Flat potentials with nonzero circulation

Put $s=x^2+y^2>0$, $P=-y{s}^{-1}$, and $Q=x{s}^{-1}$. Iterating the positive-base power rule in [F2] gives derivatives of every order of $u^{-1}$ on $u>0$; the chain and product rules therefore show that $P,Q$ are smooth on the punctured plane. Also by [F2], ${\mathrm{\partial }}_{x}Q=s^{-1}-2x^2{s}^{-2}$ and ${\mathrm{\partial }}_{y}P=-s^{-1}+2y^2{s}^{-2}$, so [F1] gives $dA=(2s^{-1}-2(x^2+y^2)s^{-2})dx\wedge dy=0$.

By [F9], the coordinate functions of $\gamma (t)=(\cos (2\pi t),\sin (2\pi t))$ are smooth on all of $\mathbb{R}$, using composition with the affine function $2\pi t$ and the chain rule in [F2]. By [F10], ${\cos }^2(2\pi t)+{\sin }^2(2\pi t)=1$, so this extension takes values in the punctured plane. It therefore supplies the neighbourhood-smooth extension required for a singular one-simplex in [F3]. The zero values in [F10] and periodicity in [F11] give $\gamma (0)=\gamma (1)=(1,0)$, hence $\mathrm{\partial }\gamma =0$. Differentiating by [F10] and the chain rule gives ${\gamma }^{\ast }dx=-2\pi \sin (2\pi t)dt$ and ${\gamma }^{\ast }dy=2\pi \cos (2\pi t)dt$. Thus the pulled-back numerator is $2\pi ({\sin }^2(2\pi t)+{\cos }^2(2\pi t))dt=2\pi dt$, and its denominator is $1$, so ${\gamma }^{\ast }A=2\pi dt$. In the positive affine coordinate on the standard one-simplex, [F5] and [F12] reduce its integral to the rectangle integral of the constant $2\pi$ on $[0,1]$. For any partition $0=t_0<\cdots <t_m=1$, every subinterval has infimum and supremum $2\pi$, so both Darboux sums equal ${\sum }_{i=0}^{m-1}2\pi (t_{i+1}-t_i)=2\pi$. Their lower supremum and upper infimum therefore agree at $2\pi$, proving ${\int }_{\gamma }A=2\pi >0$ by [F11]; [F4] identifies this as the period. If a globally defined smooth $\chi$ satisfied $A=d\chi$, then [F6] would give ${\int }_{\gamma }A={\int }_{\gamma }d\chi ={\int }_{\mathrm{\partial }\gamma }\chi =0$, contradicting that positive value. Thus $A$ is not globally exact.

Every point of $B$ has the whole segment to a chosen centre in $B$, so $B$ is star-shaped. The form $A$ is closed by step 1.1; [F7] therefore supplies a smooth local primitive on $B$.

If $A=d{\chi }_1=d{\chi }_2$ on $B$, then $d({\chi }_1-{\chi }_2)=0$. The zero-form case of [F1] makes both coordinate partial derivatives of ${\chi }_1-{\chi }_2$ zero, so its total derivative vanishes throughout the convex open set $B$; [F8] makes it constant.