Line Integrals of Vector Fields — Question 8

Question 8

Let $C$ be the counterclockwise boundary of the rectangle $0\le x\le 2$, $0\le y\le 1$. For $$\mathbf F(x,y)=\langle y^2,x\rangle,$$ compute the circulation $\displaystyle\oint_C\mathbf F\cdot d\mathbf r$ directly from the four edges.

Tasks

1. Compute each oriented edge contribution.

2. Add the four contributions.

3. Check which signs are forced by the traversal direction.

Question 8 – Solution

Strategy. Write $\mathbf F\cdot d\mathbf r=y^2dx+xdy$ and exploit the constant coordinate on each edge.

Step 1: Bottom and right Along the bottom, $y=0$ and $dy=0$, so the contribution is $0$. Along the right edge, $x=2$ and $y:0\to 1$, giving $$\int_0^1 2\,dy=2.$$

Step 2: Top and left Along the top, $y=1$ and counterclockwise travel has $x:2\to 0$: $$\int_2^0 1\,dx=-2.$$ On the left, $x=0$, and $dx=0$, so the contribution is $0$.

Step 3: Sum $$\boxed{\oint_C\mathbf F\cdot d\mathbf r=0+2-2+0=0}.$$

Verification The right-edge contribution is positive because both the field’s vertical component and $dy$ are positive. The top contribution is negative because $y^2>0$ while $dx<0$; their equal magnitudes cancel.