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  "body": "\n## Definition\n\nLet $F\\left(D^k u, \\dots, D u, u, x\\right) = 0$ be a [[partial differential equation|GC5BDZ6QBC]] in an open set $\\Omega \\subseteq \\R^n$.\nHere $D^j u$ is the collection of the partial derivatives $\\partial^\\alpha u$ with $\\abs{\\alpha} = j$, in [[multi-index notation|0PJXBR6QA3]].\nIts *order* is the largest integer $m$ such that $F$ actually depends on some derivative $\\partial^\\alpha u$ with $\\abs{\\alpha} = m$.\nFor a system of equations, the order is the largest order of a derivative that occurs in some equation of the system.\n\n## Remarks\n\nExample: the [[minimal surface equation|3GV2E2YTZX]] in two variables involves $\\partial_x^2 u$, $\\partial_x \\partial_y u$ and $\\partial_y^2 u$ but no third derivatives, so it has order $2$.\nNon-example of a misreading: $(\\partial_x u)^2 + (\\partial_y u)^2 = 1$ has order $1$, not $2$, because the square is a power of a first derivative and not a second derivative.\nThis is the same notion as the [[order of a differential equation|0XK0G19SK8]], applied to equations in several variables.\nA mixed derivative $\\partial^\\alpha u$ counts with its total order $\\abs{\\alpha} = \\alpha_1 + \\dots + \\alpha_n$.\n\n## Explanations\n\n### L1 human\n\nThe order of a PDE is the largest total number of differentiations applied to the unknown in any term.\n",
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