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  "body": "\n## Definition\n\nLet $t_{\\min} \\le t_0 \\le t_{\\max}$ be real numbers, let $J = [t_{\\min}, t_{\\max}]$, let $x_0 \\in \\mathbb{R}^n$, and let $a, L, K \\ge 0$.\nLet $B = \\{x \\in \\mathbb{R}^n : \\|x - x_0\\| \\le a\\}$, and let $f : J \\times \\mathbb{R}^n \\to \\mathbb{R}^n$ be a function.\nThe function $f$ satisfies the Picard–Lindelöf hypotheses at $(t_0, x_0)$ with constants $a, L, K$ if these four conditions hold.\n(1) For each $t \\in J$, the map $x \\mapsto f(t, x)$ is $K$-Lipschitz on $B$ ([[Lipschitz condition|055VVSN00V]]).\n(2) For each $x \\in B$, the map $t \\mapsto f(t, x)$ is continuous on $J$.\n(3) For each $t \\in J$ and $x \\in B$, $\\|f(t, x)\\| \\le L$.\n(4) $L \\cdot \\max(t_{\\max} - t_0,\\ t_0 - t_{\\min}) \\le a$.\n\n## Remarks\n\nConditions (1) and (2) together imply that $f$ is continuous on $J \\times B$.\nCondition (4) makes the time interval short enough. A curve that starts at $x_0$ with speed at most $L$ cannot leave $B$ during the time interval $J$.\nUnder these hypotheses, the [[Picard–Lindelöf theorem|0P6KRD5B7A]] gives a unique solution of $x' = f(t, x)$, $x(t_0) = x_0$ on $J$.\nA [[$C^1$ vector field satisfies these hypotheses|03Z9V0Y78A]] on a small time interval.\n\n## Explanations\n\n### L1 human\n\n\nThese are the conditions that make the proof of existence and uniqueness work.\nThe field must be continuous in time, Lipschitz in space and bounded near the start point.\nThe time interval must be so short that the solution stays in the region where we control the field.\n",
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  "name": "Picard–Lindelöf hypotheses",
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