Suppose I have an $(x,y)$ coordinate system. I have a zero level set of a smooth function $H(x,y)$, i.e. $H(x,y)=0$. Suppose this level set is parametrised by a parameter $t$, i.e. $(x(t), y(t))$.
I want to apply implicit function theorem to show that I can use the functions $(H, t)$ as a system of well-defined coordinates in some neighbourhood of this zero level set.
How do I set up the implicit function theorem for this and what assumptions I need to make? And in particular, under what conditions is $H$ $\mathbf{not}$ a good coordinate?
Edit. Further, from what geometrical considerations can I infer whether $(H,t)$ is a good systems of coordinates? (i.e. non-vanishing of gradient of $H$, etc...?)
If $\nabla H(x_0,y_0)\ne (0,0)$, one of the partial derivatives is $\ne 0$ and by the implicit function theorem: $$ \partial_x H(x_0,y_0)\ne 0\implies x\hbox{ can be written locally as function of } y: H(X(y),y) = 0 $$ $$ \partial_y H(x_0,y_0)\ne 0\implies y\hbox{ can be written locally as function of } x: H(x,Y(x)) = 0 $$ In any case, you have your first condition. For the second condition, suppose wolg $\partial_y H(x_0,y_0)\ne 0$. You want $$(x,y)\longmapsto (x,H(x,y))$$ be locally a diffeomorphism. Calculating the differential at $(x_0,y_0)$: $$ \pmatrix{ 1&0\cr \partial_x H(x_0,y_0)&\partial_y H(x_0,y_0) } $$ and this matrix is inversible: $$ \left|\matrix{ 1&0\cr \partial_x H(x_0,y_0)&\partial_y H(x_0,y_0) }\right| = \partial_y H(x_0,y_0)\ne 0. $$ Now, apply the inverse function theorem.