# Integral surface

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

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

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 \eqref{*} 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

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

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

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