Let $U,V$ be open sets of $\mathbb{R^2}$ and $f:V \rightarrow U$ a diffeomorphism between them. Let $S$ be a regular surface and $(U, \varphi)$ an atlas of $S$. Show $(V,\varphi \circ f )$ is also an atlas of $S$.
Let $p\in S$. We have to show there exists an open subset $W \subset S$ such that $p \in W$ verifying:
- $\varphi \circ f: V \rightarrow W$ is an homeomorphism.
- $d(\varphi \circ f)_q : \mathbb{R^2} \rightarrow \mathbb{R^3}$ is injective $\forall q \in V$.
$(U, \varphi)$ is an atlas so there exists $W' \subset S$ with $p \in W'$ such that $\varphi:U \rightarrow W'$ is homeomorphic and $d(\varphi)_q$ is injective $\forall q \in U$.
If we take $W=W'$ then $\varphi \circ f: V \rightarrow W'$ is an composition of homeomorphisms and hence homeomorphism, but I don't know how to prove the injectivity of $d(\varphi \circ f)_q$.