Good day!
As in my other question I am referring to the book "Differential Equations and Dynamical Systems" by Lawrence Perko, chapter 3.12. I have a question regarding Lemma 2:
Lemma 2. If $v$ and $w$ are two continuous vector fields defined on a Jordan Curve $C$ which never have opposite directions or are zero on $C$, then $I_v(C)=I_w(C)$.
What is the definition of "never having opposite directions" for the vector fields $v,w\in C^1(\mathbb{R}^2)$ (or $v,w\in C^1(E)$ where $E$ is an open subset of $\mathbb{R}^2$). I couldn't find a definition in Perko's book for this.
KittyL is right: not having opposite direction means $v/\|v\| \ne - w/\|w\|$.
It's worth noting that both conditions (nonvanishing and not pointing in opposite directions) can be combined into one: $$ t v + (1-t) w \ne 0 \quad \text{ for } t\in [0,1] $$ The latter, not coincidentally, is exactly what you need to argue $I_v=I_w$ by means of straight-line homotopy.