Potential functions, curl and domain holes

Try

Let φ(x,y)=x²+xy. Its gradient is F=(2x+y,x). Along any smooth path r(t), the chain rule gives dφ(r(t))/dt=F(r(t))·r′(t). Integrating from a starting point A to an ending point B yields φ(B)−φ(A). For A=(0,1) and B=(2,1), the difference is 6. The path can curve or take a detour as long as it stays where the potential is defined.

Predict

Predict the integral around a closed path for this field. The endpoints agree, so the potential difference is zero. This is stronger than observing zero on one circle: the same argument works for every piecewise smooth closed path in the domain. Conversely, a single closed path with nonzero circulation rules out any global potential on that domain. The logical directions deserve care because checking a few loops cannot establish a universal property.

Experiment

Construct a potential for F=(3x²+y, x+2y). Integrate the first component with respect to x to obtain φ=x³+xy+h(y). The added function h(y), not just a numerical constant, represents everything that differentiating in x could have hidden. Differentiating in y gives x+h′(y). Matching the second component gives h′=2y, hence h=y²+C.

Verify both components after construction. A one-component antiderivative is only a candidate until every derivative matches. Then use the potential to compute work without choosing a path, and check one simple path directly as a consistency test.

Understand

A field is conservative on a domain when it is the gradient of a single-valued differentiable potential there. The fundamental theorem of line integrals gives endpoint differences. On a path-connected open domain, path independence is equivalent to zero circulation around every closed piecewise smooth curve and to existence of a potential, with suitable regularity.

In the plane, the scalar curl is Q_x−P_y. A C¹ conservative field has zero curl. A standard sufficient converse requires a simply connected open domain: every loop can be contracted within it. In three dimensions, a C¹ curl-free field on a simply connected open domain also admits a potential. Domain conditions cannot be dropped merely because the derivative formula simplifies to zero.

Apply

Consider F=(−y/(x²+y²),x/(x²+y²)) away from the origin. Direct differentiation gives Q_x=P_y, so its planar curl is zero on its domain. On a circle r=(Rcos t,Rsin t), the dot product F(r)·r′ is 1, yielding circulation 2π. There is no global single-valued potential on the punctured plane.

The origin is not a harmless missing evaluation point: its removal creates a hole around which a loop cannot shrink inside the domain. Locally an angular potential can be chosen on a region with a cut, but moving continuously around the full circle changes the angle by 2π. A local formula need not define one global function.

Continue

Optional depth derives a potential from path independence: fix a base point and define φ(x) as the integral along any path to x. Path independence makes the definition unambiguous; comparing nearby points along a short coordinate segment recovers the field components as partial derivatives. This explains why global path behavior and local differentiation are connected.

Keep a diagnostic checklist tied to a concrete field: identify its domain, compute curl, inspect topology, and either construct a potential or exhibit a nonzero loop. The next theorem relates planar circulation to accumulated curl over a region, making boundary orientation and excluded holes explicit.