\quad p\in M, \mathbf{v}\in T_{p}M\rbrace[/itex]?

I guess my real struggle about the concept is why we can treat the points [itex]p\in M[/itex] and tangent vectors [itex]\mathbf{v}\in T_{p}M[/itex] as independent variables? Intuitively, I understand that for each point on an [itex]n[/itex]-dimensional manifold there is an associated [itex]n[/itex]-dimensional tangent space, and as such, a space constructed from the union of all such tangent spaces at each point on the manifold should then be [itex]2n[/itex]-dimensional. I also see how, at any given point [itex]p\in M[/itex] there exists an infinite number of vectors to choose from in the tangent space to that point, and as such given a point [itex]p\in M[/itex] one can essentially independently choose any vector from the tangent space at that point. However, I don't see how this would work the other way around, as surely one needs to pick a point in the manifold before one can choose a vector, as otherwise one does not know which tangent space it belongs to?! (Or is it this why the tangent bundle is constructed as a disjoint union, such that one can pick a vector first and then distinguish it from any other vector by picking a point on the underlying manifold to 'attach' it to a particular tangent space [itex]T_{p}M[/itex]?)