91Ó°ÊÓ

The vector field line integral, $$F(x, y) = y i − x j$$, with $$C$$ the circle of radius $$2$$, centered at the origin, and is traversed counterclockwise starting at $$(2, 0)$$.

We have, the circle, $$C$$ of radius $$2$$ is centered at the origin, and traversed counterclockwise starting at $$(2, 0)$$.

$$\implies r(t)= \langle 2cost, 2sint \rangle$$ $$(0\leq t\leq 2\pi)$$

Here, we have $$x=2cost$$ and $$y=2sint$$

Differentiating, we get

$$dr= \langle -2sint, 2cost \rangle dt$$

Evaluating the vector field line integral over the indicated curves, we get

$$\int F(x,y)\cdot dr=\int (yi-xj)\cdot (-2sint,2cost)dt$$

$$\implies \int F(x,y)\cdot dr=\int_{0}^{2\pi }(-4sin^{2}t-4cos^{2}t)dt$$

Solving further, we get

$$\int F(x,y)\cdot dr= -4\int_{0}^{2\pi }dt$$

$$\implies \int F(x,y)\cdot dr= -4(2\pi)$$

$$\implies \int F(x,y)\cdot dr=-8 \pi$$