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Thread: Calculus over the reals

  1. #1 Calculus over the reals 
    Moderator Moderator AlexP's Avatar
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    What allows us to do calculus over ? I believe I've read the phrase 'complete, ordered field' with respect to it. I know completeness is necessary, and that there are other ways to complete the rationals to make calculus possible (p-adic numbers). I can see how the order and field properties are needed too. Must in this case also be a metric space in order to make definitions meaningful?

    Basically, exactly what type of structure is when we do calculus over it?


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  3. #2  
    Forum Ph.D.
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    http://en.wikipedia.org/wiki/Haar_measure

    Look up the above - it may be the answer to your question.


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  4. #3 Re: Calculus over the reals 
    . DrRocket's Avatar
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    Quote Originally Posted by AlexP
    What allows us to do calculus over ? I believe I've read the phrase 'complete, ordered field' with respect to it. I know completeness is necessary, and that there are other ways to complete the rationals to make calculus possible (p-adic numbers). I can see how the order and field properties are needed too. Must in this case also be a metric space in order to make definitions meaningful?

    Basically, exactly what type of structure is when we do calculus over it?
    The real numbers are the only complete Archimedian field.

    The formalism requires a metric space, but there are more general ways to handle limits, so that aspect is more incidental than essential. However, calculus is based on the metric associated with the absolute value norm, and that is important.

    You can do polynomial calculus with very little, but to do what most people think of as calculus requires completeness in the usual metric

    The real numbers are the basis on which other analytic structures, and manifolds, are built.

    I noticed a link to Haar measure in another post. Lebesque measure is a Haar measure on the real numbers, but the importance of Haar measure lies in the application to arbitrary locally compact abelian groups (or in the case of left and right invariant measures to non-abelian locally compact groups) and abstract harmonic analysis. Until you study convolution algebras on groups, you can safely ignore Haar measure issues.
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