Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Fundamental Lemma of Variational Calculus

  1. Feb 2, 2008 #1
    I was just going through the derivation of the euler - lagrange equation that rests on the proof of the fundamental lemma of the calculus of variations which states the following:

    If a function g(x) vanishes at the endpoints i.e g(a) = 0, g(b) = 0 and is continuously differentiable within the interval, and a second function f(x) is smooth within the interval and if
    integral[ f(x)g(x) ] from a to b = 0,

    then f(x) is identically zero within the interal (a,b).

    This result is simple and intuitive if g(x) is positive within (a,b) but what happens if we have f(x) = k and g(x) = sin(x). Then g(x) = 0 at the endpoints 0 and 2pi but

    integral[ k * sin(x) ] from a to b = 0 for k not equal to zero.

    I would appreciate it if someone explained this to me and cleared my doubts as regarding the lemma. thanks a lot :)
  2. jcsd
  3. Feb 2, 2008 #2


    User Avatar
    Science Advisor

    That "lemma" is clearly not true. Are you sure it doesn't say "[itex]\int f(x)g(x)dx[/itex]= 0 for every such funcition g"? Or perhaps "g(x)= 0 only at the endpoints"?
  4. Feb 2, 2008 #3
  5. Feb 2, 2008 #4


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    To say "for every function g" and "for any function g" is the same thing. But "for every function g", and "for a function g" is not the same thing.
  6. Feb 3, 2008 #5
    Oh man. I must have meant to say "some", and it came out as "any".
    Last edited: Feb 3, 2008
  7. Feb 3, 2008 #6
    oh i see...i made a mistake in reading it and interpreted it as being true for some function g rather than for every function belonging to C(infinity). thanks a lot you guys.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook