Past exam of the mathematical methods II course of the University of Oxford / 2013 / 9

Assume the stated continuous derivatives on an open simply connected region. If $N_x-M_y=0$, Green's theorem gives zero circulation around every simple closed curve and its enclosed region; a piecewise smooth closed curve can be decomposed into such loops, giving the same conclusion.

Conversely, suppose every closed-curve integral vanishes. Apply Green's theorem to an arbitrarily small disk about any interior point $P$. Its integral of $N_x-M_y$ is zero. Divide by the disk area and shrink its radius to zero; continuity makes the average approach $(N_x-M_y)(P)$, so that value must vanish. ThusSimple connectivity is needed for the global sufficiency statement, ensuring that a loop does not enclose a missing portion of the domain. Local necessity already follows from small disks.

On the punctured plane $x^2+y^2>0$, write $M=-y/(x^2+y^2)$, $N=x/(x^2+y^2)$. Direct differentiation givesThere is no $z$ dependence or $z$ component, soThe origin is excluded: the field is undefined there. A zero classical curl away from the origin does not state that the field is smooth on an entire disk containing it.

Along a path avoiding the origin, polar coordinates givewhere the angle is followed continuously along the path. For a closed path its total change is an integer number of turns, soFor a simple counterclockwise loop, the value is $2\pi$ if the origin lies inside and zero if it lies outside. Clockwise orientation reverses the nonzero sign. The winding number formula also handles repeated or self-intersecting loops. A curve passing through the origin is outside the ordinary line integral's domain.

There is no contradiction. The domain on which the field is smooth is the punctured plane, which is not simply connected. A loop enclosing the origin cannot be spanned by a planar region entirely inside that domain, so the smoothness hypothesis needed to apply Green's theorem to its full interior fails. A loop that bounds a region avoiding the origin does have zero circulation.

Locally the field is the gradient of the polar angle, but that angle cannot be made a single-valued continuous potential on the whole punctured plane. The nonzero period $2\pi$ around the hole measures this global obstruction, despite the field having zero local curl.