{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0A6PMR20N4"
    }
  ],
  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be an open set, let $f \\colon U \\to \\mathbb{R}^m$, let $a \\in U$, let $v \\in \\mathbb{R}^n$, and let $L \\in \\mathbb{R}^m$.\nLet $g(t) = f(a + t v)$ for real $t$ near $0$. The function $f$ has the directional derivative $L$ at $a$ in the direction $v$ when $g$ is differentiable at $t = 0$ with $g'(0) = L$.\nEquivalently, $\\dfrac{f(a + t v) - f(a)}{t} \\to L$ as $t \\to 0$.\n\n## Remarks\n\nThis definition states a property of $f$, $a$, $v$ and $L$. The [[directional derivative|0Y7PVW4Q44]] $D_v f(a)$ is then the vector $L$, which is unique.\nOnly the values of $f$ on the line through $a$ in the direction $v$ matter. So $f$ can have a directional derivative in each direction at $a$ and still fail to be continuous at $a$, see [[this counterexample|0TNZC55X3J]].\n\n## Explanations\n\n### L1 human\n\nThe function $g(t) = f(a + t v)$ is the function $f$ seen along the straight line through $a$ with direction $v$.\nTo have a directional derivative means that this function of one variable has a derivative at the start of the line.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "06KWGHDN2J",
    "kind": "definition",
    "msc": [
      "26B05"
    ],
    "name": "Having a directional derivative",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Directional_derivative"
      }
    ],
    "tag": "05EGFBZTGC",
    "uses": [
      "0XPR1E85W0"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/05EGFBZTGC/",
  "links": [
    {
      "field": "defines",
      "target": "06KWGHDN2J"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/05EGFBZTGC.md",
  "msc": [
    "26B05"
  ],
  "name": "Having a directional derivative",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/05E/05EGFBZTGC.md",
  "status": null,
  "tag": "05EGFBZTGC",
  "uri": "m2://05EGFBZTGC",
  "url": "https://math2.tamnd.com/wiki/having-a-directional-derivative/",
  "writing": {
    "tag": "05EGFBZTGC",
    "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
      }
    ]
  }
}