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:(adsbygoogle = window.adsbygoogle || []).push({});

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 :)

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# Fundamental Lemma of Variational Calculus

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