Discussion
The partial differential equation :
\(f(x, y, z, p, q) = 0\) and
\(g(x, y, z, p, q) = 0\)
are compatible on a domain D if :
Answer: B
The equations are compatible on D if \(\dfrac{\partial(f, g)}{\partial(p, q)} \neq 0\) and \(dz = p\,dx + q\,dy\) is integrable.
Compatibility means the two equations have a common one-parameter family of solutions. The Jacobian condition \(\partial(f, g)/\partial(p, q) \neq 0\) is what lets us solve \(f = 0, g = 0\) for \(p\) and \(q\) as functions \(p = \phi(x, y, z)\), \(q = \psi(x, y, z)\) by the implicit function theorem. The common solutions are then the integrals of \(dz = \phi\,dx + \psi\,dy\), which exist when this Pfaffian equation is integrable; the integrability condition reduces to \([f, g] = 0\) in Jacobi bracket form.
The Jacobians with respect to \((x, y)\), \((p, x)\) or \((q, y)\) do not let us solve for \(p\) and \(q\), so they are not the right condition.
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