{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "6MB2G4T913"
    }
  ],
  "body": "\n## Summary\n\nThe homogeneity equation is the [[functional equation|SRAN4EKWS0]] $f(tx) = t^k f(x)$, required for all positive scalars $t$ and all points $x$.\nIts solutions are the positively homogeneous functions of degree $k$.\nThey are the functions that scale in a fixed way when all inputs are scaled by the same factor.\n[[Homogeneous polynomials|2C3PAQ7H50]] of degree $k$ are the basic examples.\n\n## Explanations\n\n### L1 human\n\nDoubling every input of a homogeneous function of degree $k$ multiplies the output by $2^k$.\nThe area of a rectangle, $f(a, b) = ab$, is homogeneous of degree $2$: doubling both sides multiplies the area by $4$.\nOn a line, the solution is fixed by its values at $1$ and $-1$, since $f(x) = x^k f(1)$ for $x \u003e 0$.\nIn several variables the solution can be any function on the unit sphere, extended along rays.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "39B22",
      "26B35"
    ],
    "name": "Homogeneity equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Homogeneous_function"
      }
    ],
    "specializes": [
      "SRAN4EKWS0"
    ],
    "tag": "Y7BD5ECENG"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/Y7BD5ECENG/",
  "links": [
    {
      "field": "specializes",
      "target": "SRAN4EKWS0"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/Y7BD5ECENG.md",
  "msc": [
    "39B22",
    "26B35"
  ],
  "name": "Homogeneity equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Y7B/Y7BD5ECENG.md",
  "status": null,
  "tag": "Y7BD5ECENG",
  "uri": "m2://Y7BD5ECENG",
  "url": "https://tamnd.github.io/math2/wiki/homogeneity-equation/",
  "writing": {
    "tag": "Y7BD5ECENG",
    "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
      }
    ]
  }
}