# Non-Integrability of a Pfaffian - Geometric Interpretation?

1. Jul 27, 2013

### bolbteppa

The question of Solving a Pfaffian ODE can be interpreted as the question of finding the family of surfaces $U = c$ perpendicular to a surface $f$ generated by the vector field

$$F(x,y,z) = (P(x,y,z),Q(x,y,z),R(x,y,z))$$

At each point, the gradient of the family of surfaces $U = c$ will either be parallel or equal to $F$.

Thus
$$\vec{\nabla}u = (∂u/∂x,∂u/∂y,∂u/∂z) = λ(P(x,y,z),Q(x,y,z),R(x,y,z))$$ should hold at each point.
This is just kind of basic & general from the geometry of the situation, & motivates the idea of a potential.

In standard terminology, if the field F is potential we have that
$$\vec{\nabla}u = (∂u/∂x,∂u/∂y,∂u/∂z) = (P(x,y,z),Q(x,y,z),R(x,y,z))$$
If the field is not potential we may be able to find an integrating factor such that
$$\vec{\nabla}u = (∂u/∂x,∂u/∂y,∂u/∂z) = μ(x,y,z)(P(x,y,z),Q(x,y,z),R(x,y,z))$$
In other words, the integrating factor is a function that, when evaluated at each point, merely acts as a scale factor so that $F$ living on a surface becomes equal to the gradient of the family of surfaces orthogonal to it. In this case we can indeed claim that a family of surfaces lies orthogonal to our vector field, & we can solve for it (using, by my last count, 8 different methods).

One can show that an integrability condition $<F, \vec{\nabla} \times F> = 0$ must hold in order for a pfaffian to be integrable. In other words, it seems as though the integrability condition must hold in order for there to exist a family of surfaces orthogonal to the surface in which the vector field lives, however according to my book even in the case when the integrability condition fails one can find families of surfaces orthogonal to the vector field, by eliminating some variable & reducing the pfaffian to an ode (the orthogonal surfaces in this case are cylinders).

What's going on here? How can the integrability condition fail yet there still exist families of surfaces (cylinders) orthogonal to the vector field when the whole point of the method is that they can only be found when the integrability condition holds? Thanks for reading.