Stokes' Theorem — Question 1

Question 1 – Solution

Strategy. Span the circle by its flat disk; the prescribed counterclockwise direction selects the upward normal.

Step 1: Curl and orientation $$\nabla\times\mathbf F =\begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\ \partial_x&\partial_y&\partial_z\\-y&x&z\end{vmatrix} =\langle 0,0,2\rangle.$$ Let $S$ be the unit disk with $\mathbf n=\mathbf k$.

Step 2: Apply Stokes $$\boxed{\oint_C\mathbf F\cdot d\mathbf r =\iint_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS =\iint_S2\,dA=2\pi}.$$

Step 3: Direct check With $\mathbf r(t)=\langle\cos t,\sin t,0\rangle$, $$\mathbf F(\mathbf r(t))=\langle-\sin t,\cos t,0\rangle =\mathbf r'(t).$$ Therefore $\int_0^{2\pi}\mathbf F\cdot\mathbf r'\,dt=\int_0^{2\pi}1\,dt=2\pi$.

Verification The boundary is counterclockwise from the tip of $\mathbf k$, so the orientation pairing is correct.