How can I solve the following?
Let $F_1=(-y,x,z)$ and $F_2=(y,x,z)$. Calculate for each force field the work done in moving a particle around the circle in the $(x,y)$ plane. Which of the two force fields is conservative?
I know that the work done by a force $F$ on an object which undergoes an infinitesimal vector displacement $dr$ can be written as $dW = F \cdot dr$, where $dr = i\, dx+j\, dy+k\, dz$. Since the particle is moving around the circle in the $(x,y)$ plane then our integral should be from $0$ to $2\pi$ and we integrate with respect to $\theta$.
You have to use Stoke's theorem here, the line integral represents the work $W(C)$ done in moving a particle around the circle in the counterclockwise direction under the influence of the vector field $F_1$ and $F_2$.
So you need to find
$$\int_{D} \text{curl} \ F\ dS = \int_{C} F \dot \ t ds$$
Where $\text{curl} F = \nabla \times F_1 = \begin{vmatrix}i&j&k\\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial x}\\ F_x&F_y&F_z\end{vmatrix}$ taking the field $F_1$ for example. In particular if $\text{curl} F= 0$ then $W(C) = 0$.