Stokes' Theorem — Question 6

Question 6 – Solution

Strategy. Encode planar area as the curl flux of a field whose vertical curl is $1$.

Step 1: Choose the field Let $$\mathbf F=\left\langle-\frac y2,\frac x2,0\right\rangle.$$ Then $\nabla\times\mathbf F=\mathbf k$. If $D$ is the region enclosed by positively oriented $C$, Stokes gives $$\operatorname{area}(D)=\iint_D1\,dA =\oint_C\mathbf F\cdot d\mathbf r =\boxed{\frac 12\oint_C(x\,dy-y\,dx)}.$$

Step 2: Parametrize the ellipse Use $$x=a\cos t,\qquad y=b\sin t, \qquad 0\le t\le 2\pi.$$ Then $$x\,dy-y\,dx =ab(\cos^2t+\sin^2t)\,dt=ab\,dt.$$ Therefore $$\boxed{\operatorname{area}(D)=\frac 12\int_0^{2\pi}ab\,dt=\pi ab}.$$

Verification Setting $a=b=R$ recovers the circle area $\pi R^2$, including the correct scale and orientation sign.