# Paper 3, Section II, C

(a) Let

$\mathbf{F}=(z, x, y)$

and let $C$ be a circle of radius $R$ lying in a plane with unit normal vector $(a, b, c)$. Calculate $\nabla \times \mathbf{F}$ and use this to compute $\oint_{C} \mathbf{F} \cdot d \mathbf{x}$. Explain any orientation conventions which you use.

(b) Let $\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be a smooth vector field such that the matrix with entries $\frac{\partial F_{j}}{\partial x_{i}}$ is symmetric. Prove that $\oint_{C} \mathbf{F} \cdot d \mathbf{x}=0$ for every circle $C \subset \mathbb{R}^{3}$.

(c) Let $\mathbf{F}=\frac{1}{r}(x, y, z)$, where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ and let $C$ be the circle which is the intersection of the sphere $(x-5)^{2}+(y-3)^{2}+(z-2)^{2}=1$ with the plane $3 x-5 y-z=2$. Calculate $\oint_{C} \mathbf{F} \cdot d \mathbf{x}$.

(d) Let $\mathbf{F}$ be the vector field defined, for $x^{2}+y^{2}>0$, by

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

Show that $\nabla \times \mathbf{F}=\mathbf{0}$. Let $C$ be the curve which is the intersection of the cylinder $x^{2}+y^{2}=1$ with the plane $z=x+200$. Calculate $\oint_{C} \mathbf{F} \cdot d \mathbf{x}$.