I'm tring to find the circulation of F=<x^{2}-y, 4z, x^{2}> around the curve, C which is the plane z = 2 intersects z = Sqrt(x^{2}+y^{2})
All help appreciated guys!
$\displaystyle circ = \int [C] F•dr = \int [C] { (x²−y)dx+4zdy+x²dz }$
Simplify integral first if you can to reduce work
$\displaystyle \int [C] 4zdy = 8 \int [C] dy = 0$ since C is closed
$\displaystyle \int [C] x²dz = 0 $
$\displaystyle \int [C] x²dx = 0$
$\displaystyle \int [C] F•dr = \int [C] −ydx$
To find this set $\displaystyle x=2cosθ, y=2sinθ, 0≤θ≤2π$ (ac/w from above)
$\displaystyle
\int [C] −ydx = \int (−2sinθ)(−2sinθdθ), [θ=0,2π]$
$\displaystyle = \int 4sin²θdθ, [θ=0,2π] = \int 2(1−cos2θ)dθ, [θ=0,2π] = 4π$
Alternatively, $\displaystyle \int [C] –ydx $=area within C by Green’s Theorem = π2²