I'm learning a bit about smooth manifolds, and currently I'm learning about tangent bundles (just definitions mainly) and vector field.
This is my reference : Tu's Introduction to Manifolds. I was also watching a video about tangent bundles (because I was struggling with the concept).
Once I understood the definition however I realized I have an issue understanding why we need the notion of tangent bundle. I can't remember where I read this but am I right when I say that tangent bundles are necessary if we want to generalize the notion of function on a manifold?
Consider a smooth manifold $M$, if we wanted to define what a vector field is to me the definition should reflect the fact that for each $p \in M$ we have a $v \in T_p M$, therefore it should be a map.
This is probably the key why such association isn't good as definition because a map needs both domain (in this case $M$) and an image space, however my naive definition involves for each $p$ a different space $T_p M$ and this is why we need the notion of tangent bundle.
Is this observation correct?