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  "body": "\n## Statement\n\nLet $D \\subseteq \\R \\times \\R^n \\times \\R^k$ be open and let $f : D \\to \\R^n$, $(t, x, \\lambda) \\mapsto f(t, x, \\lambda)$, be continuous with continuous partial derivatives with respect to $x$ and $\\lambda$.\nFor $(t_0, x_0, \\lambda) \\in D$, let $t \\mapsto \\varphi(t; t_0, x_0, \\lambda)$ be the [[maximal solution|019VSXSFMQ]] of $x' = f(t, x, \\lambda)$ with $x(t_0) = x_0$.\nLet $\\Omega$ be the set of $(t, t_0, x_0, \\lambda)$ for which $\\varphi(t; t_0, x_0, \\lambda)$ is defined.\nThen:\n\n1. The set $\\Omega$ is open and $\\varphi$ is [[continuously differentiable|06SKCHDQT5]] on $\\Omega$.\n2. The matrix $J(t) = \\partial \\varphi / \\partial x_0$ solves the variational equation $J' = A(t)\\, J$ with $J(t_0) = I$, where $A(t) = \\partial_x f(t, \\varphi(t), \\lambda)$ is the [[Jacobian matrix|047NDTEPCV]] along the solution.\n3. The matrix $K(t) = \\partial \\varphi / \\partial \\lambda$ solves $K' = A(t)\\, K + \\partial_\\lambda f(t, \\varphi(t), \\lambda)$ with $K(t_0) = 0$.\n4. The derivative with respect to the initial time is $\\partial \\varphi / \\partial t_0 = -J(t)\\, f(t_0, x_0, \\lambda)$.\n\n## Remarks\n\nProof idea: the parameter $\\lambda$ can be absorbed into the state by adding the equation $\\lambda' = 0$, so it suffices to treat initial values.\nOne shows that the difference quotients $(\\varphi(t; t_0, x_0 + h e_i) - \\varphi(t; t_0, x_0))/h$ converge, using the mean value theorem and [[Grönwall's inequality|07EJHQYR9A]] to compare them with the solution of the variational equation.\nExample: for $x' = \\lambda x$, $\\varphi(t; 0, x_0, \\lambda) = x_0 e^{\\lambda t}$, and $K = \\partial \\varphi / \\partial \\lambda = t x_0 e^{\\lambda t}$ indeed solves $K' = \\lambda K + x_0 e^{\\lambda t}$ with $K(0) = 0$.\nIf $f$ is only Lipschitz in $x$, the solution depends [[continuously|0G3DBXRRQJ]] on the data but need not be differentiable.\nFor $x' = \\abs{x}$, the map $x_0 \\mapsto \\varphi(1; 0, x_0)$ is $e x_0$ for $x_0 \\ge 0$ and $e^{-1} x_0$ for $x_0 \u003c 0$, not differentiable at $0$.\nIf $f$ is of class $C^r$, then $\\varphi$ is of class $C^r$.\n\n## Explanations\n\n### L1 human\n\nIf the equation and its parameters change a little, the solution changes a little, and in a smooth way.\nThe theorem also tells how fast the solution reacts: the rates of change solve a linear equation along the solution.\nThis is how one measures the sensitivity of a model to its inputs.\n\n### L2 human\n\nThe variational equation is the linearization of the system along a given solution.\nIt is the basic tool for stability of periodic orbits, for Lyapunov exponents and for perturbation methods.\nThe result also shows that the flow of a $C^1$ vector field is a $C^1$ map, with derivative a [[fundamental matrix|0MQ24RBG93]] of the variational equation.\n",
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