{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ and $W \\subseteq \\mathbb{R}^m$ be open, let $g \\colon U \\to W$, let $f \\colon W \\to \\mathbb{R}^k$, and let $a \\in U$.\nSuppose that $g$ is differentiable at $a$ and $f$ is differentiable at $g(a)$.\nThen the [[Jacobian matrices|0YF0VXKXGB]] satisfy\n$$J_{f \\circ g}(a) = J_f(g(a)) \\, J_g(a),$$\na product of a $k \\times m$ matrix and an $m \\times n$ matrix.\n\n## Remarks\n\nThis is the [[chain rule in several variables|0EAAS2EHDR]], written with matrices: the matrix of a composition of linear maps is the product of their matrices.\nFor $k = m = n$, taking determinants gives $\\det J_{f \\circ g}(a) = \\det J_f(g(a)) \\cdot \\det J_g(a)$.\nFor example, let $g(r, \\theta) = (r \\cos \\theta, r \\sin \\theta)$ and $f(x, y) = x^2 + y^2$. Then $J_f(g(r, \\theta)) J_g(r, \\theta) = (2r \\cos \\theta, \\, 2r \\sin \\theta) J_g(r, \\theta) = (2r, \\, 0)$, which is the Jacobian matrix of $(f \\circ g)(r, \\theta) = r^2$.\n\n## Explanations\n\n### L1 human\n\nEach Jacobian matrix turns small input steps into small output steps.\nDoing two maps one after the other means multiplying their matrices, in the same order as the maps act, from right to left.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "26B10"
    ],
    "name": "Chain rule for Jacobian matrices",
    "role": "corollary",
    "sources": [
      {
        "locator": "General rule: Vector-valued multivariate functions",
        "url": "https://en.wikipedia.org/wiki/Chain_rule"
      }
    ],
    "tag": "03GGQEAG9S",
    "uses": [
      "0EAAS2EHDR",
      "0YF0VXKXGB"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/03GGQEAG9S/",
  "links": [
    {
      "field": "uses",
      "target": "0EAAS2EHDR"
    },
    {
      "field": "uses",
      "target": "0YF0VXKXGB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/03GGQEAG9S.md",
  "msc": [
    "26B10"
  ],
  "name": "Chain rule for Jacobian matrices",
  "prov": null,
  "role": "corollary",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/03G/03GGQEAG9S.md",
  "status": null,
  "tag": "03GGQEAG9S",
  "uri": "m2://03GGQEAG9S",
  "url": "https://math2.tamnd.com/wiki/chain-rule-for-jacobian-matrices/",
  "writing": {
    "tag": "03GGQEAG9S",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}