Conceptual question: Green's Theorem and Line Integrals

In summary, Green's Theorem states that the line integral over a closed curve C is equal to the double integral of the vector field's partial derivatives over the region D bounded by C. However, this only applies to non-conservative vector fields. Line integrals over closed curves are not necessarily equal to 0, unless the vector field is conservative. In order to be considered conservative, the integral over any closed path must be 0. Therefore, if Green's Theorem is applied and yields a 0 answer, it does not necessarily mean the vector field is conservative. It may still be conservative if the integral over any other closed path is also 0.
  • #1
JHans
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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...
 
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  • #2
Line integrals over closed curves are necessarily equal to 0 when the vector field you're integrating is a conservative vector field.
 
  • #3
Meaning that, if I use Green's Theorem and get 0, then the vector field is conservative?
 
  • #4
Not necessarily. In order to be "conservative" (I would say a "total derivative") the integral over any closed path would have to be 0.
 
  • #5
JHans said:
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.
 

1. What is Green's Theorem?

Green's Theorem is a mathematical concept that relates the line integral of a two-dimensional vector field over a closed curve to the double integral of the same vector field over the region enclosed by the curve.

2. What is the significance of Green's Theorem?

Green's Theorem is significant because it allows us to calculate a line integral, which may be difficult to evaluate directly, by converting it into a double integral, which is often easier to solve.

3. How is Green's Theorem related to the concept of flux?

Green's Theorem is closely related to the concept of flux, as it can be used to calculate the net flow of a vector field through a closed curve. In this sense, Green's Theorem can be thought of as a two-dimensional version of the Divergence Theorem.

4. Can Green's Theorem be applied to three-dimensional vector fields?

No, Green's Theorem can only be applied to two-dimensional vector fields. For three-dimensional vector fields, the equivalent theorem is called the Stokes' Theorem.

5. How can Green's Theorem be used in real-world applications?

Green's Theorem has many practical applications, such as in fluid mechanics, electromagnetism, and engineering. It can be used to calculate the work done by a force field, the flow of a fluid, or the electric field around a charged object.

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