Green's Theorem in 3 Dimensions for non-conservative field

[Tom31415926535]

Homework Statement

C is the directed curve forming the triangle (0, 0, 0) to (0, 1, 1) to (1, 1, 1) to (0, 0, 0).

Let F=(x,xy,xz) Find ∫F·ds.

Homework Equations

The Attempt at a Solution

My intial instinct was to check if it was conservative. Upon calculating:

∇xF=(0,-z,y)

I concluded that it isn't conservative. How do I apply Green's theorem for a three-dimensional vector field that is non-conservative?

 

[Orodruin]

What you are after seems to be the curl theorem (also called Stokes’ theorem). It relates the circulation integral of a field with the integral over a surface of the curl of the field. Green’s identities relate surface integrals to volume integrals based on the divergence theorem.

That being said, I see no need to apply any integral theorem here. The three line integrals involved should be rather straightforward.

 

[wrobel]

The integral can easily be calculated directly