{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "3482HSGW27"
    },
    {
      "field": "uses",
      "target": "36WNFJTRRR"
    },
    {
      "field": "uses",
      "target": "3VMJGQCD7S"
    },
    {
      "field": "uses",
      "target": "X6QC23AAB9"
    },
    {
      "field": "defines",
      "target": "5N1825CM10"
    }
  ],
  "body": "\n## Summary\n\nThe energy of a smooth map between [[Riemannian manifolds|7CBT4SM83Y]] is half the integral, over the domain, of the squared norm of its differential.\nIt generalizes the Dirichlet energy of a function and the [[energy of a path|Q12AB1MJ64]].\nIts critical points are the [[harmonic maps|KZJYG9DS3M]], and its gradient is minus the [[tension field|W8F8S8DX5X]].\nIt measures how much a map stretches the domain on average.\n\n## Explanations\n\n### L1 human\n\nPicture the domain as a rubber sheet that the map stretches and lays onto the target.\nThe energy density at a point is the total squared stretching in all directions there, and the energy adds it up over the sheet.\nA map that is constant does not stretch at all and has energy $0$.\nFor example, the map $x \\mapsto 2x$ from the interval $[0, 1]$ to the line stretches everything by $2$, so its energy is $\\frac{1}{2} \\int_0^1 4 \\, dx = 2$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "58E20",
      "53C43"
    ],
    "name": "Energy of a map between Riemannian manifolds",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Harmonic_map"
      }
    ],
    "tag": "E6HDE1DFXD"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/E6HDE1DFXD/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/E6HDE1DFXD.md",
  "msc": [
    "58E20",
    "53C43"
  ],
  "name": "Energy of a map between Riemannian manifolds",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/E6H/E6HDE1DFXD.md",
  "status": null,
  "tag": "E6HDE1DFXD",
  "uri": "m2://E6HDE1DFXD",
  "url": "https://tamnd.github.io/math2/wiki/energy-of-a-map-between-riemannian-manifolds/",
  "writing": {
    "tag": "E6HDE1DFXD",
    "kind": "concept",
    "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
      }
    ]
  }
}