Chapter 17.1, Problem 30E

To determine

(a)

To show:

$\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}=0$, where $F(x,y)=\langle \frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\rangle$.

To determine

(b)

To explain:

the reason why Green’s theorem can’t be used to argue that ${\displaystyle \underset{C_R}{\int }F\cdot dr}=0$.

To determine

(c)

To prove:

${\displaystyle \underset{C_R}{\int }F\cdot dr}=0$ by direct computation of the line integral.

To determine

(d)

To explain:

the reason why we can use the Green’s theorem for concluding that ${\displaystyle \underset{C}{\int }F\cdot dr}=0$.