= Azimuthal inverse-radius vector field
{title2=$\mathbf G=\mathbf e_\phi/\rho$}
The <vector field> $\mathbf G=(-y,x,0)/(x^2+y^2)$ has zero <curl> away from its excluded axis, but has circulation $2\pi$ around a positively oriented circle enclosing that axis. <Stokes theorem> gives zero total circulation for a smooth annular surface avoiding the axis, whose inner and outer boundary circles have opposite orientations. It cannot be applied directly to a disk meeting the axis, because the field is undefined there.
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