{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe compare the Euclidean norm of a vector with the absolute values of its entries.\n\n1. Let $U$, $f$, $a$ and $L$ be as in the statement. The component $L_j$ is the map $h \\mapsto (L h)_j$. It is linear, since $L$ is linear and taking the $j$-th entry is linear.\n2. For $h$ with $a + h \\in U$, put $r(h) = f(a + h) - f(a) - L h \\in \\mathbb{R}^m$. Its entries are $r_j(h) = f_j(a + h) - f_j(a) - L_j h$.\n3. For each $v \\in \\mathbb{R}^m$ and each $j$, we have $|v_j| \\le \\|v\\|$, since $v_j^2 \\le v_1^2 + \\dots + v_m^2$.\n4. For each $v \\in \\mathbb{R}^m$, we have $\\|v\\| \\le |v_1| + \\dots + |v_m|$. Indeed, the square of the right side is $\\sum_j v_j^2$ plus nonnegative cross terms.\n5. Suppose that $f$ is [[differentiable|0135XZZQ8M]] at $a$ with $Df(a) = L$. So $\\|r(h)\\| / \\|h\\| \\to 0$ as $h \\to 0$.\n6. By step 3, $0 \\le |r_j(h)| / \\|h\\| \\le \\|r(h)\\| / \\|h\\|$ for each $h \\ne 0$. So $|r_j(h)| / \\|h\\| \\to 0$.\n7. On $\\mathbb{R}$, the Euclidean norm is the absolute value. So step 6 says that $f_j$ is differentiable at $a$ with $Df_j(a) = L_j$, for each $j$.\n8. Conversely, suppose that each $f_j$ is differentiable at $a$ with $Df_j(a) = L_j$. So $|r_j(h)| / \\|h\\| \\to 0$ for each $j$.\n9. By step 4, $0 \\le \\|r(h)\\| / \\|h\\| \\le \\sum_{j=1}^m |r_j(h)| / \\|h\\|$ for each $h \\ne 0$. The right side is a sum of $m$ terms that tend to $0$.\n10. So $\\|r(h)\\| / \\|h\\| \\to 0$, and $f$ is differentiable at $a$ with $Df(a) = L$.\n\n## Idea\n\nA vector in [[Euclidean space|0XPR1E85W0]] is small exactly when all its entries are small. The error of $f$ is the vector of the errors of its components. So $f$ is well approximated by $L$ exactly when each $f_j$ is well approximated by the matching row $L_j$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "26B05"
    ],
    "name": "Proof of the derivative of a map into R^n by components",
    "proves": "0NS0X75RS1",
    "sources": [
      {
        "locator": "Proposition 8.3.9",
        "url": "https://www.jirka.org/ra/realanal2.pdf"
      }
    ],
    "tag": "2N2EA37353",
    "uses": [
      "0135XZZQ8M",
      "0XPR1E85W0"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/2N2EA37353/",
  "links": [
    {
      "field": "proves",
      "target": "0NS0X75RS1"
    },
    {
      "field": "uses",
      "target": "0135XZZQ8M"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/2N2EA37353.md",
  "msc": [
    "26B05"
  ],
  "name": "Proof of the derivative of a map into R^n by components",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2N2/2N2EA37353.md",
  "status": null,
  "tag": "2N2EA37353",
  "uri": "m2://2N2EA37353",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-derivative-of-a-map-into-r-n-by-components/",
  "writing": {
    "tag": "2N2EA37353",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}