# Fundemental Lemma of the Calculus of Variations

1. Nov 26, 2009

### Niles

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

$$\int_a^b f(t)h(t)dt = 0$$

for all $h\in C_{0,0}^1([a,b], R)$, where $C_{0,0}^1([a,b], R)$ 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. Nov 26, 2009

### HallsofIvy

Staff Emeritus
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
3. Nov 27, 2009

### Niles

Thank you. Yes, for some reason I didn't type in the conclusion, even though I did write it down in class. Thanks.