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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f = (f_1, \\dots, f_m) \\colon U \\to \\mathbb{R}^m$, and let $a \\in U$.\nSuppose that each [[partial derivative|09EGC6F8PT]] $\\partial f_i / \\partial x_j$ exists at $a$, for $1 \\le i \\le m$ and $1 \\le j \\le n$.\nThe Jacobian matrix of $f$ at $a$ is the $m \\times n$ matrix\n$$J_f(a) = \\begin{pmatrix} \\dfrac{\\partial f_1}{\\partial x_1}(a) \u0026 \\cdots \u0026 \\dfrac{\\partial f_1}{\\partial x_n}(a) \\\\ \\vdots \u0026 \\ddots \u0026 \\vdots \\\\ \\dfrac{\\partial f_m}{\\partial x_1}(a) \u0026 \\cdots \u0026 \\dfrac{\\partial f_m}{\\partial x_n}(a) \\end{pmatrix},$$\nwhose entry in row $i$ and column $j$ is $\\partial f_i / \\partial x_j (a)$.\nOther notations are $Df(a)$, $f'(a)$ and $\\dfrac{\\partial (f_1, \\dots, f_m)}{\\partial (x_1, \\dots, x_n)}$.\n\n## Remarks\n\nIf $f$ is [[differentiable|0135XZZQ8M]] at $a$, then $Df(a)(h) = J_f(a) \\, h$ for each column vector $h \\in \\mathbb{R}^n$.\nRow $i$ of $J_f(a)$ is the transpose of the [[gradient|0ABK9V9Y64]] of $f_i$ at $a$. For $m = 1$, the Jacobian matrix is the gradient written as a row.\nFor the polar map $f(r, \\theta) = (r \\cos \\theta, r \\sin \\theta)$, the Jacobian matrix is $\\begin{pmatrix} \\cos \\theta \u0026 -r \\sin \\theta \\\\ \\sin \\theta \u0026 r \\cos \\theta \\end{pmatrix}$.\nFor $m = n$, its determinant is the [[Jacobian determinant|0ZCSMYHTBP]].\n\n## Explanations\n\n### L1 human\n\nThe Jacobian matrix lists how each output changes with each input: one row for each output, one column for each input.\nIt is the multivariable form of the derivative. It turns a small step $h$ in the input into the approximate step $J_f(a) h$ in the output.\n",
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