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## Homework Statement

**F**= y

^{2}z

^{3}

**i**+ 2xyz

^{3}

**j**+3xy

^{2}z

^{2}

**k**

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 x

^{2}+ y

^{2}= 16

## Homework Equations

∫ F (dot) dr = ∫∫ (∇x

**F**) (dot)

**k**dA

## The Attempt at a Solution

z = 2x + 3y ==> z = (8cosθ + 12sinθ)

**r**(θ) = 4cosθ

**i**+ 4sinθ

**j**+ (8cosθ + 12sinθ)

**k**

d

**r**= -4sinθ

**i**+ 4cosθ

**j**+ (-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 ∫∫ (∇x

**F**) (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,

**is correct.**

__0 circulation__Thanks in advance.