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Changing the direction of an integral

  1. Feb 16, 2009 #1
    Very simple question: Suppose I have a region, D, defined by an outer boundary [tex]\partial D[/tex], but with a hole (like an annulus, for example), defined by [tex]\partial D_2[/tex].

    If I were to apply Green's Theorem to this region, coupled with a vector field, it would give me something like,

    [tex]\text{Area Integral} = \int_{\partial D} + \int_{\partial D_2}[/tex]

    where by the conditions of the theorem, the outer contour should go counterclockwise and the inner contour should go clockwise.

    My question is, can we do this?

    [tex]\text{Area Integral} = \int_{\partial D} - \int_{\partial D_3}[/tex]

    where now [tex]\partial D_3[/tex] is the same contour as [tex]\partial D_2[/tex] but going clockwise.
  2. jcsd
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