{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f \\colon U \\to \\mathbb{R}$, let $a \\in U$, and let $g \\in \\mathbb{R}^n$.\nThe function $f$ has the gradient $g$ at $a$ when\n$$\\lim_{h \\to 0} \\frac{f(a + h) - f(a) - g \\cdot h}{\\|h\\|} = 0,$$\nwhere the limit is over $h \\ne 0$ with $a + h \\in U$.\nEquivalently, $f$ is [[differentiable|0135XZZQ8M]] at $a$ and $Df(a)(h) = g \\cdot h$ for each $h \\in \\mathbb{R}^n$.\n\n## Remarks\n\nThis definition states a property of $f$, $a$ and $g$. When it holds, the vector $g$ is unique, and it is the [[gradient|0NS2QYVC4Y]] $\\nabla f(a)$.\nThe same definition works in each real inner product space, with the inner product in place of the dot product.\n\n## Explanations\n\n### L1 human\n\nThe function has the gradient $g$ at $a$ when the plane through $(a, f(a))$ with slopes given by $g$ hugs the graph of $f$ near $a$.\nThe error of this approximation must be small compared with the distance from $a$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0ABK9V9Y64",
    "kind": "definition",
    "msc": [
      "26B12"
    ],
    "name": "Having a gradient",
    "sources": [
      {
        "locator": "Relationship with derivative",
        "url": "https://en.wikipedia.org/wiki/Gradient"
      }
    ],
    "tag": "0BXE0S1XER",
    "uses": [
      "0135XZZQ8M",
      "0XPR1E85W0"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0BXE0S1XER/",
  "links": [
    {
      "field": "defines",
      "target": "0ABK9V9Y64"
    },
    {
      "field": "uses",
      "target": "0135XZZQ8M"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0BXE0S1XER.md",
  "msc": [
    "26B12"
  ],
  "name": "Having a gradient",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0BX/0BXE0S1XER.md",
  "status": null,
  "tag": "0BXE0S1XER",
  "uri": "m2://0BXE0S1XER",
  "url": "https://math2.tamnd.com/wiki/having-a-gradient/",
  "writing": {
    "tag": "0BXE0S1XER",
    "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
      }
    ]
  }
}