{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0ZC78099PA"
    }
  ],
  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be an open set, let $f \\colon U \\to \\mathbb{R}^m$, let $a \\in U$, and let $v \\in \\mathbb{R}^n$.\nThe directional derivative of $f$ at $a$ in the direction $v$ is the limit\n$$D_v f(a) = \\lim_{t \\to 0} \\frac{f(a + t v) - f(a)}{t},$$\nwhen this limit exists in $\\mathbb{R}^m$.\nOther notations for it are $\\partial_v f(a)$ and $\\nabla_v f(a)$.\n\n## Remarks\n\nFor $v = e_i$, the directional derivative is the [[partial derivative|09EGC6F8PT]] $\\partial f / \\partial x_i$ at $a$.\nSome books require that $v$ is a unit vector, so that $D_v f(a)$ is a rate of change per unit of distance.\nIf $D_v f(a)$ exists and $c \\in \\mathbb{R}$, then $D_{cv} f(a)$ exists and equals $c \\, D_v f(a)$.\nWhen $f$ is [[differentiable|0HHHDCXZV4]] at $a$, the directional derivatives are the values of the derivative: $D_v f(a) = Df(a)(v)$, see [[this statement|0A6PMR20N4]].\n\n## Explanations\n\n### L1 human\n\nStand at the point $a$ on the graph of $f$ and walk in the direction $v$.\nThe directional derivative is the slope that you feel at the first step.\nFor $f(x, y) = x^2 + y^2$ at $a = (1, 0)$ and $v = (0, 1)$, you get $f(1, t) = 1 + t^2$, so the slope is $0$. You start along the tangent of the circle $x^2 + y^2 = 1$, where $f$ is constant.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "06KWGHDN2J",
    "kind": "definition",
    "msc": [
      "26B05"
    ],
    "name": "Directional derivative",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Directional_derivative"
      }
    ],
    "tag": "0Y7PVW4Q44",
    "uses": [
      "0XPR1E85W0",
      "09EGC6F8PT"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0Y7PVW4Q44/",
  "links": [
    {
      "field": "defines",
      "target": "06KWGHDN2J"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    },
    {
      "field": "uses",
      "target": "09EGC6F8PT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0Y7PVW4Q44.md",
  "msc": [
    "26B05"
  ],
  "name": "Directional derivative",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0Y7/0Y7PVW4Q44.md",
  "status": null,
  "tag": "0Y7PVW4Q44",
  "uri": "m2://0Y7PVW4Q44",
  "url": "https://math2.tamnd.com/wiki/directional-derivative-definition/",
  "writing": {
    "tag": "0Y7PVW4Q44",
    "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
      }
    ]
  }
}