= Magnetic flux quantization of an Abelian Higgs vortex
{title2=$\int B\,d^2x=2\pi\deg(\Phi/|\Phi|)$}
When $|\Phi|\to1$ and its <gauge covariant derivative> decays fast enough, on a large circle write $\Phi=\rho e^{i\chi}$ locally. The tangential derivative gives $A=d\chi-\operatorname{Im}(\overline\Phi D\Phi)/\rho^2$. The integrated error tends to zero, so <Stokes theorem> turns total <magnetic flux> into the phase change $2\pi N$. This integer is the <winding number> of the normalized <Higgs field> on the circle. For positive <Bogomolny vortex equations> it is the total zero multiplicity.
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