{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "K9BPYVQNZ9"
    },
    {
      "field": "defines",
      "target": "TDQ4YBMW9C"
    }
  ],
  "body": "\n## Summary\n\nThe extremal length of a family of curves in a plane domain measures how long the curves are compared with the area they sweep out.\nIt is defined by testing all possible ways of measuring length with a weight function $\\rho$, and it is invariant under conformal maps.\nIntroduced by Beurling and developed by Ahlfors, it is a flexible tool for estimates in geometric function theory and for the theory of quasiconformal maps.\nIts reciprocal is the [[conformal modulus|2FQAKBVBX5]] of the family.\n\n## Explanations\n\n### L1 human\n\nThink of a rectangle of width $a$ and height $b$, and of all the curves that cross it from left to right.\nThese curves are long when $a$ is large and plentiful when $b$ is large, and the ratio $a/b$ captures both effects; it is the extremal length of the family.\nStretching the rectangle by a common factor does not change $a/b$, and in fact no conformal map changes the extremal length.\nThe definition replaces the Euclidean length by any weighted length and takes the most favourable weight.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "30C75",
      "31A15"
    ],
    "name": "Extremal length",
    "same_as": {
      "wikidata": "Q5422297",
      "wikipedia": "Extremal length"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Extremal_length"
      }
    ],
    "tag": "HX72KT9D0G"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/HX72KT9D0G/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/HX72KT9D0G.md",
  "msc": [
    "30C75",
    "31A15"
  ],
  "name": "Extremal length",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HX7/HX72KT9D0G.md",
  "status": null,
  "tag": "HX72KT9D0G",
  "uri": "m2://HX72KT9D0G",
  "url": "https://math2.tamnd.com/wiki/extremal-length/",
  "writing": {
    "tag": "HX72KT9D0G",
    "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
      }
    ]
  }
}