= Solution
Since $|\Phi|\to1$, the <Higgs field> is nonzero outside a sufficiently large disk. On a large positively oriented circle write $\Phi=\rho e^{i\chi}$ locally, where $\rho>0$. Its normalized phase is a map from the circle to the unit circle. The phase need not be a globally single-valued real function, but its total change is $2\pi N$ for an integer <winding number> $N$. This integer is the same on all sufficiently large circles because the intervening annulus contains no zeros.
The <gauge covariant derivative> gives
$$
D\Phi=e^{i\chi}[d\rho+i\rho(d\chi-A)],\qquad
A=d\chi-\frac{\operatorname{Im}(\overline\Phi D\Phi)}{\rho^2}.
$$
Integrate around the circle. The assumed rapid decay makes the integral of the second term tend to zero. For example, uniform $|D\Phi|=o(R^{-1})$ and $\rho\to1$ on radius-$R$ circles suffice. Hence
$$
\lim_{R\to\infty}\oint_{|x|=R}A_jdx^j=2\pi N.
$$
By <Stokes theorem>, $\oint_{|x|=R}A=\int_{|x|\leq R}B\,d^2x$. Integrability of $B$ allows the limit, giving the <magnetic flux quantization of an Abelian Higgs vortex>:
$$
\boxed{\int_{\mathbb R^2}B\,d^2x=2\pi N.}
$$
The <topological charge> $N$ is the degree of $\Phi/|\Phi|$ at infinity, counting net phase winding around the cores. For isolated zeros it is the sum of their signed local winding numbers. In the positive <Abelian Higgs vortex> construction, each core of multiplicity $m_a$ has local phase $m_a\arg(z-z_a)$, so
$$
\boxed{N=\sum_am_a,}
$$
the number of vortices counted with multiplicity. Antivortices have the opposite sign. The integer is invariant under smooth gauge transformations defined throughout the disk, whose boundary winding is zero.
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