arcs transversal to $C^k$ maps

Question

In reference to:

"if $M \subset\mathbb{R}^2$ and $f\in C^k(M,\mathbb{R}^2)$, then for each regular $x\in M$, we can find an arc $\Sigma$ containing x which is transversal to f"

what is meant by an arc being transversal to f?

Answer 1

It means that the image of the differential of $f$ together with the tangent space of $\Sigma$ at each point of intersection generate the whole space (the plane in this case). Note that here you should write "tranverse" and not "tranversal".