Stokes' Theorem — Question 3

Question 3 – Solution

Strategy. Stokes’ Theorem permits any smooth oriented spanning surface with the same oriented boundary; use the equatorial disk.

Step 1: Boundary orientation The outward normal on the upper hemisphere induces counterclockwise motion around the equator when viewed from $+z$.

Step 2: Choose the disk Let $D$ be $x^2+y^2\le 9$, $z=0$, with upward normal. It has the same oriented boundary as the hemisphere. Since $$\nabla\times\mathbf F=\langle 0,0,2\rangle,$$ Stokes’ Theorem gives $$\boxed{\oint_C\mathbf F\cdot d\mathbf r =\iint_D2\,dA=2(9\pi)=18\pi}.$$

Step 3: Explain the replacement Both the hemisphere and disk are orientable, share $C$, and lie where $\mathbf F$ has continuous derivatives. Each curl flux therefore equals the same boundary circulation.

Verification On the equator, $\mathbf F=\langle-y,x,0\rangle$ is tangent in the counterclockwise direction and has magnitude $3$; multiplying by the circumference $6\pi$ also gives $18\pi$.