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.(adsbygoogle = window.adsbygoogle || []).push({});

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...

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# Conceptual question: Green's Theorem and Line Integrals

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