Parametrize $\{(x,y,z)\in\mathbb{R}^3\mid x+y^2+z^2=1, x\geq0\}$

Question

Let $M =\{(x,y,z)\in\mathbb{R}^3\mid x+y^2+z^2=1,x\geq0\}$.

It seems to me that this manifold is a "cone" since we have $y^2+z^2=1-x$ for $x\in[0,1]$ which, geometrically, is a circle in the $yz$ plane starting at radius $1$ at $x=0$ and shrinking to radius $0$ at $x=1$. Furthermore, I suspect that the boundary is just $\partial M=\{(0,y,z)\in\mathbb{R}^3\mid y^2+z^2=1\} \cup \{(1,0,0)\}$ but I am not sure how to prove that because I am having difficulty parametrizing this manifold (i.e. finding a diffeomorphism between $\mathbb{R}^2$ and $M$).

Answer 1

$x + y^2 + z^2 = 1$ describes a paraboloid.

How about $ y = r \cos t, z = r \sin t, x = 1-r^2$

Answer 2

We're going to parametrize the surface in terms of T.

Let $M =\{(x,y,z)\in\mathbb{R}^3|x+y^2+z^2=1,x\geq0\}$.

let $t \in [0,1]$

let $x = t, y = (1-t)^\frac{1}{2} \cos\theta, z = (1-t)^\frac{1}{2} \sin \theta $ Then we can check that indeed $x + y^2 + z^2 = t + (1-t)(\cos ^2 \theta + \sin ^2 \theta ) = 1$ $ \forall t \in [0,1] $.

For questions like these try to think how you could best "naturally" assign coordinates. for this example, you may first want to give the height, and then once you have your height sorted, you may want to think of where in the circle you are.