# Understanding John Lee's Intuition Regarding Failure of Exactness of Covector Fields in the Punctured Plane

I am trying to understand what is meant by the following wording on page 297 of John Lee's Introduction to Smooth Manifolds, the 2nd Edition:

"The key to constructing a potential function in Theorem 11.49 is that we can reach every point $$x \in M$$ by a definite path from $$c$$ to $$x$$, chosen to vary smoothly as $$x$$ varies. This is what fails in the case of the closed covector field $$\omega$$ on the punctured plane (Example 11.43): because of the hole, it is impossible to choose a smoothly varying family of paths starting at a fixed base point and reaching every point of the domain exactly once."

Theorem 11.49: If $$U$$ is a star-shaped open subset of $$\mathbb{R}^{n}$$ or $$\mathbb{H}^{n}$$, then every closed covector field on $$U$$ is exact.

In Example 11.43, the covector field is

$$\omega = \frac{xdy - ydx}{x^{2} + y^{2}}$$

and the book demonstrates that integrating this over the unit cirlce gives $$2\pi$$. My understanding is (a bit informally) that path independence of a line integral is equivalent to the line integral over any loop being 0 which is the same as every loop being homotopic to a point, and this is what goes wrong. Is Lee saying there is another way to see what goes wrong? Or is this a different wording for what I just mentioned? Thanks in advance.

So you look for a covector like $$\omega$$ that might actually care what path you take to get to a point. Check: does it matter what path you take to get to a point when you try to construct a potential? Equivalently, are there any loops with nonzero path integral? and lo and behold, there are!