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Math Help - Wedge Product of Differentials in Integration

  1. #1
    Newbie Phantasma's Avatar
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    Wedge Product of Differentials in Integration

    Can someone explain the difference between \iint f(x,y) \, dx\wedge dy and \iint f(x,y) \, dx \, dy?

    In general, is \iint\limits_{D} f(x,y) \, dx \, dy = \iint\limits_{D} f(x,y) \, dy \, dx = \iint\limits_{D} f(x,y) \, dx\wedge dy?

    Does Fubini's Theorem disregard orientation, instead working solely with a measure space without assuming additional structure?

    I've been trying to figure this out, and now I have a headache from trying to read stuff about it because it uses some weird physics prefix notation.

    Any help is greatly appreciated. Thank you.
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  2. #2
    MHF Contributor

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    Re: Wedge Product of Differentials in Integration

    There is no difference except, as you suggest, orientation. \int\int f(x,y)dx\wedge dy is the integral on a surface with the normal oriented in the direction of dx \times dy. \int\int f(x,y) dy\wedge dx is the integral on a surface with the normal oriented in the direction of dy\times dx. In "Calculus III" integration, the orientation is assumed to be given as part of the limits of integration.
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