Line Integrals of Vector Fields — Question 3

Question 3

Let $C$ travel from $A=(0,0)$ to $B=(2,0)$ and then from $B$ to $C=(2,1)$. For $$\mathbf F(x,y)=\langle-y,x\rangle,$$ evaluate $\displaystyle\int_C\mathbf F\cdot d\mathbf r$.

Tasks

1. Parametrize each segment with the stated orientation.

2. Compute the contribution from each segment.

3. Explain geometrically why the first contribution vanishes.

Question 3 – Solution

Strategy. Split the piecewise path and add the two oriented line integrals.

Step 1: Horizontal segment Use $\mathbf r_1(t)=\langle t,0\rangle$, $0\le t\le 2$. Then $$\mathbf F(\mathbf r_1(t))=\langle 0,t\rangle,\qquad \mathbf r_1'(t)=\langle 1,0\rangle,\qquad \mathbf F\cdot\mathbf r_1'=0.$$

The field is vertical along this segment while the motion is horizontal, so the vectors are perpendicular.

Step 2: Vertical segment Use $\mathbf r_2(s)=\langle 2,s\rangle$, $0\le s\le 1$. Thus $$\mathbf F(\mathbf r_2(s))=\langle-s,2\rangle,\qquad \mathbf r_2'(s)=\langle 0,1\rangle,$$ and $$\int_{C_2}\mathbf F\cdot d\mathbf r=\int_0^1 2\,ds=2.$$

Step 3: Add Hence $$\boxed{\int_C\mathbf F\cdot d\mathbf r=0+2=2}.$$

Verification Only the upward component on the vertical segment contributes, and that component is the constant $2$ over a distance of $1$.