Let P,dx+Q,dy+R,dz=0 be a Pfaffian differential equation and consider: (I) Every Pfaffian differential equation is integrable. (II) It is integrable iff bar Xcdotcurlbar X=0, where bar X=(P,Q,R). (III) It is integrable iff P(partial Q/partial z-partial R/partial y)+Q(partial R/partial x-partial P/partial z)+R(partial P/partial y-partial Q/partial x)=0. Then:

Q72Section II (Mathematics)72 of 180

PDE — integrability of Pfaffian equations

Let $P\,dx+Q\,dy+R\,dz=0$ be a Pfaffian differential equation and consider: (I) Every Pfaffian differential equation is integrable. (II) It is integrable iff $\bar X\cdot\operatorname{curl}\bar X=0$ , where $\bar X=(P,Q,R)$ . (III) It is integrable iff $P\bigl(\frac{\partial Q}{\partial z}-\frac{\partial R}{\partial y}\bigr)+Q\bigl(\frac{\partial R}{\partial x}-\frac{\partial P}{\partial z}\bigr)+R\bigl(\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}\bigr)=0$ . Then:

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