Namespaces
Variants
Actions

Difference between revisions of "Interior differential operator"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
''with respect to a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i0518201.png" />''
+
{{TEX|done}}
 +
''with respect to a surface $\Sigma_m$''
  
A [[Differential operator|differential operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i0518202.png" /> such that for any function for which it is defined its value at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i0518203.png" /> can be calculated from only the values of this function on the smooth surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i0518204.png" /> defined in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i0518205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i0518206.png" />. An interior differential operator can be computed using derivatives in directions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i0518207.png" /> which lie in the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i0518208.png" />. If one introduces coordinates such that on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i0518209.png" />,
+
A [[Differential operator|differential operator]] $L(u)$ such that for any function for which it is defined its value at a point $M\in\Sigma_m$ can be calculated from only the values of this function on the smooth surface $\Sigma_m$ defined in the space $E^n$, $m<n$. An interior differential operator can be computed using derivatives in directions $l$ which lie in the tangent space to $\Sigma_m$. If one introduces coordinates such that on $\Sigma_m$,
  
<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/i051820/i05182010.png" /></td> </tr></table>
+
$$x_{m+1}=x_{m+1}^0,\dots,x_n=x_n^0,$$
  
then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i05182011.png" />, provided it is interior with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i05182012.png" />, will not contain, after suitable transformations, derivatives with respect to the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i05182013.png" /> (the so-called exterior or extrinsic derivatives). For instance, the operator
+
then the operator $L(u)$, provided it is interior with respect to $\Sigma_m$, will not contain, after suitable transformations, derivatives with respect to the variables $x_{m+1},\dots,x_n$ (the so-called exterior or extrinsic derivatives). For instance, the operator
  
<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/i051820/i05182014.png" /></td> </tr></table>
+
$$L(u)=\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}$$
  
is an interior differential operator with respect to any smooth surface containing a straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i05182015.png" />, and with respect to any one of these lines. If the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i05182016.png" /> is an interior differential operator with respect to a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i05182017.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i05182018.png" /> is said to be a [[Characteristic|characteristic]] of the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i05182019.png" />.
+
is an interior differential operator with respect to any smooth surface containing a straight line $x-x_0=y-y_0=z-z_0$, and with respect to any one of these lines. If the operator $L(u)$ is an interior differential operator with respect to a surface $\Sigma_{n-1}$, then $\Sigma_{n-1}$ is said to be a [[Characteristic|characteristic]] of the differential equation $L(u)=0$.
  
An operator is sometimes called interior with respect to a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051820/i05182020.png" /> if, at the points of this surface, the leading order of the extrinsic derivatives is lower than the order of the operator.
+
An operator is sometimes called interior with respect to a surface $\Sigma_m$ if, at the points of this surface, the leading order of the extrinsic derivatives is lower than the order of the operator.

Latest revision as of 12:55, 4 September 2014

with respect to a surface $\Sigma_m$

A differential operator $L(u)$ such that for any function for which it is defined its value at a point $M\in\Sigma_m$ can be calculated from only the values of this function on the smooth surface $\Sigma_m$ defined in the space $E^n$, $m<n$. An interior differential operator can be computed using derivatives in directions $l$ which lie in the tangent space to $\Sigma_m$. If one introduces coordinates such that on $\Sigma_m$,

$$x_{m+1}=x_{m+1}^0,\dots,x_n=x_n^0,$$

then the operator $L(u)$, provided it is interior with respect to $\Sigma_m$, will not contain, after suitable transformations, derivatives with respect to the variables $x_{m+1},\dots,x_n$ (the so-called exterior or extrinsic derivatives). For instance, the operator

$$L(u)=\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}$$

is an interior differential operator with respect to any smooth surface containing a straight line $x-x_0=y-y_0=z-z_0$, and with respect to any one of these lines. If the operator $L(u)$ is an interior differential operator with respect to a surface $\Sigma_{n-1}$, then $\Sigma_{n-1}$ is said to be a characteristic of the differential equation $L(u)=0$.

An operator is sometimes called interior with respect to a surface $\Sigma_m$ if, at the points of this surface, the leading order of the extrinsic derivatives is lower than the order of the operator.

How to Cite This Entry:
Interior differential operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interior_differential_operator&oldid=33241
This article was adapted from an original article by B.L. Rozhdestvenskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article