In the textbook "An Introduction to Manifolds" written by Loring W.Tu, I am confused about a proof of a theorem. The theorem aims to prove that if $g:N \to \mathbb{R}$ is a smooth map on the manifold $N$ then, a nonempty regular level set $S = g^{-1}(c)$ is a regular sub manifold of a manifold of co-dimension 1.
In the proof, assume $f = g - c$, and assume $\frac{\partial f}{\partial x^1}$ not equals to zero, then $(U_p,f,x^2,\dots,x^n)$ formed a coordinate system. And after we proved that there existed a neighborhood $U_p$ of $p$ on the level set $U_p \bigcap S$ is defined by setting the first coordinate f equal to zero. Then $S$ is a regular submanifold of dimension $n-1$ in $N$.
Here is the proof:
My question is that can we set other coordinates to be zero? If can't, why we must have to set $f$ equal to zero?
You can substitute $f$ for the $i^{\text{th}}$ coordinate $x^i$ precisely when $\dfrac{\partial f}{\partial x^i}\ne 0$. This is what's required to insure that the Jacobian determinant is nonzero.