Could anyone explain the following fact in detail?
Suppose $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is smooth and $\nabla f\neq 0$ on the level set $M = \{f=0\}$, so that $M$ is a smooth hypersurface. A smooth regular curve $\phi:(-\epsilon,\epsilon)\rightarrow\mathbb{R}^n$ has m-order contact with the hypersurface $M$ at $\phi(0) \iff (f\circ \phi)^{(i)}(0) = 0, 0\leq i \leq m$.
Currently, I am a little confused about how to choose the curve $\psi$ in M s.t. $$\phi^{(i)}(0) = \psi^{(i)}(0), 0\leq i\leq m$$ in $\impliedby$ case.