{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "04A2XEX4D4"
    },
    {
      "field": "uses",
      "target": "0NE2W2XPQJ"
    },
    {
      "field": "uses",
      "target": "0VXYV3NQQ1"
    }
  ],
  "body": "\n## Definition\n\nLet $D \\subseteq \\mathbb{R}^n$, let $f \\colon D \\to \\mathbb{R}$ be a [[function of several variables|0JPCHTXWG0]], and let $c \\in \\mathbb{R}$.\nThe level set of $f$ at the value $c$ is the set\n$$L_c(f) = \\{x \\in D : f(x) = c\\}.$$\nIt is the [[preimage|01AHG4G9RT]] $f^{-1}(\\{c\\})$ of the one-point set $\\{c\\}$.\nFor $n = 2$ a level set is also called a level curve, and for $n = 3$ it is also called a level surface.\n\n## Remarks\n\nFor $f(x, y) = x^2 + y^2$, the level set at $c \u003e 0$ is the circle of radius $\\sqrt{c}$ about the origin. The level set at $c = 0$ is the point $(0, 0)$, and the level set at $c \u003c 0$ is empty.\nThe level sets of $f$ for different values of $c$ are disjoint, and their union is the domain $D$.\nAt a point where $f$ is differentiable with a nonzero [[gradient|0ABK9V9Y64]], the gradient is [[orthogonal to the level set|0VXYV3NQQ1]].\nA level set of a function of two variables is a [[contour line|0QD83RAN8G]] of its graph.\n\n## Explanations\n\n### L1 human\n\nOn a map of a hill, a level set is the set of all places at one fixed height.\nFor the bowl $f(x, y) = x^2 + y^2$, each level set at a positive height is a circle, and the circles grow as the height grows.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0B2RZ0778N",
    "kind": "definition",
    "msc": [
      "26B99"
    ],
    "name": "Level set",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Level_set"
      }
    ],
    "tag": "0Y5Q8Q9WJ7",
    "uses": [
      "0JPCHTXWG0",
      "01AHG4G9RT"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0Y5Q8Q9WJ7/",
  "links": [
    {
      "field": "defines",
      "target": "0B2RZ0778N"
    },
    {
      "field": "uses",
      "target": "0JPCHTXWG0"
    },
    {
      "field": "uses",
      "target": "01AHG4G9RT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0Y5Q8Q9WJ7.md",
  "msc": [
    "26B99"
  ],
  "name": "Level set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0Y5/0Y5Q8Q9WJ7.md",
  "status": null,
  "tag": "0Y5Q8Q9WJ7",
  "uri": "m2://0Y5Q8Q9WJ7",
  "url": "https://math2.tamnd.com/wiki/level-set-definition/",
  "writing": {
    "tag": "0Y5Q8Q9WJ7",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}