Consider the vector field

$$\mathbf{F}=\left(-y /\left(x^{2}+y^{2}\right), x /\left(x^{2}+y^{2}\right), 0\right)$$

defined on all of $\mathbb{R}^{3}$ except the $z$ axis. Compute $\boldsymbol{\nabla} \times \mathbf{F}$ on the region where it is defined.

Let $\gamma_{1}$ be the closed curve defined by the circle in the $x y$-plane with centre $(2,2,0)$ and radius 1 , and $\gamma_{2}$ be the closed curve defined by the circle in the $x y$-plane with centre $(0,0,0)$ and radius 1 .

By using your earlier result, or otherwise, evaluate the line integral $\oint_{\gamma_{1}} \mathbf{F} \cdot \mathrm{d} \mathbf{x}$.

By explicit computation, evaluate the line integral $\oint_{\gamma_{2}} \mathbf{F} \cdot \mathrm{d} \mathbf{x}$. Is your result consistent with Stokes' theorem? Explain your answer briefly.