{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0DQ8ERJZNS"
    }
  ],
  "body": "\n## Definition\n\nLet $D \\subseteq \\mathbb{R}^2$ be a bounded [[Jordan measurable|0565DC8RMK]] closed set, and let $r \\colon D \\to \\mathbb{R}^3$ be a [[parametrised surface|05788J51WZ]] that is injective on the interior of $D$.\nThe surface area of $S = r(D)$ is\n$$\\operatorname{area}(S) = \\iint_D \\| r_u(u, v) \\times r_v(u, v) \\| \\, du \\, dv.$$\n\n## Remarks\n\nThe integrand is the length of the [[normal vector|0BJDQ2HBH1]]. It is the area of the parallelogram spanned by $r_u$ and $r_v$, which approximates the image of a small parameter rectangle.\nFor the graph $z = f(x, y)$ over $D$, the area is $\\iint_D \\sqrt{1 + f_x^2 + f_y^2} \\, dx \\, dy$.\nFor the unit sphere, $\\| r_\\varphi \\times r_\\theta \\| = \\sin \\varphi$, and the area is $\\int_0^{2\\pi} \\int_0^\\pi \\sin \\varphi \\, d\\varphi \\, d\\theta = 4\\pi$.\nThe value does not depend on the parametrisation. A piecewise smooth surface gets the sum of the areas of its pieces. Inscribed polyhedra do not give a good definition: the Schwarz lantern is a counterexample.\n\n## Explanations\n\n### L1 human\n\nTo find the area of a curved surface, cut the parameter domain into small rectangles.\nEach one maps to a small curved patch, which is nearly a parallelogram, and you add up the areas of these parallelograms.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0S29Y4ZW0Y",
    "kind": "definition",
    "msc": [
      "28A75"
    ],
    "name": "Surface area",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Surface_area"
      },
      {
        "locator": "Surface area",
        "url": "https://en.wikipedia.org/wiki/Parametric_surface"
      }
    ],
    "tag": "0WQE3G7B88",
    "uses": [
      "05788J51WZ",
      "0BJDQ2HBH1",
      "0NKBMFC18R",
      "0565DC8RMK"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0WQE3G7B88/",
  "links": [
    {
      "field": "defines",
      "target": "0S29Y4ZW0Y"
    },
    {
      "field": "uses",
      "target": "05788J51WZ"
    },
    {
      "field": "uses",
      "target": "0BJDQ2HBH1"
    },
    {
      "field": "uses",
      "target": "0NKBMFC18R"
    },
    {
      "field": "uses",
      "target": "0565DC8RMK"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0WQE3G7B88.md",
  "msc": [
    "28A75"
  ],
  "name": "Surface area",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0WQ/0WQE3G7B88.md",
  "status": null,
  "tag": "0WQE3G7B88",
  "uri": "m2://0WQE3G7B88",
  "url": "https://math2.tamnd.com/wiki/surface-area-definition/",
  "writing": {
    "tag": "0WQE3G7B88",
    "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
      }
    ]
  }
}