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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^n$ be open, let $T \u003e 0$, and write the variables in $U \\times (0, T)$ as $(x, t)$.\nConsider a [[linear PDE|6DZETZ5T9G]] in $U \\times (0, T)$ of the following form.\n$$\\partial_t^2 u - \\sum_{i,j=1}^{n} a_{ij}(x, t) \\, \\partial_i \\partial_j u + \\sum_{i=1}^{n} b_i(x, t) \\, \\partial_i u + b_0(x, t) \\, \\partial_t u + c(x, t) \\, u = f(x, t).$$\nHere all coefficients are real and $a_{ij} = a_{ji}$.\nThe equation is a *second-order linear hyperbolic equation* if its spatial part is [[uniformly elliptic|B4TTNHSTRX]].\nThat is, there is $\\theta \u003e 0$ with $\\sum_{i,j} a_{ij}(x, t) \\xi_i \\xi_j \\ge \\theta \\abs{\\xi}^2$ for all $(x, t)$ and all $\\xi \\in \\R^n$.\n\nMore generally, let $P$ be a linear operator of order $m$ in the variables $(x, t)$, with [[principal symbol|F31T650DKY]] $\\sigma_m$.\nIt is *hyperbolic with respect to the time direction* at a point $y$ if both conditions hold:\n\n1. $\\sigma_m(y, (0, \\dots, 0, 1)) \\ne 0$;\n2. for every $\\xi \\in \\R^n$ the polynomial $\\tau \\mapsto \\sigma_m(y, (\\xi, \\tau))$ has only real roots.\n\nIt is *strictly hyperbolic* if, in addition, these $m$ roots are distinct for every $\\xi \\ne 0$.\n\n## Remarks\n\nExample: the [[wave equation|5EHD2XBQDG]] $\\partial_t^2 u - c^2 \\Delta u = 0$ with $c \u003e 0$ is hyperbolic with $a_{ij} = c^2 \\delta_{ij}$; its symbol $\\tau^2 - c^2 \\abs{\\xi}^2$ has the distinct real roots $\\tau = \\pm c \\abs{\\xi}$ for $\\xi \\ne 0$.\nNon-example: the equation $\\partial_t^2 u + \\Delta u = 0$ is not hyperbolic; its symbol $\\tau^2 + \\abs{\\xi}^2$ has the non-real roots $\\tau = \\pm i \\abs{\\xi}$, and the equation is elliptic.\nA first-order system $\\partial_t u + \\sum_j A_j(x, t) \\partial_j u = 0$ is called hyperbolic when every matrix $\\sum_j \\xi_j A_j$ with $\\xi \\in \\R^n$ has real eigenvalues and a basis of eigenvectors; see also [[strict hyperbolicity|P3VXR8FW17]].\n\n## Explanations\n\n### L1 human\n\nA hyperbolic equation is built like the wave equation: a second time derivative minus a uniformly elliptic operator in space.\n",
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