---
tag: 5B8NFYB2XQ
kind: concept
name: First variation
msc: [49K05, 49J50, 46G05]
sources:
  - url: https://en.wikipedia.org/wiki/First_variation
same_as:
  wikidata: Q1362077
  wikipedia: First variation
---

## Summary

The first variation of a [[functional|1EQM0DJ4NC]] $J$ at $y$ in the direction $\eta$ is the rate of change of $J(y + t\eta)$ at $t = 0$.
It is the analogue for functionals of the [[directional derivative|06KWGHDN2J]].
At a minimiser it vanishes in every admissible direction, and for [[integral functionals|JRQZ803Q5Y]] this condition leads to the [[Euler–Lagrange equation|PS0VV149ET]].

## Explanations

### L1 human

Push a function $y$ a little in the direction of another function $\eta$, and watch how the value of the functional changes.
The first variation is the speed of that change at the start.
At the bottom of a valley every direction is flat to first order, so at a minimiser the first variation is zero.
For $J(y) = \int_0^1 y^2 \, dx$, the first variation at $y$ in the direction $\eta$ is $2 \int_0^1 y \eta \, dx$.