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  "body": "\n## Summary\n\nThe generator of a [[strongly continuous semigroup|EVF180QJXY]] $(T(t))_{t \\ge 0}$ of operators is the derivative of $T(t)$ at $t = 0$.\nIt is usually an unbounded, closed, densely defined operator $A$.\nThe semigroup solves the abstract Cauchy problem $u'(t) = Au(t)$, $u(0) = x$, through $u(t) = T(t)x$, and one writes $T(t) = e^{tA}$.\nThe generator determines the semigroup uniquely, so questions about evolution equations become questions about one operator.\n\n## Explanations\n\n### L1 human\n\nFor a number $a$, the function $t \\mapsto e^{ta}$ has derivative $a$ at $t = 0$.\nA semigroup is a family of operators that behaves like $e^{ta}$, and its generator plays the role of $a$.\nFor the heat equation, the semigroup moves an initial temperature forward in time, and the generator is the Laplacian.\nFor the shift $(T(t)f)(s) = f(s + t)$, the generator is the derivative $d/ds$.\nThe generator is often a differential operator, which is why it is defined only on a dense set of nice vectors.\n",
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