If we have a unit sphere denoted by $S^2$ , in $\mathbb{R}^3$, and some function $z$ that maps $z:S^2 \rightarrow (0,\infty)$ which is defined to be smooth and has $+$ value. Now given some defined set $\Lambda$:

$$\Lambda=\{{z(x)x: x\in S^2}\}$$

Question 1)If $Z(u_1,u_2):U\rightarrow S^2$ is a parametrisation of $S^2$ (smooth), how can one show that another parametrisation on $\Lambda$ defined by $F:U \rightarrow \Lambda$, $(u_1,u_2) \rightarrow$ $z(Z(u_1,u_2))Z(u_1,u_2)$, then $\Lambda$ is a regular surface?

Question 2)Now, we have coinciding parametrisations of $S^2$ which are given by $Z_i(u_1,u_2):U_i \rightarrow S^2$, and $F_i:U_i\rightarrow \Lambda$ the parametrization of $\Lambda$ caused by $Z_i$. Prove that $F_1^{-1}(F_2)=Z_1^{-1}(Z_2)$ for $i=1,2$.

Is $\Lambda$ and $S^2$ diffeomorhpic? If so give an explicit form of the diffeomorphism.

Question 3)

Question 4)Let us now define some map $\Psi: \mathbb{R}^3 -\{(0,0,0)\} \rightarrow$ $\mathbb{R}^3 - \{(0,0,0)\}$, which is given by:

$$\Psi(x) = \frac{x}{|x|^2}$$

Denote $\Lambda^*=\Psi(\Lambda)$.

a)Is $\Lambda^*$ a regular surface?

b)For some map $\psi:\Lambda \rightarrow \Lambda ^*$which is some restriction on $\Psi$ on $\Lambda$, prove that $\psi$ is a diffeomorphism.

c)For any $c \in \Lambda$, the tangent map $d\psi_c:T_c \Lambda \rightarrow T_{\psi(c)} \Lambda^*$ at $c$ (which is a map from the tangent plane of $\Lambda$ at $c$ to the tangent plane at $\Lambda ^*$ at $\psi(c)$), prove that:

$$d\psi_c(W) =\frac{|c|^2 W-2(c.W)c}{|c|^4}$$ where $c.W$ is the dot product in $\mathbb{R}^3$.

The chapter explains nicely on the concepts, then the problems all look something like this, and I just cant touch them yet. It would be extremely helpful if someone can show me how to do this, and maybe I will learn ways to doing these kind of problems. I have asked other forums too as I need help getting familiar with problems like these.