Integral surface

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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


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


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


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.


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



[a1] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)
[a2] K. Rektorys (ed.) , Survey of applicable mathematics , Iliffe (1969) pp. Sect. 18.7
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Integral surface. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by N.N. Ladis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article