Differential geometry: $S_1,S_2$ are regular surfaces, $f:S_1\to\mathbb{R}^{3}$ is smooth, then $(df)_p (T_pS_1) \subseteq T_f(p)S_2$.

Question

If $S_1$ and $S_2$ are two regular surfaces and $f: S_1\to\mathbb{R}^{3}$ is smooth function and $f(S_1)\subseteq S_2$ and $p$ is a point in $S_1$, then $(df)_p (T_pS_1) \subseteq T_{f(p)}S_2$.

How can we prove that implication?

Answer 1

Let $v \in T_p S_1$ be a tangent vector. Then there is a curve $\gamma$ in $S_1$ such that $\gamma(0)=p$ and $\gamma'(0)=v$. By definition, we have $(df)_p(v) = (f \circ \gamma)'(0)$. Now note that $f\circ \gamma$ is a curve on $S_2$ satisfying $(f\circ \gamma)(0)=f(p) \in S_2$. So, by the very definition of a tangent vector, its velocity $(f\circ \gamma)'(0)\in T_{f(p)}S_2$ is a tangent vector on $S_2$ at the point $f(p)$. Note that it is a vector in $T_{f(p)}S_2$ and not in $T_pS_2$ as you wrote in your question.