{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $\\mathbb{K}$ be $\\R$ or $\\C$.\nLet $E$ and $F$ be smooth [[vector bundles|P4DZM975BV]] over $M$ with fibres $\\mathbb{K}^r$ and $\\mathbb{K}^s$.\nWrite $\\Gamma(E)$ for the space of smooth [[sections|WWX7H5W1RP]] of $E$.\nLet $m \\ge 0$ be an integer.\nConsider a linear map $P \\colon \\Gamma(E) \\to \\Gamma(F)$.\nIt is a *differential operator of order at most $m$* if the following holds.\nLet $(U, x_1, \\ldots, x_n)$ be any chart of $M$ over which $E$ and $F$ are trivial.\nIn the trivialisations, there are smooth $A_\\alpha \\colon U \\to \\mathbb{K}^{s \\times r}$ for $\\abs{\\alpha} \\le m$ with\n$$(P u)(x) = \\sum_{\\abs{\\alpha} \\le m} A_\\alpha(x) \\, \\partial^\\alpha u(x) \\quad \\text{for all } u \\in \\Gamma(E) \\text{ and } x \\in U,$$\nwhere $u$ is written on $U$ as a function $U \\to \\mathbb{K}^r$.\nThe operator has *order $m$* if moreover some $A_\\alpha$ with $\\abs{\\alpha} = m$ is not identically zero.\n\n## Remarks\n\nThe condition does not depend on the charts and trivialisations.\nA change of coordinates or of trivialisation turns a sum of this form into another one of the same order.\nA differential operator is local: the value of $Pu$ near a point depends only on $u$ near that point.\nBy the Peetre theorem, linear local operators are, locally, differential operators.\n\nExample: the [[exterior derivative|0ZHSTAG8ZY]] $d \\colon \\Gamma(\\Lambda^k T^* M) \\to \\Gamma(\\Lambda^{k+1} T^* M)$ is a differential operator of order $1$.\nIn a chart, $d(\\sum_I f_I \\, dx_I) = \\sum_I \\sum_{j} \\partial_j f_I \\, dx_j \\wedge dx_I$, whose coefficients are constant matrices.\n\nNon-example: on the circle $S^1 = \\R / 2\\pi\\Z$, the map $u \\mapsto \\bigl(\\int_0^{2\\pi} u(t) \\, dt\\bigr) \\cdot 1$ on functions is linear but not a differential operator.\nIt is not local: changing $u$ on a small interval changes $Pu$ everywhere.\n\n## Explanations\n\n### L1 human\n\nA differential operator between bundles is, in every coordinate patch, a matrix of partial derivatives with smooth coefficients.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "CGBX1N9V8H",
    "kind": "definition",
    "msc": [
      "58J99",
      "47F05",
      "35A99"
    ],
    "name": "Differential operator between vector bundles",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Differential_operator"
      }
    ],
    "tag": "QQ5F8B8DYC",
    "uses": [
      "F45VQ47GPV",
      "P4DZM975BV",
      "WWX7H5W1RP",
      "6DZETZ5T9G",
      "0ZHSTAG8ZY"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/QQ5F8B8DYC/",
  "links": [
    {
      "field": "defines",
      "target": "CGBX1N9V8H"
    },
    {
      "field": "uses",
      "target": "F45VQ47GPV"
    },
    {
      "field": "uses",
      "target": "P4DZM975BV"
    },
    {
      "field": "uses",
      "target": "WWX7H5W1RP"
    },
    {
      "field": "uses",
      "target": "6DZETZ5T9G"
    },
    {
      "field": "uses",
      "target": "0ZHSTAG8ZY"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/QQ5F8B8DYC.md",
  "msc": [
    "58J99",
    "47F05",
    "35A99"
  ],
  "name": "Differential operator between vector bundles",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QQ5/QQ5F8B8DYC.md",
  "status": null,
  "tag": "QQ5F8B8DYC",
  "uri": "m2://QQ5F8B8DYC",
  "url": "https://tamnd.github.io/math2/wiki/differential-operator-between-vector-bundles-definition/",
  "writing": {
    "tag": "QQ5F8B8DYC",
    "kind": "definition",
    "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
      }
    ]
  }
}