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.

I = - 1/2

Step-by-step 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