Integral surface

The surface in -dimensional space defined by an equation , where the function is a solution of a partial differential equation. For example, consider the linear homogeneous first-order equation

(*)

Here is the unknown and are given functions of the arguments . Suppose that in some domain of -dimensional space the functions are continuously differentiable and do not vanish simultaneously, and suppose that the functions are functionally independent first integrals in of the system of ordinary differential equations in symmetric form

Then the equation of every integral surface of (*) in can be expressed in the form

where is a continuously-differentiable function. For a Pfaffian equation

which is completely integrable in some domain of three-dimensional space and does not have any singular points in , each point of is contained in an integral surface. These integral surfaces never intersect nor are they tangent to one another at any point.

References

[1]	W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)

References

[a1]	E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)
[a2]	K. Rektorys (ed.) , Survey of applicable mathematics , Iliffe (1969) pp. Sect. 18.7