Conceptual question: Green's Theorem and Line Integrals

[JHans]

Alright, I have a conceptual question regarding Green's Theorem that I'm hoping someone here can explain. We recently learned in my college class that, by Green's Theorem, if C is a positively-oriented, piecewise-smooth, simple closed curve in the plane and D is the region bounded by C, then the line integral over the curve is equal to the double integral of the vector field's partial derivatives over the region D. Sorry I can't put that in mathematical notation, but I hope those of you familiar with Green's Theorem understand what I'm saying.

My question, though, is that aren't line integrals over closed curves equal to 0? Why, then, do these applications of Green's Theorem yield numerical answers other than 0? If I understand it correctly, only line integral of conservative vector fields over closed curves equal 0. Does this mean that, if I apply Green's Theorem and get 0 as an answer, the vector field is conservative?

I hope someone can elaborate on this a little bit. I find vector calculus in general to be a little confusing...

 

[nicksauce]

Line integrals over closed curves are necessarily equal to 0 when the vector field you're integrating is a conservative vector field.

 

[JHans]

Meaning that, if I use Green's Theorem and get 0, then the vector field is conservative?

 

[HallsofIvy]

Not necessarily. In order to be "conservative" (I would say a "total derivative") the integral over any closed path would have to be 0.

 

[nicksauce]

Meaning that, if I use Green's Theorem and get 0, then the vector field is conservative?

Conservative defining a vector field as F(x,y) = 0 when x,y <= 1, and something else otherwise. Then if you integrate over a path that that is only defined in for x,y <= 1, you'll get 0, but you could still get a non-zero answer if you integrate over some other path.