{
  "aliases": [
    "stokes-cartan-theorem"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0AKRKVHW94"
    },
    {
      "field": "uses",
      "target": "0HR3N02317"
    },
    {
      "field": "uses",
      "target": "0JX86PSSA9"
    }
  ],
  "body": "\n## Statement\n\nLet $\\Omega$ be an oriented smooth manifold of dimension $n$ with boundary $\\partial \\Omega$, and give $\\partial \\Omega$ the induced orientation.\nLet $\\omega$ be a smooth [[differential form|0QHTX7WQEY]] of degree $n - 1$ on $\\Omega$ with compact support, and let $d\\omega$ be its [[exterior derivative|004FRCXH6M]]. Then\n$$\\int_\\Omega d\\omega = \\int_{\\partial \\Omega} \\omega.$$\n\n## Remarks\n\nFor $\\Omega = [a, b]$ and a function $f$, the theorem is the fundamental theorem of calculus, $\\int_a^b f'(x) \\, dx = f(b) - f(a)$. The boundary point $a$ gets the negative orientation.\nFor a curve and a 0-form, it is the [[gradient theorem|0SD237FEJD]]. For a plane region and a 1-form, it is [[Green's theorem|0HR3N02317]].\nFor a surface in $\\mathbb{R}^3$ and a 1-form, it is the classical [[Stokes' theorem|0AKRKVHW94]]. For a solid in $\\mathbb{R}^3$ and a 2-form, it is the [[divergence theorem|0JX86PSSA9]].\nThe orientation and the compact support are needed so that both integrals are defined.\n\n## Explanations\n\n### L1 human\n\nThe integral of a derivative over a region equals the integral of the original form over the boundary of the region.\nOne formula covers the fundamental theorem of calculus and the three integral theorems of vector calculus.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "stokes-cartan-theorem"
    ],
    "kind": "statement",
    "msc": [
      "58C35"
    ],
    "name": "Generalised Stokes theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q10861254",
      "wikipedia": "Generalized Stokes theorem"
    },
    "sources": [
      {
        "locator": "Formulation for smooth manifolds with boundary",
        "url": "https://en.wikipedia.org/wiki/Generalized_Stokes_theorem"
      },
      {
        "locator": "Introduction",
        "url": "https://en.wikipedia.org/wiki/Generalized_Stokes_theorem"
      },
      {
        "locator": "Classical (vector calculus) case",
        "url": "https://en.wikipedia.org/wiki/Generalized_Stokes_theorem"
      }
    ],
    "tag": "0HPHKCGEQ0",
    "uses": [
      "004FRCXH6M",
      "0QHTX7WQEY",
      "0EKZW08FQ5",
      "0SD237FEJD",
      "0HR3N02317",
      "0AKRKVHW94",
      "0JX86PSSA9"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0HPHKCGEQ0/",
  "links": [
    {
      "field": "uses",
      "target": "004FRCXH6M"
    },
    {
      "field": "uses",
      "target": "0QHTX7WQEY"
    },
    {
      "field": "uses",
      "target": "0EKZW08FQ5"
    },
    {
      "field": "uses",
      "target": "0SD237FEJD"
    },
    {
      "field": "uses",
      "target": "0HR3N02317"
    },
    {
      "field": "uses",
      "target": "0AKRKVHW94"
    },
    {
      "field": "uses",
      "target": "0JX86PSSA9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0HPHKCGEQ0.md",
  "msc": [
    "58C35"
  ],
  "name": "Generalised Stokes theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0HP/0HPHKCGEQ0.md",
  "status": null,
  "tag": "0HPHKCGEQ0",
  "uri": "m2://0HPHKCGEQ0",
  "url": "https://math2.tamnd.com/wiki/generalised-stokes-theorem/",
  "writing": {
    "tag": "0HPHKCGEQ0",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}