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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 $\theta$.

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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 $W(C) = 0$.

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    $\begingroup$ Would the curl be $ \nabla \times (y,x,z)\\ = \left| \begin{array}{ccc} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ y & x &z\end{array} \right| ?$ $\endgroup$ – Robben Oct 2 '14 at 23:18
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    $\begingroup$ Thank you. I will attempt it now using your method. $\endgroup$ – Robben Oct 2 '14 at 23:22
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    $\begingroup$ Check this link here, to get an intuitive perspective over this. $\endgroup$ – Aaron Maroja Oct 2 '14 at 23:23
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    $\begingroup$ Aaron, I got the curl to be $2k$ do I now integrate from $0$ to $2\pi$? And is $dS$ the surface which is the circle? $\endgroup$ – Robben Oct 2 '14 at 23:46
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    $\begingroup$ You need to parametrize $x=r\ \text{cos} \theta, y = r\ \text{sin} \ \theta, z=0$. Then do the dot product with the $\text{curl} F$. $\endgroup$ – Aaron Maroja Oct 3 '14 at 2:11

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