I want to write a particular $C^{\infty}-$diffeomorphism formally, but I need help as to clear up the notation for it to be more canonical and easy to understand. Furthermore, I don't know how to define $Df_{x}$ properly and formally and need help with a satisfying notation too.
The $C^{\infty}-$diffeomorphism in question is (bear in mind I don't know the canonical notation, so I'll try to explain what I mean by each symbol):
$f \colon \mathbb{R}^3/z^+ \to \mathbb{R}^3$, in which $S^{2}(r)-\{p\} \mapsto \pi(g(r))$.
Where
$\bullet$ $\mathbb{R}^3/z^+$ means $\mathbb{R}^3 - \{ (x,y,z) \in \mathbb{R}^3 \ \colon \ z > 0 \}$;
$\bullet$ $S^2(r) - \{ p \}$ means $S^2$ with radius $r > 0$ minus the pole $p = (0,0,r) \in S^{2}(r)$;
$\bullet$ $\pi(g(r))$ means the plane $\{ (x,y,z) \in \mathbb{R}^3 \ \colon \ z = g(r) \}$, where $g \colon \mathbb{R}^+ \to \mathbb{R}$, with $g(r) = \begin{cases}r, \ \ \ \ r \geqslant 1,\\-1/r+2, \ \ \ \ r < 1.\end{cases}$
As for how to define $Df_{x}$, my needs is for every path in the domain of $f$ to map to a path in the codomain of $f$ such that the velocity at any point $x$ in the path is preserved. Is that possible, and if so, how do I write the function in "non hand-wavey" terms?
And as a final note, if the question is poorly writen, I would be happy to attempt to reword it better.