Chapter 16.3, Problem 35E

Let F(x, y) = $\frac{-yi+xj}{x^2+y^2}$

(a) Show that $\partial P\text{/}\partial y=\partial Q\text{/}\partial x$.

(b) Show that ∫_C F · dr is not independent of path. [Hint: Compute ∫ $C_1$ F · dr and ∫ $C_2$ F · dr, where C₁ and C₂ are the upper and lower halves of the circle x² + y² = 1 from (1, 0) to (‒1, 0).] Does this contradict Theorem 6?