One-to-one mapping from $\mathbb{R}^4$ to $\mathbb{R}^3$ I'm trying to define a mapping from $\mathbb{R}^4$ into $\mathbb{R}^3$ that takes the flat torus to a torus of revolution. 
Where the flat torus is defined by $x(u,v) = (\cos u, \sin u, \cos v, \sin v)$.
And the torus of revolution by $x(u,v) = ( (R + r \cos u)\cos v, (R + r \cos u)\sin v, r \sin u)$.
I think an appropriate map would be:
$f(x,y,z,w) = ((R + r x)z, (R + r x)w, r y)$ where $R$, $r$ are constants greater than $0$.
But now I'm having trouble showing this is one-to-one.

Shaun's answer is insufficient since there are immersions which are not 1-1. For example, the figure 8 is an immersed circle. Also, the torus covers itself and all covering maps are immersions. http://en.wikipedia.org/wiki/Immersion_(mathematics)
Your parametrization of the torus of rotation is the the same as in http://en.wikipedia.org/wiki/Torus You just have to notice that the minimal period in both coordinates of the $uv$ plane is the same $2\pi$ in the case of both the flat and rotated tori.
