Stokes' Theorem — Question 5

Question 5

Let $S$ be the annulus $1\le x^2+y^2\le 4$ in the plane $z=0$, oriented upward. Its boundary consists of an outer circle $C_o$ and inner circle $C_i$. For $$\mathbf F=\left\langle-\frac y2,\frac x2,0\right\rangle,$$ evaluate the circulation around the fully oriented boundary $\partial S$.

Tasks

1. Determine the induced direction on each boundary component.

2. Apply Stokes’ Theorem to the annulus.

3. Verify the result from the two separate circle integrals.

Question 5 – Solution

Strategy. For an upward annulus, the outer boundary is counterclockwise and the inner boundary is clockwise.

Step 1: Orient the boundary The right-hand rule gives $$\partial S=C_o-C_i^{\mathrm{CCW}}.$$ Equivalently, traverse the outer circle counterclockwise and the inner circle clockwise.

Step 2: Apply Stokes Since $\nabla\times\mathbf F=\mathbf k$, $$\boxed{\oint_{\partial S}\mathbf F\cdot d\mathbf r =\iint_S1\,dA=\pi(2^2-1^2)=3\pi}.$$

Step 3: Check components For a counterclockwise circle of radius $R$, $$\oint\mathbf F\cdot d\mathbf r=\pi R^2.$$ Thus the outer circle contributes $4\pi$, while the clockwise inner circle contributes $-\pi$. Their sum is $3\pi$.

Verification Assigning both circles counterclockwise would give $5\pi$ and violate the induced-boundary convention.