A question about atlas for an ellipsoid using $S^2$ atlas and another question regarding diffeomorphisms between two connected open sets in $R^n$

Question

I have two questions: $\quad$ 1) I am trying to show that an ellipsoid is a 2- dimensional smooth manifold in $R^3$ by using the following $S^2$ atlas: ${A = \{(U_i,f_i),(V_i,g_i) | 1\leq i\leq 3 \text{ }\}}$ where: ${U_i = \{(x_1,x_2,x_3)\in S^2 | x_i>0 \text{ }\}}$ , ${V_i = \{(x_1,x_2,x_3)\in S^2 | x_i<0 \text{ }\}}$, $f_i$ and $g_i$ are the projections of $(x_1,x_2,x_3)\in S^2$ to $R^2$ without the $x_i$ coordinate. I thought about taking the atlas of the ellipsoid to be ${B = \{(U'_i,f_i\circ\theta),(V'_i,g_i\circ\theta) | 1\leq i\leq 3 \text{ }\}}$ where $\theta$ is the diffeomorphism from $E$ to $S^2$ such that $\theta(x_1,x_2,x_3)=(\frac{x_1}{a}, \frac{x_2}{b}, \frac{x_3}{c})$, ${U'_i = \{(x_1,x_2,x_3)\in E |x_i>0 \text{ }\}}$ and ${V'_i = \{(x_1,x_2,x_3)\in E | x_i<0 \text{ }\}}$ Am I correct? $\quad$ 2) Every diffeomorphism $f$ between two connected open sets in $R^n$ preserves orientation or reverse orientation. How can I show that $det(Df_x)>0$ or $det(Df_x)<0$?

Answer 1

I would do it like this: the equation of the ellipsoid $S$ is $f (x,y, z) = ax^2 +by^2+cz^2−1 = 0;\ a,b,c,>0$, so $S=f^{-1}(0).$ The only critical point of $f$ is $(0,0,0)$, which, of course, does not lie on the ellipsoid. So every point of $S$ is a regular point. Let $p=(x,y,z)\in S$ be such that $x\neq 0.$ Then $\partial f_x(p)\ne 0$ and so the Jacobian matrix of the map $(x,y,z)\mapsto (f(x,y,z),y,z)$ is invertible. The inverse function theorem now applies to give a neighborhood $p\in U\subseteq \mathbb R^3$ such that $(U,(f,y,z))$ is a chart in $\mathbb R^3.$ And since $f$ vanishes on $U\cap S,$ we see that $(U,(f,y,z))$ is an adaptive chart for $S$ and so $(U\cap S,(y,z))$ is a chart for $S$. Similarly, if $y\neq 0$, or $z\neq 0$,we get charts $(V\cap S,(x,z))$ and $(W\cap S,(x,y))$, respectively, for $S$.Since each $p\in S$ must have at least one of $x,y$ or $z$ not equal to zero, we see that we get adapted charts in $\mathbb R^3$ that cover $S$, so $S$ is a regular submanifold of $\mathbb R^3$ of dimension $2$.

For the second question, note that $\det$ is continuous on the connected open set $U$ and since $f$ is a diffeomorphism, $x\in U\Rightarrow \det Df_x\neq 0$.