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− | The surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i0516701.png" />-dimensional space defined by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i0516702.png" />, where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i0516703.png" /> is a solution of a partial differential equation. For example, consider the linear homogeneous first-order equation | + | {{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 |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i0516704.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
| + | $$X_1\frac{\partial u}{\partial x_1}+\ldots+X_n\frac{\partial u}{\partial x_n}=0.\tag{*}$$ |
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− | Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i0516705.png" /> is the unknown and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i0516706.png" /> are given functions of the arguments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i0516707.png" />. Suppose that in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i0516708.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i0516709.png" />-dimensional space the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i05167010.png" /> are continuously differentiable and do not vanish simultaneously, and suppose that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i05167011.png" /> are functionally independent first integrals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i05167012.png" /> of the system of ordinary differential equations in symmetric form | + | 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 |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i05167013.png" /></td> </tr></table>
| + | $$\frac{dx_1}{X_1}=\ldots=\frac{dx_n}{X_n}.$$ |
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− | Then the equation of every integral surface of (*) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i05167014.png" /> can be expressed in the form | + | Then the equation of every integral surface of \ref{*} in $G$ can be expressed in the form |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i05167015.png" /></td> </tr></table>
| + | $$u=\Phi(\phi_1,\dots,\phi_{n-1}),$$ |
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− | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i05167016.png" /> is a continuously-differentiable function. For a [[Pfaffian equation|Pfaffian equation]] | + | where $\Phi$ is a continuously-differentiable function. For a [[Pfaffian equation|Pfaffian equation]] |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i05167017.png" /></td> </tr></table>
| + | $$P(x,y,z)dx+Q(x,y,z)dy+R(x,y,z)dz=0,$$ |
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− | which is completely integrable in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i05167018.png" /> of three-dimensional space and does not have any singular points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i05167019.png" />, each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051670/i05167020.png" /> is contained in an integral surface. These integral surfaces never intersect nor are they tangent to one another at any point. | + | 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. |
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| ====References==== | | ====References==== |
Revision as of 19:11, 9 October 2014
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
$$X_1\frac{\partial u}{\partial x_1}+\ldots+X_n\frac{\partial u}{\partial x_n}=0.\tag{*}$$
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
$$\frac{dx_1}{X_1}=\ldots=\frac{dx_n}{X_n}.$$
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
$$P(x,y,z)dx+Q(x,y,z)dy+R(x,y,z)dz=0,$$
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) |
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 |
How to Cite This Entry:
Integral surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_surface&oldid=19162
This article was adapted from an original article by N.N. Ladis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article