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

## 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.

## 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?

## Answers and Replies

Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
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
Science Advisor
The integral can easily be calculated directly