If I have,
\begin{align} f(u,v) &= (x(u,v),y(u,v),z(u,v))\\ &=(( \sqrt{2} + \cos(u)) \cos(v) , (\sqrt{2} + \cos(u)) \sin(v) , \sin(u)). \end{align}
where $(u,v) \in \mathbb{R^2}$
I showed that the image set $f(\mathbb{R^2})$ is a submanifold by observing that it is an open set of $\mathbb{R^3}$.
Now I want to show that $f(\mathbb{R^2})$ is diffeomorphic to $T^2$
I thought about showing that the function $g(f(\mathbb{R^2}))$ is differentiable by using the chain rule, but to prove that the inverse is differentiable is too much work I think, and I also know bijectivity won't work here, I don't know what should be the 'obvious' way to start this off, any hints?