# Thread: Finding the Circulation of a force vector around a cone

1. ## Finding the Circulation of a force vector around a cone

I'm tring to find the circulation of F=<x2-y, 4z, x2> around the curve, C which is the plane z = 2 intersects z = Sqrt(x2+y2)
All help appreciated guys!

2. ## Re: Finding the Circulation of a force vector around a cone

Have you at least determined what the curve is?

3. ## Re: Finding the Circulation of a force vector around a cone

$\displaystyle circ = \int [C] F•dr = \int [C] { (x²−y)dx+4zdy+x²dz }$

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