I have a question maybe more relevant to an English language section of StackExchange, but I doubt that anybody but a Mathematician could properly answer my question.
Let $\mathcal M$ be a smooth manifold, let $X$ be a smooth vector field on $\mathcal M$ and let $\Sigma$ be a smooth hypersurface of $\mathcal M$. Let $t\mapsto \gamma(t,x)$ be the integral curve of $X$ originating at $x\in \Sigma$. The vector field $X$ is tranverse to $\Sigma$ at $x\in \Sigma$ whenever $\dot \gamma(0,x)\notin T_x(\Sigma)$: we may say as well in that case that the curve $\gamma$ is transverse to $\Sigma$.
When $\gamma$ is not transverse to $\Sigma$, I suppose that for some $k\ge 2$ $$ \frac{d^l\gamma}{dt^l}(0,x)\in T_x(\Sigma)\quad\text{for $1\le l\le k-1$}, \quad \frac{d^k\gamma}{dt^k}(0,x)\notin T_x(\Sigma), $$ and I want to say that [the order of contact of $\gamma$ with $\Sigma$ is exactly $k$] or that [the curve $\gamma$ is tangential to $\Sigma$ with a contact of order $k$]. Are these formulations correct?
