2 definitions of vector fields on $\mathbb R^n$: maps $\mathbb R^n \longrightarrow \mathbb R^n$ and smooth sections of $T\mathbb R^n$

Question

One definition says a vector field is a map $f:\mathbb R^n \longrightarrow \mathbb R^n$.

Another definition says a vector field is a smooth section of $T\mathbb R^n$, the tangent bundle of $\mathbb R^n$.

The intuition for a vector field on $\mathbb R^n$ is that it is a function that attaches a vector to each point in $\mathbb R^n$.

Are these 2 definitions equivalent? Do they agree with intuition?

Answer 1

The two definitions you posted aren't equivalent, since the second one has a smoothness criterion, but the first one doesn't. If you were to delete the smoothness criterion though, they would be the same. Since $\mathbb{R}^n$ can be covered by a single chart, $T\mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n$. You can then show that a section on $T\mathbb{R}^n$ will define a unique function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, and vice versa.