Complete integral

Remarks

Clause 3 says that $u$ depends on all $n$ parameters independently, so that no parameter can be removed by a change of the others.

Example: 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 $a \in \R^n$. Check: $\nabla u = a$, so $x \cdot a + f(a) = u$, and clause 2 holds. For clause 3, $D^2_{xa} u$ is the identity matrix, so the matrix has rank $n$.

Non-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. Indeed both rows of the matrix equal $(1, 0, 0)$, so its rank is $1$; the family really has only one parameter $a_1 + a_2$. By 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 $(x_2, 0, 1)$.

Envelopes 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.