Example of 1-form which is closed, but not exact
(Supplement to 2.6; Optional, for those who already took complete Calculus II)
On $\mathbb{R}^2\setminus \{0\}$ (hole is very important) consider form
$$\frac{-ydx+xdy}{x^2+y^2}.$$
So, $M(x,y)= -y/(x^2+y^2)$, $N(x,y)=x /(x^2+y^2)$.
- Prove that
$$\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}=0.$$
as $(x,y)\ne 0$.
- Consider polar angle $\theta(x,y)=\arctan (y/x)$.
- Prove that $$\frac{\partial \theta}{\partial x}=-\frac{y}{(x^2+y^2)},\qquad \frac{\partial \theta}{\partial y}=\frac{x}{(x^2+y^2)}.$$
- This provides another proof that the form is closed.
- The trouble with this primitive is that it is defined uniquely on a subdomain $\Omega\subset \mathbb{R}\setminus \{0\}$ if and only if one cannot go around $\{0\}$ as in this case increment of $\theta$ would be $2\pi$ (and thus $\theta$ would be defined up to $2\pi \mathbb{Z}$.
- Further, there is no primitive, as if $d F=d\theta$ then increment of $F$ as we go around $\{0\}$ along contour $\Gamma$ would be
$$\oint _\Gamma dF= \oint _\Gamma d\theta =2\pi$$
(and thus $F$ would be defined up to $2\pi \mathbb{Z}$.