= Local strain compatibility versus screw-dislocation circulation
{title2=$\oint\nabla w\cdot d\mathbf x=b$}
The antiplane field $w=b\arg(x+iy)/(2\pi)$ has a smooth local <gradient> on the <punctured plane> and satisfies the differential <strain> compatibility condition there. Its <circulation> around the defect axis is nevertheless $b\ne0$, excluding a globally single-valued <displacement> on that domain. A radial cut permits a branch with an axial jump $b$ between the two cut faces; joining faces with that mismatch is the Volterra construction of a <screw dislocation> with <Burgers vector> parallel to the axis. The <stress> has a singular core where <linear elasticity> requires microscopic or plastic regularization. This example separates a local differential integrability test from global closure around a hole.
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