Let $F:N\rightarrow M$ be a $C^{\infty}$ map, At each point $p\in N$, the map $F$ induces a linear map of tangent spaces, called its differential at $p$,
$F_{*}:T_p N\rightarrow T_{F(p)}M$ as follows. $X_p\in T_pN$ then $F_{*}(X_p)$ is the tangent vector in $T_{F(p)}M$ defined by $F_{*}((X_p))f=X_p(f\circ F)\in\mathbb{R}$, I understand that a tangent vector $F_{*}(X_p)$ acts on a $C^{\infty}$ map on $M$, what is the role of $X_p=\sum_{i=1}^{n}a^{i}\partial/dx_i$ here in the definition?, where $\{\partial/dx_{i}\}_{i=1}^{n}$ are basis of tangent space at $p$ of $N$
well, let the basis for tangent space at $F(p)$ of $M$ be $\{\partial/dy_i\}_{i=1}^{m}$, so then $F_{*}(X_p)=\sum_{i=1}^{m}b^i\partial/dy_i$