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  "body": "\n## Summary\n\nA weak derivative of a locally integrable function is a function that obeys the integration by parts formula against all smooth compactly supported test functions.\nIt agrees with the usual [[partial derivative|09EGC6F8PT]] for continuously differentiable functions, but it also exists for many functions with corners.\nWeak derivatives are used to define [[Sobolev spaces|M4CEZ5K9HP]] and weak solutions of partial differential equations.\n\n## Explanations\n\n### L1 human\n\nFor a smooth function $u$ and a test function $\\varphi$ that vanishes near the ends of an interval, integration by parts gives $\\int u \\varphi' = -\\int u' \\varphi$.\nThe weak derivative turns this around: $v$ is called the derivative of $u$ if $\\int u \\varphi' = -\\int v \\varphi$ for every test function $\\varphi$.\nThe function $\\abs{x}$ has a corner at $0$ and no classical derivative there.\nIts weak derivative is the sign function, which is $-1$ for $x \u003c 0$ and $1$ for $x \u003e 0$.\nThe value at the single point $0$ does not matter, since weak derivatives are only defined up to sets of measure zero.\n",
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