{
  "aliases": [
    "laplace-operator"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0DEB0JMN0W"
    },
    {
      "field": "uses",
      "target": "0H6DP8CHND"
    },
    {
      "field": "uses",
      "target": "19WY7FYPXK"
    },
    {
      "field": "uses",
      "target": "1VGB63K5HS"
    },
    {
      "field": "uses",
      "target": "2F6AJ1DM11"
    },
    {
      "field": "uses",
      "target": "2JK9YMZV2B"
    },
    {
      "field": "uses",
      "target": "3FE2AQS534"
    },
    {
      "field": "uses",
      "target": "4AWC5Q4FHE"
    },
    {
      "field": "uses",
      "target": "4BT3Y8TCDN"
    },
    {
      "field": "uses",
      "target": "4X0A0CQ0B0"
    },
    {
      "field": "uses",
      "target": "5J2GTEA6CV"
    },
    {
      "field": "uses",
      "target": "5WXY2T8JG0"
    },
    {
      "field": "uses",
      "target": "63ZT8D3DXD"
    },
    {
      "field": "uses",
      "target": "7E9R9X72JP"
    },
    {
      "field": "uses",
      "target": "8D56GQ4AG8"
    },
    {
      "field": "uses",
      "target": "9CWM3F4M04"
    },
    {
      "field": "uses",
      "target": "9GVQ7JXC25"
    },
    {
      "field": "uses",
      "target": "9PV9CN8X0Y"
    },
    {
      "field": "uses",
      "target": "9Y01DS79PR"
    },
    {
      "field": "uses",
      "target": "9ZV3T5C6B7"
    },
    {
      "field": "uses",
      "target": "A0F09CMEJM"
    },
    {
      "field": "uses",
      "target": "AFBY76MAX7"
    },
    {
      "field": "uses",
      "target": "BE2CBPAZK6"
    },
    {
      "field": "uses",
      "target": "CD7JZ43JBA"
    },
    {
      "field": "uses",
      "target": "CM7D3ES438"
    },
    {
      "field": "uses",
      "target": "CQ7E0V6T2G"
    },
    {
      "field": "uses",
      "target": "CS10HH24VT"
    },
    {
      "field": "uses",
      "target": "D4W7DBAT55"
    },
    {
      "field": "uses",
      "target": "DHRDTHFBFA"
    },
    {
      "field": "uses",
      "target": "DKV5J6N6T6"
    },
    {
      "field": "uses",
      "target": "F3DXBJZAME"
    },
    {
      "field": "uses",
      "target": "F68QFEYFY0"
    },
    {
      "field": "uses",
      "target": "FRKAPMVZ7M"
    },
    {
      "field": "uses",
      "target": "G82RA93DWJ"
    },
    {
      "field": "uses",
      "target": "GETN4MCAAN"
    },
    {
      "field": "uses",
      "target": "HD74ZBGCJ2"
    },
    {
      "field": "uses",
      "target": "J05VKG4XQB"
    },
    {
      "field": "uses",
      "target": "JWNRB34ZGV"
    },
    {
      "field": "uses",
      "target": "K76033B5G1"
    },
    {
      "field": "uses",
      "target": "KSA55H8Q8N"
    },
    {
      "field": "uses",
      "target": "M435YDQWQ4"
    },
    {
      "field": "uses",
      "target": "N5JN15Z5J6"
    },
    {
      "field": "uses",
      "target": "NXWRDGX8T5"
    },
    {
      "field": "uses",
      "target": "PR9A84ZPT1"
    },
    {
      "field": "uses",
      "target": "QDXXWA1QGF"
    },
    {
      "field": "uses",
      "target": "RYQJH4RYB4"
    },
    {
      "field": "uses",
      "target": "S0QKEJQ1WM"
    },
    {
      "field": "uses",
      "target": "S977PT8S3S"
    },
    {
      "field": "uses",
      "target": "SNC7NR587P"
    },
    {
      "field": "uses",
      "target": "SZ2C28JG5K"
    },
    {
      "field": "uses",
      "target": "T6WEFSKYHZ"
    },
    {
      "field": "uses",
      "target": "VNB1BFDEYK"
    },
    {
      "field": "uses",
      "target": "W0F3YWAJ23"
    },
    {
      "field": "uses",
      "target": "W6QS1ME39J"
    },
    {
      "field": "uses",
      "target": "X3VH8H1XV3"
    },
    {
      "field": "uses",
      "target": "XDK7SQW3F6"
    },
    {
      "field": "uses",
      "target": "XS0Q09SV2T"
    },
    {
      "field": "uses",
      "target": "YATEFAKZYB"
    },
    {
      "field": "uses",
      "target": "Z8NJ4KHGAT"
    },
    {
      "field": "uses",
      "target": "ZARKMD0C0H"
    },
    {
      "field": "uses",
      "target": "ZFC8N4CFC4"
    },
    {
      "field": "uses",
      "target": "ZZD7X71HVN"
    },
    {
      "field": "defines",
      "target": "0XQ2ZVN1PZ"
    },
    {
      "field": "about",
      "target": "2JK9YMZV2B"
    },
    {
      "field": "about",
      "target": "QDXXWA1QGF"
    },
    {
      "field": "about",
      "target": "RYQJH4RYB4"
    },
    {
      "field": "about",
      "target": "X3VH8H1XV3"
    }
  ],
  "body": "\n## Summary\n\nThe Laplacian of a twice differentiable function on $\\mathbb{R}^n$ is the [[divergence|0MWV6ZTK6S]] of its [[gradient|0ABK9V9Y64]].\nIt measures how the value of the function at a point differs from its average value on small spheres around the point.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "aliases": [
      "laplace-operator"
    ],
    "kind": "concept",
    "msc": [
      "35J05"
    ],
    "name": "Laplacian",
    "same_as": {
      "wikidata": "Q203484",
      "wikipedia": "Laplace operator"
    },
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Laplace_operator"
      }
    ],
    "tag": "0E0B6RA8K0",
    "uses": [
      "0MWV6ZTK6S",
      "0ABK9V9Y64"
    ]
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/0E0B6RA8K0/",
  "links": [
    {
      "field": "uses",
      "target": "0MWV6ZTK6S"
    },
    {
      "field": "uses",
      "target": "0ABK9V9Y64"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/0E0B6RA8K0.md",
  "msc": [
    "35J05"
  ],
  "name": "Laplacian",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0E0/0E0B6RA8K0.md",
  "status": null,
  "tag": "0E0B6RA8K0",
  "uri": "m2://0E0B6RA8K0",
  "url": "https://tamnd.github.io/math2/wiki/laplacian/",
  "writing": {
    "tag": "0E0B6RA8K0",
    "kind": "concept",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "at least two examples",
        "have": 0,
        "want": 2
      },
      {
        "what": "edge not_instance_of in",
        "note": "at least one non-example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge specializes both",
        "note": "the more general notions and the special cases",
        "have": 0,
        "want": 1
      }
    ]
  }
}