I want to show :
This is from "Mathematical analysis" by Andrew Browder. This is not a manifolds text so we have only defined submanifolds on $R^n$ using the local immersion definition. I know there are at least 4 more equivalent definitions. The definition of tangent space is defined by smooth curves. Also smooth mapping on a set is defined to be a smooth extension $F$ to an open set such that $F$ and $f$ agrees on $M$ and a previous theorem showed that the differential (jacobian matrix) applied to tangent spaces are well defined.
So by definition, an element of $h\in T_p(M)$ is a smooth curve $\gamma$ that maps to $M$ such that $\gamma(0)=p$ and $\gamma'(0)=h$. So I want to show $dF_p(h)=h' \in T_{f(p)}(N)$. Now this is where I have been stuck for a while, because we want a curve $\gamma_N$ that maps to $N$ such that $\gamma_N(0)=f(p)$ and $\gamma_N'(0)=h'$, but I can't think of any good reason why such a curve should exist.
Best Answer
By the definition, $h\in T_p(M)$ if there exists a smooth curve $\gamma$ that maps to $M$ such that $\gamma(0)=p$ and $\gamma'(0)=h$. It is important to note that this also means that if you are given any smooth curve $\gamma$ that maps to $M$ such that $\gamma(0)=p$, then (by definition) $\gamma'(0)$ is an element of $T_pM$.
Given $h\in T_p(M)$, consider a smooth curve $\gamma$ that maps to $M$ such that $\gamma(0)=p$ and $\gamma'(0)=h$. Then the smooth curve $\alpha=f\circ\gamma$ maps to $N$, because $f(M)\subset N$. Moreover, $$ \alpha(0)=f(\gamma(0))=f(p) $$ and $$ \alpha'(0)=df_p\gamma'(0)=df_ph. $$ By the first paragraph we know that $df_ph\in T_{f(p)}N$. Thus, we have shown that $$ df_pT_p(M)\subset T_{f(p)}N. $$