Chapter 18.1, Problem 30E

(a)

To determine

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

(b)

To determine

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

(c)

To determine

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

(d)

To determine

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