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  "body": "\n## Definition\n\nLet $\\Omega$ be an [[open set|02BH3MCBQB]] in $\\R^n$ with $n \\ge 2$, and let $k \\ge 1$ be an integer.\nFor a function $u \\colon \\Omega \\to \\R$ of class $C^k$, write $D^j u(x)$ for the collection of all [[partial derivatives|09EGC6F8PT]] $\\partial^\\alpha u(x)$ of order $\\abs{\\alpha} = j$, in [[multi-index notation|0PJXBR6QA3]].\nLet $F$ be a given function of these derivatives and of $x$.\nA *partial differential equation* of order at most $k$ in $\\Omega$ is an equation of the following form.\n$$F\\left(D^k u(x), D^{k-1} u(x), \\dots, D u(x), u(x), x\\right) = 0 \\quad \\text{for } x \\in \\Omega.$$\nThe function $u$ is the unknown.\nA *classical solution* is a function $u \\in C^k(\\Omega)$ that satisfies the equation at every point of $\\Omega$.\nA *system* of partial differential equations is defined in the same way with $u$ and $F$ taking values in $\\R^N$ and $\\R^M$.\n\n## Remarks\n\nExample: with $n = 2$ and variables $(x, y)$, the equation $\\partial_x^2 u + \\partial_y^2 u = 0$ is a partial differential equation.\nThe function $u(x, y) = x^2 - y^2$ is a classical solution on $\\R^2$, since $\\partial_x^2 u = 2$ and $\\partial_y^2 u = -2$.\nNon-example: $u'(x) = u(x)$ for $u \\colon \\R \\to \\R$ involves derivatives with respect to one variable only, so it is an [[ordinary differential equation|0NT64JV96A]].\nComplex-valued unknowns are treated in the same way.\nMany PDEs also have solutions in a weaker sense, such as [[weak solutions|SJZX81A783]], which need not be of class $C^k$.\n\n## Explanations\n\n### L1 human\n\nA partial differential equation is a condition on a function of several variables and its partial derivatives, which must hold at every point.\nThe unknown is a whole function, not a number.\n",
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