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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\mathbb{R}^n \\times \\mathbb{R}^m$ be open, and let $F \\colon \\Omega \\to \\mathbb{R}^m$ be of [[class C^1|0TCBZMEYGN]], written $F(x, y)$ with $x \\in \\mathbb{R}^n$ and $y \\in \\mathbb{R}^m$.\nLet $(a, b) \\in \\Omega$ with $F(a, b) = 0$.\nSuppose that the $m \\times m$ matrix $D_y F(a, b) = \\left( \\partial F_i / \\partial y_j (a, b) \\right)_{i, j}$ is invertible.\nThen there are open sets $U \\subseteq \\mathbb{R}^n$ and $V \\subseteq \\mathbb{R}^m$ with $a \\in U$ and $b \\in V$, and a unique function $g \\colon U \\to V$ with the following property. For $(x, y) \\in U \\times V$, $F(x, y) = 0$ if and only if $y = g(x)$.\nThe function $g$ is of class $C^1$, and $Dg(x) = -\\big( D_y F(x, g(x)) \\big)^{-1} D_x F(x, g(x))$ for each $x \\in U$.\n\n## Remarks\n\nHere $D_x F$ is the $m \\times n$ matrix of partial derivatives with respect to $x$. Together, $D_x F$ and $D_y F$ form the [[Jacobian matrix|0YF0VXKXGB]] of $F$.\nFor $F(x, y) = x^2 + y^2 - 1$ and $(a, b)$ with $b \\ne 0$, the solution near $(a, b)$ is $y = \\pm \\sqrt{1 - x^2}$. At $(\\pm 1, 0)$, $\\partial F / \\partial y = 0$, and no such function exists.\nOne proof applies the [[inverse function theorem|0K9HSJFCMJ]] to $(x, y) \\mapsto (x, F(x, y))$. If $F$ is of class $C^k$, then $g$ is of class $C^k$.\nThe theorem has a long history. Cauchy gave the first rigorous form, and Dini extended the real version to any number of variables.\n\n## Explanations\n\n### L1 human\n\nAn equation $F(x, y) = 0$ often defines $y$ as a function of $x$, even when you cannot solve it with a formula.\nThe theorem says this works near a solution, provided $F$ really depends on $y$ there, in the sense that $D_y F$ is invertible.\n",
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