We can define the tangent bundle in two ways, depending on how we define our tangent spaces. If our tangent spaces are derivations on the space of smooth germs at $p \in \mathcal M$ (where $\mathcal M$ is a smooth manifold without boundary), then the tangent bundle is $$T\mathcal M := \bigcup_{p\in \mathcal M} T_p\mathcal M$$ since the $T_p\mathcal M$ are already disjoint under this definition. Alternatively, if we define $T_p\mathcal M$ to be the space of derivations at $p$, then $$T\mathcal M := \coprod _{p \in \mathcal M} T_p\mathcal M =\bigcup_{p\in \mathcal M}\{p\}\times T_p\mathcal M.$$ My questions are:
- If we define $T_p\mathcal M$ in the second way, how is it that we can say $T\mathcal M$ is a vector bundle with typical fibre $T_p\mathcal M$?
- What are the advantages/disadvantages of each definition?