Question

Use Stokes’ Theorem to find Z C xydx − yzdz, where C is the boundary curve of {(x, y, z): x + y + z = 1, x, y, z ≥ 0}, oriented clockwise as seen from the origin.

Answer 1

I = -1/2

Explanation:

We apply the equation

I = ∫C (F). dS = ∫∫S curl F. dS

curl (F) = ∇×F = ∇ × (xy, 0, -yz) = (−z, 0 , −x)

Calculate the normal vector n = (1, 1, 1) of the plane

then

I = ∫∫S (−z, 0 , −x).(1, 1, 1) dS = ∫∫S (-z-x) dS = ∫∫S (-(1-(x+y))-x) dS

I = ∫∫S (-1+y) dS

if dS = dxdy

and 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 we have

I = -1/2