{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f \\colon U \\to \\mathbb{R}$ be [[differentiable|0135XZZQ8M]] on $U$, and let $x, y \\in U$ be points such that the segment $\\{x + t (y - x) : 0 \\le t \\le 1\\}$ lies in $U$.\nThen there is a number $c \\in (0, 1)$ such that\n$$f(y) - f(x) = \\nabla f\\big(x + c (y - x)\\big) \\cdot (y - x).$$\n\n## Remarks\n\nThe proof applies the one-variable mean value theorem to $g(t) = f(x + t(y - x))$ on $[0, 1]$, with $g'(t) = \\nabla f(x + t(y - x)) \\cdot (y - x)$ by the [[chain rule for a curve|0A8TJ0VHC7]].\nThe theorem is false for maps into $\\mathbb{R}^m$ with $m \\ge 2$. For $f(t) = (\\cos t, \\sin t)$ on $[0, 2\\pi]$, $f(2\\pi) - f(0) = 0$, but $f'(t) \\ne 0$ for each $t$. The [[mean value inequality|02CYB184RC]] replaces it.\nBy the Cauchy–Schwarz inequality, it gives $|f(y) - f(x)| \\le \\sup_{z} \\|\\nabla f(z)\\| \\, \\|y - x\\|$, with $z$ on the segment.\n\n## Explanations\n\n### L1 human\n\nOn a straight path from $x$ to $y$, the average slope of $f$ equals the actual slope at some point between them.\nThis reduces the problem to one variable, by looking at $f$ only along the segment.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "26B05"
    ],
    "name": "Mean value theorem in several variables",
    "role": "theorem",
    "sources": [
      {
        "locator": "Mean value theorem in several variables",
        "url": "https://en.wikipedia.org/wiki/Mean_value_theorem"
      }
    ],
    "tag": "03HGK7EW6R",
    "uses": [
      "0135XZZQ8M",
      "0NS2QYVC4Y",
      "0A8TJ0VHC7"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/03HGK7EW6R/",
  "links": [
    {
      "field": "uses",
      "target": "0135XZZQ8M"
    },
    {
      "field": "uses",
      "target": "0NS2QYVC4Y"
    },
    {
      "field": "uses",
      "target": "0A8TJ0VHC7"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/03HGK7EW6R.md",
  "msc": [
    "26B05"
  ],
  "name": "Mean value theorem in several variables",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/03H/03HGK7EW6R.md",
  "status": null,
  "tag": "03HGK7EW6R",
  "uri": "m2://03HGK7EW6R",
  "url": "https://math2.tamnd.com/wiki/mean-value-theorem-in-several-variables/",
  "writing": {
    "tag": "03HGK7EW6R",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}