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) |
Comments
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 |
Integral surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_surface&oldid=33518