I'm currently doing a geometry course, and I'm struggling a bit on the practical side of how to show a map is a smooth homeomorphism. Can we go through the following example from one of my problem sheets, which is made up from pretty typical and not too "messy" functions?
I want to show that
$$\sigma(u,v) = (u+v, 2uv, u^2+v^2)$$
is a smooth homeomorphism from the set $U=\{(u,v) \in \mathbb{R}^2: u>v\}$ to its image.
I think that we need to show that is smooth bijection, and has a smooth inverse, and that we could do this by showing that it maps all open sets to open sets.
I'm not really sure how to show this in a way that is efficient and accurate, though, can someone provide me with the general method for showing something like this?