{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "004FRCXH6M"
    },
    {
      "field": "uses",
      "target": "00N3T18QNA"
    },
    {
      "field": "uses",
      "target": "03HGK7EW6R"
    },
    {
      "field": "uses",
      "target": "04A2XEX4D4"
    },
    {
      "field": "uses",
      "target": "05R9NF2J49"
    },
    {
      "field": "uses",
      "target": "05RCT1522R"
    },
    {
      "field": "uses",
      "target": "0BJDQ2HBH1"
    },
    {
      "field": "uses",
      "target": "0F405477F8"
    },
    {
      "field": "uses",
      "target": "0FP4BVDAAQ"
    },
    {
      "field": "uses",
      "target": "0JC5X4HWR2"
    },
    {
      "field": "uses",
      "target": "0M2JRW1J0N"
    },
    {
      "field": "uses",
      "target": "0P30YYZN85"
    },
    {
      "field": "uses",
      "target": "0SD237FEJD"
    },
    {
      "field": "uses",
      "target": "0VKKP8X6Q6"
    },
    {
      "field": "uses",
      "target": "0VXYV3NQQ1"
    },
    {
      "field": "uses",
      "target": "0XQ2ZVN1PZ"
    },
    {
      "field": "uses",
      "target": "0ZC78099PA"
    }
  ],
  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f \\colon U \\to \\mathbb{R}$, and let $a \\in U$ be a point at which $f$ is [[differentiable|0135XZZQ8M]].\nThe gradient of $f$ at $a$ is the unique vector $\\nabla f(a) \\in \\mathbb{R}^n$ such that\n$$Df(a)(v) = \\nabla f(a) \\cdot v \\quad \\text{for each } v \\in \\mathbb{R}^n,$$\nwhere $\\cdot$ is the dot product of [[Euclidean space|0XPR1E85W0]].\nIn coordinates, it is the vector of [[partial derivatives|09EGC6F8PT]]\n$$\\nabla f(a) = \\left(\\frac{\\partial f}{\\partial x_1}(a), \\dots, \\frac{\\partial f}{\\partial x_n}(a)\\right).$$\nOther notations are $\\operatorname{grad} f(a)$ and $\\nabla f(a)$, where $\\nabla$ is the [[del operator|0H6DP8CHND]].\n\n## Remarks\n\nThe vector exists and is unique because each linear map $\\mathbb{R}^n \\to \\mathbb{R}$ is $v \\mapsto w \\cdot v$ for exactly one $w \\in \\mathbb{R}^n$.\nFor $f(x, y) = x^2 + 3 x y$, the gradient is $\\nabla f(x, y) = (2x + 3y, \\, 3x)$.\nA gradient is a column vector, and the derivative $Df(a)$ is the row vector of the same entries. Some books do not make this distinction.\nThe map $a \\mapsto \\nabla f(a)$ is a [[vector field|0299VCW1TH]] on $U$, the gradient field of $f$.\n\n## Explanations\n\n### L1 human\n\nThe gradient collects all slopes of $f$ along the axes into one arrow.\nOn a map of a hill, the gradient at a point is an arrow that points straight uphill. The arrow is longer where the hill is steeper.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "0ABK9V9Y64",
    "kind": "definition",
    "msc": [
      "26B12"
    ],
    "name": "Gradient",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Gradient"
      },
      {
        "locator": "Cartesian coordinates",
        "url": "https://en.wikipedia.org/wiki/Gradient"
      }
    ],
    "tag": "0NS2QYVC4Y",
    "uses": [
      "0135XZZQ8M",
      "09EGC6F8PT",
      "0XPR1E85W0"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0NS2QYVC4Y/",
  "links": [
    {
      "field": "defines",
      "target": "0ABK9V9Y64"
    },
    {
      "field": "uses",
      "target": "0135XZZQ8M"
    },
    {
      "field": "uses",
      "target": "09EGC6F8PT"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0NS2QYVC4Y.md",
  "msc": [
    "26B12"
  ],
  "name": "Gradient",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0NS/0NS2QYVC4Y.md",
  "status": null,
  "tag": "0NS2QYVC4Y",
  "uri": "m2://0NS2QYVC4Y",
  "url": "https://math2.tamnd.com/wiki/gradient-definition/",
  "writing": {
    "tag": "0NS2QYVC4Y",
    "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
      }
    ]
  }
}