This is homework so no answers please
The problem is: $A=\{(x,y,z)\in \mathbb{R}^{3}: x^{4}+y^{2}+z^{2}=1\}$ is diffeomorphic to 2-sphere $\mathbb{S}^{2}$.
Any mistakes:
Consider $f:\mathbb{R}^{3}\setminus \{0\}\to \mathbb{R}^{3}$ defined as $f(x,y,z)=(\sqrt{x},y,z)$. This is a smooth map so its restriction to $f':A\to \mathbb{S}^{2}$ is also smooth with smooth inverse $f(x,y,z)=(x^{2},y,z)$. Bijectivity follows easily.
Yeah so the problem is A will not be contained in $\mathbb{R}^{3}\setminus \{0\}$. I want to avoid going through the coordinate representations. Let alone that I will have to show A is a submanifold.
Any hints?
Thanks