Suppose that $X$ is a manifold with boundary and suppose that $x$ is a boundary point. Define the upper half space $H_x(X)$ in $T_x(X)$ to be the image of $\mathbb{H^k}$ under $d \phi_0:\mathbb{R^k} \to T_x(X)$ where $\phi:U \to X$ is a local parametrization with $\phi(0)=x$. I need to prove that $H_x(X)$ does not depend on the choice of local parametization.
My Try :
Suppose $\psi:V \to X$ be another parametrization with $\psi(0)=x$ as well. Can I assume that by shrinking(?) $U$ and $V$ we'd have $\phi(U)=\psi(V)$ ? Why can I assume such a thing ?
Assuming that, I can say $h=\psi^{-1} \circ \phi : U \to V$ is a diffeomorphism and the result follows from differentiating $h$.
Thank you in advance