I am self studying Frankel's The Geometry of Physics, early in the text he poses the example:
The x axis of the xy plane can be described as the locus of the quadratic $F(x,y) := y^2=0$. Both partial derivatives vanish on the locus, the x axis.
Is it correct to say that this is because the partial with respect to $x$ is $0$ always, while the partial with respect to $y$ is $0$ only along the $x$ axis (ie when $y=0$)
Also, what would be a manifold in this setting? Since all along the $x$ axis the partial with respect to $x$ is $0$, then the jacobian criteria is violated and it is not a manifold; however, the text says that the $x$ axis is a 1-manifold.
Any help would be greatly appreciated, thank you!