Let $\mathbf{F}(\mathbf{x})$ be a vector field defined everywhere on the domain $G \subset \mathbb{R}^{3}$.

(a) Suppose that $\mathbf{F}(\mathbf{x})$ has a potential $\phi(\mathbf{x})$ such that $\mathbf{F}(\mathbf{x})=\nabla \phi(\mathbf{x})$ for $\mathbf{x} \in G$. Show that

$$\int_{\gamma} \mathbf{F} \cdot \mathbf{d} \mathbf{x}=\phi(\mathbf{b})-\phi(\mathbf{a})$$

for any smooth path $\gamma$ from a to $\mathbf{b}$ in $G$. Show further that necessarily $\nabla \times \mathbf{F}=\mathbf{0}$ on $G$.

(b) State a condition for $G$ which ensures that $\nabla \times \mathbf{F}=\mathbf{0}$ implies $\int_{\gamma} \mathbf{F} \cdot \mathbf{d x}$ is pathindependent.

(c) Compute the line integral $\oint_{\gamma} \mathbf{F} \cdot \mathbf{d} \mathbf{x}$ for the vector field

$$\mathbf{F}(\mathbf{x})=\left(\begin{array}{c} \frac{-y}{x^{2}+y^{2}} \\ \frac{x}{x^{2}+y^{2}} \\ 0 \end{array}\right)$$

where $\gamma$ denotes the anti-clockwise path around the unit circle in the $(x, y)$-plane. Compute $\nabla \times \mathbf{F}$ and comment on your result in the light of (b).