Chapter 15.3, Problem 36ES

Use the result in Exercise 34(b).

Let $\text{F}\left(x,y\right)=\frac{y}{x^2+y^2}\text{i}-\frac{x}{x^2+y^2}\text{j}.$

(a) Show that ${\displaystyle {\int }_{C_1}\text{F}\cdot d\text{r}\ne {\displaystyle {\int }_{C_2}\text{F}\cdot d\text{r}}}$ if $C_1\text{and}{C}_2$ are the semicircular paths from $\left(1,0\right)\text{to}\left(-1,0\right)$ given by $\begin{array}{l}{C}_1:x=\cos t,y=\sin t\left(0\le t\le \pi \right)\\ C_2:x=\cos t,y=-\sin t\left(0\le t\le \pi \right)\end{array}$

(b) Show that the components of F satisfy Formula (9).

(c) Do the result in parts (a) and (b) contradict Theorem 15.3.3? Example.