The basis of a tangent space

Question

I don't know how to start the second part. Because I can't find a chart for this submanifold, is there anyway to do this problem without an explicit chart?

Answer 1

Yes, the tangent space $T_pM$ can be identified with $\ker df|_p$ where you consider $df|_p$ as a linear map. More explicitly, treating $df|_p$ as a vector $(\nabla f)(p)$, the vector $(\nabla f)(p)$ is the normal vector to the tangent plane of $M$ and so, to find $T_pM$, you just need to find two linearly independent vectors that are orthogonal to $(\nabla f)(p)$.