Example of A Non-integrable System
Not every Pfaffian system is completely integrable in the Frobenius sense. For example, consider the following one-form on R3 - (0,0,0)
If dθ were in the ideal generated by θ we would have, by the skewness of the wedge product
But a direct calculation gives
which is a nonzero multiple of the standard volume form on R3. Therefore, there are no two-dimensional leaves, and the system is not completely integrable.
On the other hand, the curve defined by
is easily verified to be a solution (i.e. an integral curve) for the above Pfaffian system for any nonzero constant c.
Read more about this topic: Integrability Conditions For Differential Systems
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