Problem
Problem 1.80 Consider the differentiable map $\varphi: \mathbb{R}^{4} \rightarrow \mathbb{R}^{2}$ given by
$$
u=x^{2}+y^{2}+z^{2}+t^{2}-1, \quad v=x^{2}+y^{2}+z^{2}+t^{2}-2 y-2 z+5 .
$$
(i) Find the set of points of $\mathbb{R}^{4}$ where $\varphi$ is not a submersion, and its image.
(ii) Calculate a basis of $\operatorname{ker} \varphi_{*(0,1,2,0)}$.
(iii) Calculate the image by $\varphi_{*}$ of $(1,0,2,1) \in T_{(1,2,0,1)} \mathbb{R}^{2}$ and the image by $\varphi^{*}$ of $(\mathrm{d} u+2 \mathrm{~d} v)_{(-1,5)} \in T_{(-1,5)}^{*} \mathbb{R}^{2}$, choosing the point $(0,0,0,0)$ in $\varphi^{-1}((-1,5))$.