= Flux around a removable point singularity
{title2=$\left|\int_{\partial B_\varepsilon}\mathbf F\cdot\mathbf n\,dS\right|=O(\varepsilon^{d-1+\alpha})$}
A <continuous extension> of a <vector field> across one point need not be <differentiable>, so it does not automatically meet the classical hypotheses of the <divergence theorem>. Instead apply the <divergence theorem> after deleting a ball $B_\varepsilon$ around that point. If $\|\mathbf F\|\leq C\varepsilon^\alpha$ on its boundary in $d$ dimensions, the magnitude of the inner <flux> is at most a constant times $\varepsilon^{d-1+\alpha}$. When $d-1+\alpha>0$ this tends to zero. If the <divergence> is also <integrable>, passing to the limit recovers the outer <flux> from the volume <integral>. For example, a planar <vector field> with $\|\mathbf F\|\leq r$ has an inner <flux> of order $\varepsilon^2$, even when its extension is not <differentiable> at the origin.
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