Green's Theorem — Question 3

Question 3

The planar field $$\mathbf F(x,y)=\langle x^2,xy\rangle$$ acts on the disk $D:(x-1)^2+y^2\le 1$. Let $C=\partial D$ be counterclockwise. Find the outward flux of $\mathbf F$ across $C$.

Tasks

1. State the flux form of Green’s Theorem.

2. Compute the divergence and its integral over $D$.

3. Use symmetry or the centroid of the disk to finish.

Question 3 – Solution

Strategy. Convert outward flux to the double integral of the planar divergence.

Step 1: Flux form For $\mathbf F=\langle M,N\rangle$, $$\oint_C\mathbf F\cdot\mathbf n\,ds =\oint_C M\,dy-N\,dx =\iint_D(M_x+N_y)\,dA.$$ Here $$M_x+N_y=2x+x=3x.$$

Step 2: Integrate The disk has area $\pi$ and centroid $(1,0)$. Therefore its average $x$-coordinate is $1$, so $$\iint_Dx\,dA=1\cdot\pi=\pi.$$ Consequently, $$\boxed{\oint_C\mathbf F\cdot\mathbf n\,ds =\iint_D3x\,dA=3\pi}.$$

Verification The disk lies in $0\le x\le 2$, so the divergence $3x$ is nonnegative. A positive outward flux is therefore consistent with the sign of the integrand.