I was confused when I prove the next Proposition appeared in the book Introduction to smooth manifolds by Lee.
Suppose $M$ is a smooth manifold with or without boundary ,$S\subset M$ is an immersed or embedded submanifold,and $p\in S$.A vector $v\in T_pM$ is in $T_pS$ iff there is a smooth curve $\gamma:J\to M$ whose image is contained in $S$,and which is also smooth as a map into $S$,s.t. $\gamma(0)=p,\gamma'(0)=v$.
If $v\in T_pM$ is in $T_pS$.Firstly,there exist a cuvre $\gamma:J\to S$ s.t. $\gamma(0)=p,\gamma'(0)=v$.Consider $\tilde{\gamma}=\iota\circ \gamma$ which is curve of $M$ and $\tilde{\gamma}(0)=p$.I need correspond $\tilde{v}=\tilde{\gamma}'(0)$ to $v=\gamma'(0)$.Note that for all $f\in C^\infty(M)$ $$\tilde{v}f=v(f|_S)$$ But I think it's not enough to say that $\tilde{v}\in T_pS$ and $\tilde{v}=v\in T_pS$,because not every $g\in C^\infty (S)$ can be extends to the smooth function in $M$ or restricted for some smooth function of $M$.
When we talk about $v\in T_pM$ is in $T_pS$ ,whether to say there exist $\tilde{v}\in T_pS$ s.t. $v=d\iota_p(\tilde{v})\in T_pM$.We think $v\in T_pM$ is in $T_pS$ as $v\in T_pM$, $v\sim\tilde{v}\in T_pM$.In this identification,above discuss is that $\tilde{v}\in T_pM$ and $\tilde{v}=v\in T_pS$,so we complete this correspond.
Do I above thinking is right?Thank for you help.