Azimuthal inverse-radius vector field ($\mathbf{G}={\mathbf{e}}_{\varphi }/\rho$)

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.