{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "05RCT1522R"
    },
    {
      "field": "uses",
      "target": "0JC5X4HWR2"
    },
    {
      "field": "uses",
      "target": "0W5NJQ8MKY"
    }
  ],
  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, and let $f \\colon U \\to \\mathbb{R}$ be twice differentiable on $U$.\nThe Laplacian of $f$ is the function $\\Delta f \\colon U \\to \\mathbb{R}$,\n$$\\Delta f = \\operatorname{div}(\\nabla f) = \\nabla \\cdot \\nabla f,$$\nthe [[divergence|02NSMQVQWK]] of the [[gradient|0NS2QYVC4Y]] of $f$. It is also written $\\nabla^2 f$.\n\n## Remarks\n\nIn Cartesian coordinates, $\\Delta f$ is the [[sum of the unmixed second partial derivatives|05RCT1522R]] of $f$.\nThe operator $\\Delta$ is linear, and it maps functions of [[class C^k|0TCBZMEYGN]] to functions of class $C^{k-2}$ for $k \\ge 2$.\nLet $f$ be of class $C^2$, and let $\\bar{f}_r$ be the average of $f$ on the sphere of radius $r$ about $x$. Then $\\Delta f(x)$ is $2n$ times the limit of $(\\bar{f}_r - f(x)) / r^2$ as $r \\to 0$. So $\\Delta f(x) \u003e 0$ means that $f(x)$ is below its nearby averages.\nThe Laplacian appears in the heat equation $\\partial_t u = \\Delta u$, the wave equation and Laplace's equation $\\Delta u = 0$.\n\n## Explanations\n\n### L1 human\n\nThe Laplacian compares the value of a function at a point with its values nearby.\nIf $f$ at the point is lower than the average around it, the Laplacian is positive. If it is higher, the Laplacian is negative.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0E0B6RA8K0",
    "kind": "definition",
    "msc": [
      "35J05"
    ],
    "name": "Laplacian",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Laplace_operator"
      },
      {
        "locator": "Averages",
        "url": "https://en.wikipedia.org/wiki/Laplace_operator"
      }
    ],
    "tag": "0XQ2ZVN1PZ",
    "uses": [
      "02NSMQVQWK",
      "0NS2QYVC4Y",
      "0TCBZMEYGN"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0XQ2ZVN1PZ/",
  "links": [
    {
      "field": "defines",
      "target": "0E0B6RA8K0"
    },
    {
      "field": "uses",
      "target": "02NSMQVQWK"
    },
    {
      "field": "uses",
      "target": "0NS2QYVC4Y"
    },
    {
      "field": "uses",
      "target": "0TCBZMEYGN"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0XQ2ZVN1PZ.md",
  "msc": [
    "35J05"
  ],
  "name": "Laplacian",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0XQ/0XQ2ZVN1PZ.md",
  "status": null,
  "tag": "0XQ2ZVN1PZ",
  "uri": "m2://0XQ2ZVN1PZ",
  "url": "https://math2.tamnd.com/wiki/laplacian-definition/",
  "writing": {
    "tag": "0XQ2ZVN1PZ",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}