= Radial-gauge potential for a divergence-free magnetic field
{title2=$\mathbf A(\mathbf x)=\int_0^1s\,\mathbf B(s\mathbf x)\times\mathbf x\,ds$}
For a smooth <divergence>-free field on $\mathbb R^3$, the <differential form> $F=\iota_{\mathbf B}(dx^1\wedge dx^2\wedge dx^3)$ is closed. The radial homotopy formula gives a global <magnetic vector potential> $\mathbf A(\mathbf x)=\int_0^1s\,\mathbf B(s\mathbf x)\times\mathbf x\,ds$ with <curl> $\mathbf B$. For constant $\mathbf B$, this reduces to $\mathbf A=\tfrac12\mathbf B\times\mathbf x$.
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