= Solution
The <magnetic field> is $B=\nabla\times A$. The magnetostatic <Ampère-Maxwell equation> is $\nabla\times B=\mu_0J$, so the <curl of the curl identity> gives
$$
\nabla(\nabla\cdot A)-\nabla^2A=\mu_0J.
$$
A <gauge transformation> $A\mapsto A+\nabla\chi$ does not change $B$. We may therefore choose the <Coulomb gauge> $\nabla\cdot A=0$ by solving the <Poisson equation> $\nabla^2\chi=-\nabla\cdot A$. It follows component by component in a <Cartesian coordinate system> that
$$
\boxed{\nabla^2A=-\mu_0J}.
$$
The free-space <Green function> of the <Laplacian> then gives
$$
\boxed{A(r)=\frac{\mu_0}{4\pi}\int_{\mathbb R^3}
\frac{J(r')}{|r-r'|}\,d^3r'}.
$$
For a localized steady <current density>, <conservation of electric charge> gives $\nabla'\cdot J=0$; <integration by parts> confirms directly that this integral has zero <divergence> and hence obeys the Coulomb gauge.
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