Multivariable CalculusSuperchallengeVector Calculus

Vector Calculus · Superchallenge problem

Let $\mathbf F = \left(\dfrac{-y}{x^2+y^2},\; \dfrac{x}{x^2+y^2},\; 0\right)$ and let $C$ be the closed curve

\mathbf r(t) = \big((2 + \cos 5t)\cos 2t,\; (2 + \cos 5t)\sin 2t,\; \sin 5t\big),\quad 0\le t\le 2\pi.

Evaluate $\displaystyle\oint_C\mathbf F\cdot d\mathbf r$ . Give your answer in terms of $\pi$ .