In the the book Differential Equations with Applications and Historical Notes, I came across a problem (link to Google Books, Problem 1) asking us to prove the statement in the title using a geometric argument. I understands the part (a) of the question, that if C is a closed path of the system, then its index would be 1 as the vector should rotate exactly once if the path is traversed once.
However, I have a hard time understanding part (b). Isn't the question itself contradictory? I mean it says "if C is a path (and it is closed as assumed in the question) that contains no critical points", but aren't we trying to prove the exact opposite (a closed path must contain at least one critical points)? And how can we go from part (b) to prove the statement? Can anyone give me some insights on this problem? Any help would be appreciated.
Edit: Picture of the problem
