Use Stokes's Theorem to evaluate the integral of (-y+z)dz+z(dy)+4x(dz) over C

Let C be the intersection of the cylinder x^2+y^2 = 9 and the plane x+y+z = 0, oriented clockwise as viewed from high above the xy-plane.

Use Stokes's Theorem to evaluate the integral of (-y+z)dz+z(dy)+4x(dz) over C.

I think you might have meant the first differential to be dx--otherwise, why write it with the k component split like that.

Stokes theorem says that (provided F is sufficiently smooth and C bounds the surface S)

∫ F∙dr = ∫∫ ∇ x F∙n dS
C . . . . .S

The curl of F=<-y+z, z, 4x> (again assuming you meant (-y + z) dx) is

∇ x F = <-1, -3, 1>.

We can use any surface that is bounded by C, so I'd just go with the plane

It is kind of weird that the contour is oriented clockwise when viewed from above because the right hand rule indicates that the normal to the plane region (x + y + z = 0) inside the cylinder will have a negative k component. Oh well. Letting g(x,y,z) = x + y + z = 0 be this surface, the normal is

n = -∇g/||∇g|| = <-1, -1, -1>/√(3).

And dS = ||∇g|| dA = √(3) dA where dA is the differential area element in the xy-plane. I'd use polar coordinates:

0 ≤ r ≤ 3, 0 ≤ Θ ≤ 2π, dA = r dr dΘ

2π .3
∫ . . ∫ 3 r dr dΘ = 27π.
0 . 0