Proof
Let f satisfy the hypotheses. Let r be any smooth function that is 0 at a and b and positive on (a, b); for example, . Let . Then h is of class on, so
The integrand is nonnegative, so it must be 0 except perhaps on a subset of of measure 0. However, by continuity if there are points where the integrand is non-zero, there is also some interval around that point where the integrand is non-zero, which has non-zero measure, so it must be identically 0 over the entire interval. Since r is positive on (a, b), f is 0 there and hence on all of .
Read more about this topic: Fundamental Lemma Of Calculus Of Variations
Famous quotes containing the word proof:
“If we view our children as stupid, naughty, disturbed, or guilty of their misdeeds, they will learn to behold themselves as foolish, faulty, or shameful specimens of humanity. They will regard us as judges from whom they wish to hide, and they will interpret everything we say as further proof of their unworthiness. If we view them as innocent, or at least merely ignorant, they will gain understanding from their experiences, and they will continue to regard us as wise partners.”
—Polly Berrien Berends (20th century)
“When children feel good about themselves, it’s like a snowball rolling downhill. They are continually able to recognize and integrate new proof of their value as they grow and mature.”
—Stephanie Martson (20th century)
“Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?”
—Henry David Thoreau (1817–1862)