Stokes' Theorem — Question 10

Question 10 – Solution

Strategy. The apparent contradiction “zero curl but nonzero circulation” signals a failed hypothesis: the field is singular on every full disk spanning $C$.

Step 1: Curl and direct integral Differentiation gives $$\nabla\times\mathbf F=\mathbf 0\qquad(x^2+y^2>0).$$ On $C$, let $\mathbf r(t)=\langle\cos t,\sin t,0\rangle$. Then $$\mathbf F(\mathbf r(t))=\langle-\sin t,\cos t,0\rangle=\mathbf r'(t),$$ so $$\boxed{\oint_C\mathbf F\cdot d\mathbf r=\int_0^{2\pi}1\,dt=2\pi}.$$

Step 2: Locate the failed hypothesis The field is undefined at the origin, which lies on the full spanning disk. Thus $\mathbf F$ does not have continuous derivatives on an open set containing that disk, and Stokes’ Theorem cannot be applied there.

Step 3: Use an annulus Remove a radius-$\varepsilon$ disk. On the resulting annulus the curl is zero, so Stokes gives $$0=\oint_{C_{\mathrm{outer}}}\mathbf F\cdot d\mathbf r +\oint_{C_{\mathrm{inner,clockwise}}}\mathbf F\cdot d\mathbf r =2\pi-2\pi.$$

Verification The inner boundary carries exactly the circulation needed to account for the puncture; no contradiction remains.