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Math Help - Calculus and discrete mathematics

  1. #1
    MHF Contributor Bruno J.'s Avatar
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    Calculus and discrete mathematics

    I'm interested to see examples of discrete versions of the major theorems of Calculus.

    Many of the operations of calculus which can be carried out on a function of a real variable have their analogues for sequences of numbers; forward difference, summation by parts, etc.

    If we think of the integer number line as an oriented, 1-singular complex, we see immediately that many of these operations have have their counterpart on any oriented 1-singular complex. Just take an oriented graph, for example; a real-valued function on its vertices can be thought of as a discrete potential function. Starting with such a potential function, we can assign to each edge of the graph the forward difference of the values corresponding to its vertices, we obtain a sort of "gradient" on the graph. "Integrating" from a point to another (i.e summing up the values of the gradient along a path) yields the potential of the last vertex minus the potential of the first one. The value of the integral is also independent of path. So we can recover, up to an additive constant, the potential function.

    Now the most interesting theorems of Calculus are not on the line, but in n-space. How could these theorems be generalized to arbitrary n-simplices? What would Gauss' theorem, or Stokes' theorem, look like? What kind of applications, if any, would discrete formulations of these theorems yield?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Bruno J. View Post
    I'm interested to see examples of discrete versions of the major theorems of Calculus.

    Many of the operations of calculus which can be carried out on a function of a real variable have their analogues for sequences of numbers; forward difference, summation by parts, etc.

    If we think of the integer number line as an oriented, 1-singular complex, we see immediately that many of these operations have have their counterpart on any oriented 1-singular complex. Just take an oriented graph, for example; a real-valued function on its vertices can be thought of as a discrete potential function. Starting with such a potential function, we can assign to each edge of the graph the forward difference of the values corresponding to its vertices, we obtain a sort of "gradient" on the graph. "Integrating" from a point to another (i.e summing up the values of the gradient along a path) yields the potential of the last vertex minus the potential of the first one. The value of the integral is also independent of path. So we can recover, up to an additive constant, the potential function.

    Now the most interesting theorems of Calculus are not on the line, but in n-space. How could these theorems be generalized to arbitrary n-simplices? What would Gauss' theorem, or Stokes' theorem, look like? What kind of applications, if any, would discrete formulations of these theorems yield?
    Do you mean something like this?
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    Awesome! Yes!
    Thanks!
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Bruno J. View Post
    Awesome! Yes!
    Thanks!
    In fact, there is an entire branch of mat known as discrete differential geometry.
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    There is also the discrete Laplacian, and the 'Dirichlet problem' one can define for it, which is closely related to random walks.
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  6. #6
    Senior Member roninpro's Avatar
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    Re: Calculus and discrete mathematics

    I'm a little late to the party, but a professor at my school sent me this blog post: Claiming Picard’s Math May Have Gaps « Gödel’s Lost Letter and P=NP. It shows a version of Picard's theorem for finite fields.
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