Discussion
The Pfaffian differential equation :
\[ \begin{aligned} \bar{X} \cdot d\bar{r} &= P(x, y, z)\, dx + Q(x, y, z)\, dy \\ &\quad + R(x, y, z)\, dz = 0 \end{aligned} \]
is integrable if :
Answer: B
The Pfaffian equation is integrable if \(\bar{X} \cdot \text{curl}\, \bar{X} = 0\).
The equation \(P\,dx + Q\,dy + R\,dz = 0\) is integrable, meaning it has a family of solution surfaces \(u(x, y, z) = c\), exactly when there is an integrating factor \(\mu\) with \(\mu\bar{X} = \nabla u\). Taking the curl of \(\mu\bar{X}\) and dotting with \(\bar{X}\) gives the necessary condition
\[ \bar{X} \cdot \text{curl}\, \bar{X} = 0 \]
and the converse is also true (it is the standard integrability theorem). In components it reads \(P(Q_{z} - R_{y}) + Q(R_{x} - P_{z}) + R(P_{y} - Q_{x}) = 0\).
\(\bar{X} \cdot \text{curl}\, \bar{X} \ne 0\) is the condition for non-integrability. \(\bar{X} \cdot \bar{X} = |\bar{X}|^{2}\) only measures the length of the field and says nothing about integrability.
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