Fundamental lemma of calculus of variations in R^k

[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 continuous 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 R^k?

 

[HallsofIvy]

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 R^k. 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.

 

[Niles]

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