Here the problem :
1.8.6 - A vector field $\vec{v}$ on a manifold $X$ in $\mathbb{R}^N$ is a smooth map $\vec{v}:X \to \mathbb{R}^N$ such that $\vec{v}(x)$ is always tangent to X at x. Verify that the following definition (which does not explicitly mention the ambient $\mathbb{R}^N$) is equivalent : a vector field $\vec{v}$ on X is a cross section of T(X) - that is, a smooth math $\vec{v}:X \to T(X)$ such that $p \circ \vec{v}$ equals the identity map of $X$.
Definition :
(1) $T(X)$ is the tangent bundle of a manifold $X$ in $\mathbb{R}^N$. The tangent spaces to $X$ at vario8us points are vector subspaces of $\mathbb{R}^N$ that will generally overlap one another. The tangent bundle T(X) is an artifice used to pull them apart. Specifically, $T(X)$ is an artifice used to pull them apart. Specifically, T(X) is the subset of $X \times \mathbb{R}^N$ defined by $$T(X)= \{(x,v) \in X \times \mathbb{R}^N : v \in T_x(X),$$ where $T_x(X)$ is the tangent spaces at $x$ on $X$.
(2) $p$ is simply the projection map $p: T(X) \to X$, $p(x,v)=x$ is a submersion.
I can not make a rigorous link between the two definitions. I know that simply use the definitions, but formally explain how the transition from one to the other?
Given a vector field $\vec{v}$ on $X$ with respect to the first definition, consider the map $\vec{w} : X\to T(X)$ given by $\vec{w}(x) = (x,\vec{v}(x))$. Note that $\vec{w}$ is smooth and $p(\vec{w}(x)) = x$ for all $x\in X$. So $\vec{w}$ is a vector field on $X$ with respect to the second definition.
Conversely, given a vector field $\vec{w}$ on $X$ under the second definition, note that since $p\circ \vec{w} = \textrm{id}_X$, $\vec{w}(x) = (x, \vec{v}(x))$ for some $\vec{v} : X \to \Bbb R^n$. Note $\vec{v}$ is smooth since $\vec{v}$ is the composition $\pi_2\circ \vec{w}$, and both $\pi_2$ and $\vec{w}$ are smooth. Further, since $(x,\vec{v}(x))\in T(X)$, $\vec{v}(x)$ is tangent to $X$ at $x$. So $\vec{v}$ is a vector field on $X$ under the first definition.