Pfaffian equation

It is a Pfaffian system generated by a single 1-form, ${ω}$ , in a manifold $M$ . Usually it is denoted by

ω \equiv 0

Example

x y d x + z d y + z^{2} y d z = 0

$◼$

Pfaff's problem is to determine its integral manifolds of maximal dimension (@bryant2013exterior page 15).

If the Pfaffian system is completely integrable then the 1-form is called Frobenius integrable. It is a weaker condition that being closed. That is, a 1-form is Frobenius integrable if there exists $μ$ such that $μ ω$ is closed. The function $μ$ is an integrating factor.
To solve a Pfaffian equation it is usually used a property of the wedge product.

Rank

The integer $r$ defined by

(d ω)^{r} \land ω \neq 0, (d ω)^{r + 1} \land ω = 0

is called the rank of the Pfaffian equation, or Pfaff rank of $ω$ . It depends on the point $x \in M$ . In the case of constant rank it can be applied the Pfaff-Darboux theorem.