This implies that the circulation around the ellipse ##C## is also zero.

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In summary: S_1## and bottom ##S_3## surfaces are equal and opposite. Thus, the circulation of ##\vec F## around the ellipse ##C## is also zero.In summary, the circulation of the vector field F in the clockwise direction around the ellipse C is 0, as shown by the application of Stokes' theorem and the fact that the curl of F is 0, indicating that the field is conservative. This can also be seen by considering the net flux through the surface created by the intersection of the plane and cylinder, which is equal to 0.
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Homework Statement



F = y2z3i + 2xyz3j +3xy2z2k
Find the circulation of F in the clockwise direction as seen from above, around the ellipse C in which the plane
2x + 3y - z = 0 meets the cylinder x2 + y2 = 16

Homework Equations



∫ F (dot) dr = ∫∫ (∇xF) (dot) k dA

The Attempt at a Solution



z = 2x + 3y ==> z = (8cosθ + 12sinθ)
r(θ) = 4cosθi + 4sinθj + (8cosθ + 12sinθ)k
dr = -4sinθi + 4cosθj + (-8sinθ + 12cosθ)k

I'm thinking of substituting the values from r(θ) to F, but that would appear to be too much work due to the third powers and having to multiply the results together after substituting.

However, the other equation ∫∫ (∇xF) (dot) k dA appears to be easier. I computed the curl of F and got 0, which is my answer at this time. Instead of having to substitute the values of r(θ) to F (which would be a pain), the curl of F shows that the circulation is 0.

I need confirmation whether this is true and the answer, 0 circulation is correct.

Thanks in advance.
 
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  • #2
##\vec F## is indeed conservative since ##\text{curl}(\vec F) = 0##.

The line integral of a conservative vector field around any closed path (within the domain) is zero.
 
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Another cool way to see this is to use Stokes' theorem.

$$\oint_C \vec F \cdot d \vec r = \iint_S \text{curl}(\vec F) \cdot d \vec S$$

The intersection of the plane with the cylinder creates a piecewise smooth surface ##S## comprised of a slanted top surface ##S_1##, a cylindrical shell ##S_2##, and a circular bottom region ##S_3## bounded by ##z = 0##. The net flux out of this surface is zero:

$$\oint_C \vec F \cdot d \vec r = \iint_S \text{curl}(\vec F) \cdot d \vec S = \iint_{S_1} \text{curl}(\vec F) \cdot d \vec S_1 + \iint_{S_2} \text{curl}(\vec F) \cdot d \vec S_2 + \iint_{S_3} \text{curl}(\vec F) \cdot d \vec S_3 = 0$$
 

Related to This implies that the circulation around the ellipse ##C## is also zero.

What is clockwise circulation of F?

Clockwise circulation of F refers to the direction in which a vector field F is rotating in a closed loop. It is a measure of the total force acting along the path of the loop, in a direction that is perpendicular to the loop itself.

How is clockwise circulation of F calculated?

The clockwise circulation of F can be calculated using the line integral of the vector field F around the closed loop. This involves taking the dot product of the vector field F and the tangent vector of the loop at each point along the path, and then integrating over the entire loop.

What does a positive clockwise circulation of F indicate?

A positive clockwise circulation of F indicates that the vector field is rotating in a clockwise direction around the loop, meaning that the force acting along the loop is in the same direction as the loop's tangent vector at each point.

What does a negative clockwise circulation of F indicate?

A negative clockwise circulation of F indicates that the vector field is rotating in a counterclockwise direction around the loop, meaning that the force acting along the loop is in the opposite direction of the loop's tangent vector at each point.

What is the significance of clockwise circulation of F?

The clockwise circulation of F is an important concept in fluid dynamics and electromagnetism, as it helps to understand the behavior of fluids and electric/magnetic fields. It can also be used to calculate work done by a force along a closed loop, and to determine the direction of rotation of a rigid body in motion.

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