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  "body": "\n## Definition\n\nLet $\\Omega$ be an open set in $\\R^n$ and consider a [[linear PDE|6DZETZ5T9G]] of order $2$ in $\\Omega$.\n$$\\sum_{i,j=1}^{n} a_{ij}(x) \\, \\partial_i \\partial_j u + \\sum_{i=1}^{n} b_i(x) \\, \\partial_i u + c(x) \\, u = f(x).$$\nIts coefficients are real and $a_{ij} = a_{ji}$, so that $A(x) = (a_{ij}(x))$ is a real [[symmetric matrix|0VR671PHE5]].\nAt a point $x \\in \\Omega$ the equation is called:\n\n1. *elliptic* if all [[eigenvalues|0BQAA2BK3Q]] of $A(x)$ are nonzero and have the same sign;\n2. *hyperbolic* if all eigenvalues of $A(x)$ are nonzero, and exactly one of them has a sign opposite to the others;\n3. *ultrahyperbolic* if all eigenvalues are nonzero, and at least two are positive and at least two are negative;\n4. *parabolic* if exactly one eigenvalue of $A(x)$ is zero and the others are nonzero and have the same sign.\n\nThe equation has one of these types in $\\Omega$ if it has that type at every point of $\\Omega$.\n\n## Remarks\n\nThe quadratic form $\\xi \\mapsto \\sum a_{ij}(x) \\xi_i \\xi_j$ is the [[principal symbol|F31T650DKY]] at $x$, so the type depends only on the second-order part.\nIn two variables, write the second-order part as $A u_{xx} + 2B u_{xy} + C u_{yy}$.\nThe equation is elliptic where $AC - B^2 \u003e 0$, hyperbolic where $AC - B^2 \u003c 0$, and parabolic where $AC - B^2 = 0$ and $(A, B, C) \\ne (0, 0, 0)$.\nExample: the [[Tricomi equation|21DD2334NR]] $u_{yy} + y \\, u_{xx} = 0$ has $AC - B^2 = y$, so it is elliptic for $y \u003e 0$, hyperbolic for $y \u003c 0$ and parabolic on the line $y = 0$.\nNon-example: for $n = 3$, a matrix $A(x)$ with eigenvalues $1, 0, 0$ fits none of the four clauses; such points are degenerate.\nThe type is invariant under a smooth change of variables, by [[Sylvester's law of inertia|0WFED3K3DQ]] applied to the transformed coefficient matrix.\n\n## Explanations\n\n### L1 human\n\nThe type of a second-order equation is read off from the signs of the eigenvalues of its matrix of second-order coefficients.\n",
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