Non-Integrability of a Pfaffian - Geometric Interpretation?

[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.

 

[jvicens]

Thank you for bringing up this interesting question about solving Pfaffian ODEs. I agree that the idea of finding a family of surfaces orthogonal to a given vector field is a useful way to approach the problem. However, as you mentioned, there are cases where the integrability condition may fail, yet there still exist families of surfaces orthogonal to the vector field.

One possible explanation for this is that the integrability condition is necessary, but not sufficient, for the existence of a family of surfaces orthogonal to the vector field. In other words, if the condition fails, we know for sure that there is no family of surfaces orthogonal to the vector field, but if it holds, there may still be other cases where a family of surfaces exists.

Another possibility is that the existence of a family of surfaces orthogonal to a vector field may not always be unique. In some cases, there may be multiple families of surfaces that are orthogonal to the vector field, even if the integrability condition holds. This could explain why there may still be families of surfaces orthogonal to the vector field in cases where the condition fails.

It is also worth noting that there are different methods for solving Pfaffian ODEs, as you mentioned. Each method may have its own limitations and assumptions, which could also impact the existence of families of surfaces orthogonal to the vector field.

In conclusion, while the integrability condition is an important factor to consider when solving Pfaffian ODEs, it may not always be the only determining factor for the existence of families of surfaces orthogonal to the vector field. Further investigation and research may be needed to fully understand these cases where the condition fails, yet families of surfaces still exist. Thank you for bringing up this thought-provoking question.