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

[Differentiate1]

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.

 

[STEMucator]

$\vec F$ 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.

 

[STEMucator]

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:

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