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
∫ 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.