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I am trying to understand how circulation works as a line integral with the curl in green's theorem. I know that a line integral describes the relationship between a vector field and a path, i.e. how much a path aligns or is going in the same direction as a vector field. This example here from this website show a loop oriented counter-clockwise. The field is oriented clockwise so it's very intuitive to assume that the line integral will be negative:

image of closed curve (circle) circulating counter clockwise in clockwise vector field

[Nykamp DQ, “Line integrals as circulation.” From Math Insight. http://mathinsight.org/line_integral_circulation] [image cited]

This also matches green's theorem when we try to solve it using the curl of the field as the field is <y,-x>. For a curve oriented clockwise however, green's theorem gives us the same answer, since it's just the integral of the curl of the vector field over the region. Why is that? I'm convinced that for a clockwise curve, the line integral should be positive since it will be going in the same direction as the field. Why doesn't green's theorem agree? Why is it still the same (negtaive)? (I understand that, using the right hand rule, the circulation of the field points towards negative k here, but doesn't that just mean the path doesn't matter? How does this work as a line integral then?)

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The double integral in Green's Theorem is only equal to the line integral over the positively oriented boundary. That is the statement of the theorem. The clockwise orientation is not the positively oriented boundary of the interior of the circle and hence the double integral over the interior does not equal the clockwise circulation.

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  • $\begingroup$ Thank you! For the clockwise circulation, would green's theorem work if we multiply our answer by a negative then? $\endgroup$
    – Elena
    Commented May 15 at 1:24
  • $\begingroup$ Yes because reversing the orientation negates the line integral $\endgroup$
    – whpowell96
    Commented May 15 at 1:29

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