I want to show that the $n$-sphere $S^n$ is obtained by gluing two copies of $\mathbf{R}^n$ along the map $\varphi : \mathbf{R}^2 \setminus \{0\} \to \mathbf{R}^2 \setminus \{0\}$ defined by $x \to \frac{x}{\|x\|^2}$.
I checked that this map is a diffeomorphism, hence $\mathbf{R}^n \coprod_\varphi \mathbf{R}^n$ is well-defined.
Intuitively (ie: with a drawing), I see why the result is true, but I don't know how to define a diffeomorphism $\mathbf{R}^n \coprod_\varphi \mathbf{R}^n \to S^n$, and how to use the gluing map.