$A=M\cap N$, $$M=\{(x,y,z)\in\Bbb R^3| x^2+y^2=1\},$$ $$N=\{(x,y,z)\in \Bbb R^3|x^2-xy+y^2-z=1\}.$$
1. Is $A$ is smooth manifold?
2. Find the points of $A$ that are farthest from the origin.
This is what I thought about doing. If I plug in $x^2+y^2=1$ into $x^2-xy+y^2-z=1$, I get $xy=-z$. Which means that the manifold is a hyperbolic paraboloid.
To show that $A$ is manifold, I should find coordinate charts on it and show that they are continuous, and the differential of them is of rank $3$.
Can someone help me to proceed from this step further? Help would be highly appreciated!
How about $(x,y,z)=(\cos t,\sin t,-\sin t\cos t)$ Check the derivative is never the zero vector.