Results 1 to 5 of 5

Math Help - Integral Theory, Fubini's Theorem

  1. #1
    Member
    Joined
    Aug 2011
    Posts
    84

    Integral Theory, Fubini's Theorem

    regarding the theoretical definition of double integrals, is it correct to think of them as a kind of double Riemann sums, sums of 2d planes? sometimes the order of integration results in simplification of the calculuation....
    once you have mastered Lebegue Integration and measure theory is Fubini's theorem and Riemann integration not useful...

    I(limit a to b) I(limit c to d) f(x,y)dydx= I (limit c to d) I (limit a to b) f(x,y)dxdy

    thanks very much!
    Last edited by mathnerd15; March 19th 2013 at 06:50 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,441
    Thanks
    1862

    Re: Integral Theory, Fubini's Theorem

    I'm not sure I like the phrase "sums of 2D planes"- you can't add planes, they aren't numbers! You can think of \int_c^d\int_a^b f(x,y)dxdy as the volume under the surface z= f(x, y) and the rectangle in the xy-plane, a< x< b, c< y< d. Using Riemann sums, we divide that region into tall thin rectangular solids, having base \Delta x\Delta y and height f(x^*, y^*) where (x^*, y^*) is some point inside one of the small rectangles. Yes, because, at each step, we have only a finite number of such rectangles, it does not matter whether we add the volume "horizontally first, then vertically" or "vertically first, then horizontally". That is the "meat" or Fubini's theorem. All the "hard" work is hidden in the limit process! And we know that works precisely because the function is "integrable".
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2011
    Posts
    84

    Re: Integral Theory, Fubini's Theorem

    thanks very much! yes it is a double Riemann sum with the number of rectangles becoming infinitely small as n goes to infinity
    so for the surface integral are you approximating a surface with a polyhedron with n faces, but what is the meaning of the integrand F(x,y,z) dot n vector delta S? wikipedia describes a parameterization of the surface by applying a new coordinate system on the surface. the dot product is the component of n along F? I'm looking at Schey's Div, Grad, Curl and Griffith's Electrodynamics
    Surface integral - Wikipedia, the free encyclopedia
    Last edited by mathnerd15; March 29th 2013 at 04:16 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Aug 2011
    Posts
    84

    Re: Integral Theory, Fubini's Theorem

    by the way is my understanding of the integral correct?
    do you mean that Riemann sums can be represented by an nxn matrix and it doesn't matter if you sum the rows or columns first?
    with Cavalieri's principle you can calculate volumes of surfaces using the A(x) area of a family of planes Px
    Last edited by mathnerd15; April 2nd 2013 at 11:08 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Aug 2011
    Posts
    84

    Re: Integral Theory, Fubini's Theorem

    it seems like a minor point but in a matrix you are adding the area of plane sides of a rectangular box
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Using Fubini's Theorem
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: October 17th 2011, 10:08 PM
  2. Applying Fubini's theorem
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 14th 2011, 01:57 PM
  3. a theorem in graph theory.. please help
    Posted in the Discrete Math Forum
    Replies: 10
    Last Post: May 8th 2011, 11:22 AM
  4. fubini's theorem
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: December 22nd 2009, 03:36 PM
  5. Fubini-Tonelli with complete measure space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: September 24th 2009, 09:41 AM

Search Tags


/mathhelpforum @mathhelpforum