If $V$ is a vector field on $M$ show that the conditions
- $V \in C^{\infty}(M,TM)$
- If $v^i : U \to \Bbb R, i =1, \dots n$, are the component functions of $V$ with respect to a chart $(U, \varphi)$, $\varphi=(x^1, \dots,x^n)$, then $v^i \in C^\infty(U)$.
are equivalent.
I know that if $(U, \varphi)$ is a chart for $M$, then $(TU, \tilde{\varphi})$ is a chart for $TM$ where $$\tilde{\varphi}(v)=(x^1(p), \dots x^n(p), v^1(c), \dots v^n(c))$$ where $c=\sum_{i=1}^n v^i \frac{\partial}{\partial x^i} \bigg|_p \in T_p M$.
It's somewhat confusing that $\tilde{\varphi}$ has components $v^i$ that are also determining the tangent vector $c$. What is the main idea here? I don't really understand the question at hand. Also what does the fact that $V$ is a vector field have to do with anything?