I am starting a course in Manifolds, and would like to see how a solution to this typical problem might appear.
Q: Consider $\mathbb{R}^2$ (with complete standard atlas containing the chart $(\mathbb{R}^2, id_{\mathbb{R}^2})$ - i.e. standard differentiable structure). Find all points $p\in{\mathbb{R}^2}$ in a neighbourhood of which the functions $x$, $x^2+y^2-1$ give a chart.
A: My attempt would be to calculate the Jacobian of the function $f(x,y)=(x,x^2+y^2-1)$ and find all points where the determinant of this Jacobian is non-zero. Would that be the correct approach?
Thank you for your help.