Coordinate chart of Hyperboloid I'm looking at this problem:
Let $H^n$ be the set $\{(x^1,\cdots,x^{n+1})\in\mathbb{R}^{n+1}\mid\sum_{i=1}^n(x^i)^2-(x^{n+1})^2=-1,x^{n+1}>0\}$ of upper part of a two-sheeted hyperboloid. Define a map
$$
\varphi:H^n\to\mathbb{R}^n,\;\;\;(x^1,\cdots,x^{n+1})\mapsto\left(\frac{x^1}{1+x^{n+1}},\cdots,\frac{x^n}{1+x^{n+1}}\right),
$$
I want to show that this map gives a local coordinate chart of $H$, with image being $D^n=\{(u^1,\cdots,u^n)\in\mathbb{R}^n\mid\sum_{i=1}^n(u^i)^2<1\}$. It is straight forward to show that this map is injective.
But when I wanted to show that the matrix $\left(\frac{\partial x}{\partial u}\right)$ is of full rank, I got stuck.
Here comes my question: do I need to check this full-rank-ness? If yes, how should I proceed?
P.S. for convenience, the inverse is
$$
(x^1,\cdots,x^n,x^{n+1})=(\frac{2u^1}{1-[(u^1)^2+\cdots+(u^n)^2]},\cdots,\frac{2u^n}{1-[(u^1)^2+\cdots+(u^n)^2]},\frac{2}{1-[(u^1)^2+\cdots+(u^n)^2]}-1)
$$
and the matrix has entries
$$
\frac{\partial x^i}{\partial u^j}=\left\{\begin{matrix}\frac{2D+4(u^i)^2}{D^2}&i=j\\ \frac{4u^iu^j}{D^2}&i\neq j\end{matrix}\right.
$$
and $\frac{\partial x^{n+1}}{\partial u^j}=\frac{4u^j}{D^2}$ where $D=1-[(u^1)^2+\cdots+(u^n)^2]$.

You have shown that the map is a bijection, the diffeomorphicity follows instantly from the smoothness of it and its inverse.
