How do you compute the circulation of this fluid (path integrals)

[randomcat]

Homework Statement

A fluid as velocity field F(x, y, z) = (xy, yz, xz). Let C denote the unit circle in the xy-plane. Compute the circulation, and interpret your answer.

Homework Equations

The Attempt at a Solution

Since the unit circle is a closed loop, I assumed that ∫ F * dr = 0
(the ∫ symbol is supposed to have a circle)
However, when I attempt to verify whether or not F = \nabla f, the two are not equal, suggesting that ∫ F * dr does not= 0
At this point I'm at a loss of how to go about calculating the circulation. Could someone please help?

 

[Dick]

Homework Statement

A fluid as velocity field F(x, y, z) = (xy, yz, xz). Let C denote the unit circle in the xy-plane. Compute the circulation, and interpret your answer.

Homework Equations

The Attempt at a Solution

Since the unit circle is a closed loop, I assumed that ∫ F * dr = 0
(the ∫ symbol is supposed to have a circle)
However, when I attempt to verify whether or not F = \nabla f, the two are not equal, suggesting that ∫ F * dr does not= 0
At this point I'm at a loss of how to go about calculating the circulation. Could someone please help?

You just want to calculate the integral. Pick a parametrization of the unit circle, like r(t)=(cos(t),sin(t),0) and work it out.

 

[hapefish]

However, when I attempt to verify whether or not F = \nabla f, the two are not equal, suggesting that ∫ F * dr does not= 0
At this point I'm at a loss of how to go about calculating the circulation. Could someone please help?

I would rephrase this to "suggesting that ∫ F * dr might not= 0" Note that IF the vector field is conservative then every closed path integral must be zero; however, the opposite is not necessarily true. There are many cases when the field is not conservative but a given closed path integral is still zero (and I suspect that this problem will end up being one of those cases).

Good Luck!