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Fundemental Lemma of the Calculus of Variations

  1. Nov 26, 2009 #1
    Hi all.

    In my notes I wrote down from the blackboard, I wrote

    [Fundemental Lemma of the Calculus of Variations] Let f : [a,b] -> R be continous and suppose that

    [tex]
    \int_a^b f(t)h(t)dt = 0
    [/tex]

    for all [itex]h\in C_{0,0}^1([a,b], R)[/itex], where [itex]C_{0,0}^1([a,b], R)[/itex] is the space of C1 parametrized curves O : [a,b] -> R that start and end in 0.

    I suspect that I missed some k's when writing this down from the blackboard. Am I correct when I say that this also works if we are in Rk?
     
  2. jcsd
  3. Nov 26, 2009 #2

    HallsofIvy

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    What you are missing is the conclusion of the lemma. You have only stated the hypotheses. IF f is a function having those properties then- f(x)= 0 of all x in [a,b].

    No, you are not correct that this also works in Rk. In higher dimensions just integrating on paths is not enough. You would also have to know that integral was 0 on all surfaces, all three dimensional regions, etc. up to the dimension of the space.
     
    Last edited: Nov 27, 2009
  4. Nov 27, 2009 #3
    Thank you. Yes, for some reason I didn't type in the conclusion, even though I did write it down in class. Thanks.
     
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