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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $m \\le n$, and let $F \\colon U \\to \\mathbb{R}^m$ be of class $C^1$.\nLet $p \\in U$ with $F(p) = 0$, and suppose that $DF(p) \\colon \\mathbb{R}^n \\to \\mathbb{R}^m$ is surjective.\nThen there are $m$ coordinates $x_{i_1}, \\dots, x_{i_m}$ and an open neighbourhood $W$ of $p$ with this property. The set $\\{x \\in W : F(x) = 0\\}$ is the graph of a $C^1$ function of the other $n - m$ coordinates.\n\n## Remarks\n\nThe derivative is surjective if and only if the [[Jacobian matrix|0YF0VXKXGB]] $J_F(p)$ has rank $m$. Then some $m$ columns of $J_F(p)$ are linearly independent, and these columns choose the coordinates.\nAfter a renumbering of coordinates, the statement is the [[implicit function theorem|0BXBADR94C]] with $y = (x_{i_1}, \\dots, x_{i_m})$.\nIf $DF(x)$ is surjective at each point of the [[level set|0Y5Q8Q9WJ7]] $F^{-1}(0)$, this set is locally a graph near each of its points. So it is a $C^1$ submanifold of dimension $n - m$.\nExample: for $F(x, y, z) = x^2 + y^2 + z^2 - 1$, $DF$ is surjective on the unit sphere. So the sphere is locally the graph of a function of two coordinates.\n\n## Explanations\n\n### L1 human\n\nThe condition that the derivative is onto means the $m$ equations are independent at the point.\nThen $m$ of the variables are determined by the others, and the solution set has $n - m$ free directions.\n",
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