View source for Integral surface

{{TEX|done}}
The surface in $(n+1)$-dimensional space defined by an equation $u=\phi(x_1,\dots,x_n)$, where the function $u=\phi(x_1,\dots,x_n)$ is a solution of a partial differential equation. For example, consider the linear homogeneous first-order equation

$$X_1\frac{\partial u}{\partial x_1}+\dotsb+X_n\frac{\partial u}{\partial x_n}=0.\tag{*}$$

Here $u$ is the unknown and $X_1,\dots,X_n$ are given functions of the arguments $x_1,\dots,x_n$. Suppose that in some domain $G$ of $n$-dimensional space the functions $X_1,\dots,X_n$ are continuously differentiable and do not vanish simultaneously, and suppose that the functions $\phi_1(x_1,\dots,x_n),\dots,\phi_{n-1}(x_1,\dots,x_n)$ are functionally independent first integrals in $G$ of the system of ordinary differential equations in symmetric form

$$\frac{dx_1}{X_1}=\dotsb=\frac{dx_n}{X_n}.$$

Then the equation of every integral surface of \ref{*} in $G$ can be expressed in the form

$$u=\Phi(\phi_1,\dots,\phi_{n-1}),$$

where $\Phi$ is a continuously-differentiable function. For a [[Pfaffian equation|Pfaffian equation]]

$$P(x,y,z)dx+Q(x,y,z)dy+R(x,y,z)dz=0,$$

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.W. [V.V. Stepanov] Stepanow,   "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Ince,   "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Survey of applicable mathematics'' , Iliffe  (1969)  pp. Sect. 18.7</TD></TR></table>