{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\R$ be [[continuous|0X3WGX95FE]].\nSuppose that\n$$\\int_a^b f(x) \\, \\eta'(x) \\, dx = 0$$\nfor every [[continuously differentiable|0VW90BBSJ4]] $\\eta \\colon [a, b] \\to \\R$ with $\\eta(a) = \\eta(b) = 0$.\nThen $f$ is constant on $[a, b]$.\n\n## Remarks\n\nLet $c = \\frac{1}{b - a} \\int_a^b f \\, dx$ be the mean of $f$, and set $\\eta(x) = \\int_a^x (f(s) - c) \\, ds$.\nThen $\\eta$ is $C^1$, $\\eta(a) = \\eta(b) = 0$ and $\\eta' = f - c$.\nThe hypothesis gives $\\int_a^b f (f - c) \\, dx = 0$, and $\\int_a^b c (f - c) \\, dx = 0$ by the choice of $c$.\nSubtracting, $\\int_a^b (f - c)^2 \\, dx = 0$, so $f = c$ by continuity.\n\nA common extension: if $f, g$ are continuous and $\\int_a^b (g \\eta + f \\eta') \\, dx = 0$ for all such $\\eta$, then $f$ is $C^1$ and $f' = g$.\nTo see it, apply the lemma to $f - G$, where $G(x) = \\int_a^x g(s) \\, ds$, after an integration by parts of $\\int_a^b g \\eta \\, dx$.\n\nExample: for $f(x) = x$ on $[0, 1]$ and $\\eta(x) = x(1 - x)$, one finds $\\int_0^1 x (1 - 2x) \\, dx = -1/6 \\ne 0$, as the lemma predicts for a non-constant $f$.\n\n## Explanations\n\n### L1 human\n\nIf $f$ gives zero against the slope of every function that starts and ends at zero, then $f$ cannot vary.\nThe reason is that one can build a test function whose slope is exactly $f$ minus its average.\n\n### L2 human\n\nThe extension above gives the Euler–Lagrange equation in integrated form, $L_p(x, y, y') = \\int_a^x L_z(s, y, y') \\, ds + C$, for a $C^1$ minimiser $y$, without assuming that $y$ has a second derivative.\nIt follows that $x \\mapsto L_p(x, y(x), y'(x))$ is $C^1$, which is the start of regularity theory in one variable.\nThe [[fundamental lemma|JNDZ86CV3Y]] is the version where $\\eta$ itself, instead of $\\eta'$, appears in the integral.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "PQ25NHV0KS"
    ],
    "kind": "statement",
    "msc": [
      "49K05"
    ],
    "name": "du Bois-Reymond lemma",
    "role": "lemma",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fundamental_lemma_of_the_calculus_of_variations"
      }
    ],
    "tag": "AATQKCWAAC",
    "uses": [
      "0X3WGX95FE",
      "0VW90BBSJ4",
      "0W29TBTY12"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/AATQKCWAAC/",
  "links": [
    {
      "field": "about",
      "target": "PQ25NHV0KS"
    },
    {
      "field": "uses",
      "target": "0X3WGX95FE"
    },
    {
      "field": "uses",
      "target": "0VW90BBSJ4"
    },
    {
      "field": "uses",
      "target": "0W29TBTY12"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/AATQKCWAAC.md",
  "msc": [
    "49K05"
  ],
  "name": "du Bois-Reymond lemma",
  "prov": null,
  "role": "lemma",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AAT/AATQKCWAAC.md",
  "status": null,
  "tag": "AATQKCWAAC",
  "uri": "m2://AATQKCWAAC",
  "url": "https://tamnd.github.io/math2/wiki/du-bois-reymond-lemma/",
  "writing": {
    "tag": "AATQKCWAAC",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}