First variation

The first variation of a functional

L

at

y

in the direction

η

is

δ L [y; η] = \frac{d}{d ε} L [y + ε η]_{ε = 0} .

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- Fundamental lemma of the calculus of variations First variation

Fundamental lemma of the calculus of variations

If a continuous function

f

satisfies

\int_{a}^{b} f (x) η (x) d x = 0

for every smooth compactly supported test function

η

, then

f = 0

on

(a, b)

.

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First variation by

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The term "first variation" is often used in the context of calculus of variations, which is a mathematical field that deals with optimizing functionals, usually integrals that depend on functions and their derivatives. The first variation is a concept that measures how a functional changes when the function is varied or perturbed slightly.

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