{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] with [[Levi-Civita connection|CAZYGE1T4E]] $\\nabla$.\nFor a smooth function $f$ on $M$, the gradient $\\operatorname{grad} f$ is the vector field with $g(\\operatorname{grad} f, X) = df(X)$ for every vector field $X$.\nFor a vector field $Y$, the divergence $\\operatorname{div} Y$ is the function $\\tr(X \\mapsto \\nabla_X Y)$.\nThe *Laplace–Beltrami operator* is the operator $\\Delta$ on smooth functions given by\n$$\\Delta f = \\operatorname{div}(\\operatorname{grad} f).$$\n\n## Remarks\n\nIn local coordinates $x^1, \\dots, x^n$, with $g_{ij}$ the metric, $g^{ij}$ its inverse and $\\abs{g} = \\det(g_{ij})$,\n$$\\Delta f = \\frac{1}{\\sqrt{\\abs{g}}} \\sum_{i,j} \\partial_i \\bigl(\\sqrt{\\abs{g}} \\, g^{ij} \\partial_j f\\bigr).$$\nEquivalently $\\Delta f = \\sum_{i,j} g^{ij} \\bigl(\\partial_i \\partial_j f - \\sum_k \\Gamma^k_{ij} \\partial_k f\\bigr)$, with $\\Gamma^k_{ij}$ the [[Christoffel symbols|FZBA1HA9ZM]].\n\nExample: on the unit sphere $S^2$ with coordinates $(\\theta, \\varphi)$ and metric $d\\theta^2 + \\sin^2\\theta \\, d\\varphi^2$, one has $\\sqrt{\\abs{g}} = \\sin\\theta$ and\n$$\\Delta f = \\frac{1}{\\sin\\theta} \\partial_\\theta(\\sin\\theta \\, \\partial_\\theta f) + \\frac{1}{\\sin^2\\theta} \\partial_\\varphi^2 f.$$\nFor $f = \\cos\\theta$ this gives $\\frac{1}{\\sin\\theta} \\partial_\\theta(-\\sin^2\\theta) = -2\\cos\\theta$.\n\nNon-example: on $\\R^2 \\setminus \\set{0}$ in polar coordinates $(r, \\theta)$, the operator $\\partial_r^2 + \\partial_\\theta^2$ is not the Laplace–Beltrami operator of the Euclidean metric $dr^2 + r^2 d\\theta^2$.\nThe formula above gives $\\Delta f = \\partial_r^2 f + \\frac{1}{r} \\partial_r f + \\frac{1}{r^2} \\partial_\\theta^2 f$.\n\nWith this sign convention $\\Delta$ is the usual Laplacian $\\sum_i \\partial_i^2$ on $\\R^n$ and $-\\Delta$ has nonnegative spectrum on a compact manifold.\nMany geometers use the opposite sign, $\\Delta = -\\operatorname{div} \\operatorname{grad}$.\nOn functions, $-\\Delta$ equals the Hodge Laplacian $\\delta d$.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "F4NWZNTRMD",
    "kind": "definition",
    "msc": [
      "58J05",
      "53C20"
    ],
    "name": "Laplace–Beltrami operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator"
      }
    ],
    "tag": "PAGN759BNR",
    "uses": [
      "7CBT4SM83Y",
      "CAZYGE1T4E",
      "FZBA1HA9ZM"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/PAGN759BNR/",
  "links": [
    {
      "field": "defines",
      "target": "F4NWZNTRMD"
    },
    {
      "field": "uses",
      "target": "7CBT4SM83Y"
    },
    {
      "field": "uses",
      "target": "CAZYGE1T4E"
    },
    {
      "field": "uses",
      "target": "FZBA1HA9ZM"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/PAGN759BNR.md",
  "msc": [
    "58J05",
    "53C20"
  ],
  "name": "Laplace–Beltrami operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PAG/PAGN759BNR.md",
  "status": null,
  "tag": "PAGN759BNR",
  "uri": "m2://PAGN759BNR",
  "url": "https://math2.tamnd.com/wiki/laplace-beltrami-operator-definition/",
  "writing": {
    "tag": "PAGN759BNR",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}