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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^n$ and $A \\subseteq \\R^n$ be open sets, and let $F \\colon \\R^n \\times \\R \\times U \\to \\R$ be a function.\nConsider the first-order [[partial differential equation|GC5BDZ6QBC]] $F(\\nabla u, u, x) = 0$ for $u \\colon U \\to \\R$.\nA *complete integral* of this equation is a function $u \\colon U \\times A \\to \\R$, written $u(x; a)$, such that:\n\n1. $u$ is $C^2$ in $(x, a)$;\n2. for each $a \\in A$, the function $x \\mapsto u(x; a)$ solves the equation on $U$;\n3. for all $(x, a) \\in U \\times A$, the $n \\times (n + 1)$ matrix\n$$\\big( D_a u \\;\\big|\\; D^2_{xa} u \\big) = \\begin{pmatrix} u_{a_1} \u0026 u_{x_1 a_1} \u0026 \\cdots \u0026 u_{x_n a_1} \\\\ \\vdots \u0026 \\vdots \u0026 \u0026 \\vdots \\\\ u_{a_n} \u0026 u_{x_1 a_n} \u0026 \\cdots \u0026 u_{x_n a_n} \\end{pmatrix}$$\nof [[partial derivatives|09EGC6F8PT]] has rank $n$.\n\n## Remarks\n\nClause 3 says that $u$ depends on all $n$ parameters independently, so that no parameter can be removed by a change of the others.\n\nExample: Clairaut's equation $x \\cdot \\nabla u + f(\\nabla u) = u$, with $f \\colon \\R^n \\to \\R$ of class $C^2$, has the complete integral $u(x; a) = a \\cdot x + f(a)$ for $a \\in \\R^n$.\nCheck: $\\nabla u = a$, so $x \\cdot a + f(a) = u$, and clause 2 holds.\nFor clause 3, $D^2_{xa} u$ is the identity matrix, so the matrix has rank $n$.\n\nNon-example: for the equation $\\partial_1 u = 0$ on $\\R^2$, the family $u(x; a) = a_1 + a_2$ satisfies clauses 1 and 2 but not clause 3.\nIndeed both rows of the matrix equal $(1, 0, 0)$, so its rank is $1$; the family really has only one parameter $a_1 + a_2$.\nBy contrast, $u(x; a) = a_1 + a_2 x_2$ is a complete integral of the same equation: the rows are $(1, 0, 0)$ and $(x_2, 0, 1)$.\n\nEnvelopes of subfamilies of a complete integral give further solutions, called general integrals, and the envelope of the whole family, when it exists, is a singular integral.\n",
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