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  "body": "\n## Proof\n\nWe show that a Jacobian matrix is the matrix of the derivative, and then we use that matrices multiply like compositions.\n\n1. Let $p, q \\ge 1$, let $V \\subseteq \\mathbb{R}^p$ be open, let $\\sigma \\colon V \\to \\mathbb{R}^q$, and let $c \\in V$. Suppose that $\\sigma$ is differentiable at $c$. Let $e_1, \\dots, e_p$ be the standard basis of $\\mathbb{R}^p$.\n2. Since [[the derivative gives the directional derivatives|0A6PMR20N4]], $\\big(\\sigma(c + t e_j) - \\sigma(c)\\big) / t \\to D\\sigma(c)(e_j)$ as $t \\to 0$, for each $j$.\n3. For each $v \\in \\mathbb{R}^q$, the entry $v_i$ satisfies $|v_i| \\le \\|v\\|$. So the $i$-th entries in step 2 converge as well: $\\big(\\sigma_i(c + t e_j) - \\sigma_i(c)\\big) / t \\to \\big(D\\sigma(c)(e_j)\\big)_i$.\n4. By the definition of the [[partial derivative|05AXG3QRKR]], step 3 says that $\\partial \\sigma_i / \\partial x_j (c)$ exists and equals $\\big(D\\sigma(c)(e_j)\\big)_i$.\n5. So the [[Jacobian matrix|0YF0VXKXGB]] $J_\\sigma(c)$ is defined. Its column $j$ is the vector $D\\sigma(c)(e_j)$. Thus $J_\\sigma(c)$ is the [[matrix|0YYMY95FWS]] of $D\\sigma(c)$ in the standard bases.\n6. Now let $U$, $W$, $g$, $f$ and $a$ be as in the statement. By the [[chain rule in several variables|0EAAS2EHDR]], $f \\circ g$ is differentiable at $a$, and $D(f \\circ g)(a) = Df(g(a)) \\circ Dg(a)$.\n7. By step 5 applied to $f \\circ g$ at $a$, to $f$ at $g(a)$ and to $g$ at $a$, the three Jacobian matrices are defined. They are the matrices of $D(f \\circ g)(a)$, $Df(g(a))$ and $Dg(a)$ in the standard bases.\n8. Since the [[matrix of a composition is the product of the matrices|05Q1B7MH0M]], step 6 gives $J_{f \\circ g}(a) = J_f(g(a)) \\, J_g(a)$.\n9. Here $J_f(g(a))$ is a $k \\times m$ matrix and $J_g(a)$ is an $m \\times n$ matrix. Indeed $f$ maps $W \\subseteq \\mathbb{R}^m$ into $\\mathbb{R}^k$, and $g$ maps $U \\subseteq \\mathbb{R}^n$ into $\\mathbb{R}^m$.\n\n## Idea\n\nThe Jacobian matrix of a differentiable map is the matrix of its derivative, because the partial derivatives are the values of the derivative on the basis vectors. The chain rule says that the derivative of a composition is a composition of linear maps. Matrices of linear maps multiply in the same order as the maps compose.\n",
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