Namespaces
Variants
Actions

Difference between revisions of "Conormal"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A term used in the theory of boundary value problems for partial differential equations (cf. [[Boundary value problem, partial differential equations|Boundary value problem, partial differential equations]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025220/c0252201.png" /> be the outward normal at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025220/c0252202.png" /> to a smooth surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025220/c0252203.png" /> situated in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025220/c0252204.png" /> with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025220/c0252205.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025220/c0252206.png" /> be a contravariant continuous tensor, usually representing the coefficients of some second-order (elliptic) differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025220/c0252207.png" />. Then the conormal (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025220/c0252208.png" />) is the vector
+
<!--
 +
c0252201.png
 +
$#A+1 = 13 n = 0
 +
$#C+1 = 13 : ~/encyclopedia/old_files/data/C025/C.0205220 Conormal
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/c/c025/c025220/c0252209.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025220/c02522010.png" />. In other words, the conormal is the contravariant description (in the space with metric defined by the tensor inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025220/c02522011.png" />) of the normal covariant vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025220/c02522012.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025220/c02522013.png" /> (in the space with Euclidean metric).
+
A term used in the theory of boundary value problems for partial differential equations (cf. [[Boundary value problem, partial differential equations|Boundary value problem, partial differential equations]]). Let  $  bold \nu = ( \nu _ {1} \dots \nu _ {n} ) $
 +
be the outward normal at a point  $  x $
 +
to a smooth surface  $  S $
 +
situated in a Euclidean space  $  E  ^ {n} $
 +
with coordinates  $  x  ^ {1} \dots x  ^ {n} $,
 +
and let  $  g  ^ {ij} $
 +
be a contravariant continuous tensor, usually representing the coefficients of some second-order (elliptic) differential operator  $  D = g  ^ {ij} ( \partial  / \partial  x  ^ {i} ) ( \partial  / \partial  x  ^ {j} ) $.  
 +
Then the conormal (with respect to  $  D $)
 +
is the vector
 +
 
 +
$$
 +
\mathbf n  = \
 +
( \nu  ^ {1} \dots \nu  ^ {n} ),
 +
$$
 +
 
 +
where  $  \nu  ^ {i} = g  ^ {ik} \nu _ {k} $.  
 +
In other words, the conormal is the contravariant description (in the space with metric defined by the tensor inverse to $  g  ^ {ij} $)  
 +
of the normal covariant vector $  bold \nu $
 +
to $  S $(
 +
in the space with Euclidean metric).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Friedman,  "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Friedman,  "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR></table>

Revision as of 17:46, 4 June 2020


A term used in the theory of boundary value problems for partial differential equations (cf. Boundary value problem, partial differential equations). Let $ bold \nu = ( \nu _ {1} \dots \nu _ {n} ) $ be the outward normal at a point $ x $ to a smooth surface $ S $ situated in a Euclidean space $ E ^ {n} $ with coordinates $ x ^ {1} \dots x ^ {n} $, and let $ g ^ {ij} $ be a contravariant continuous tensor, usually representing the coefficients of some second-order (elliptic) differential operator $ D = g ^ {ij} ( \partial / \partial x ^ {i} ) ( \partial / \partial x ^ {j} ) $. Then the conormal (with respect to $ D $) is the vector

$$ \mathbf n = \ ( \nu ^ {1} \dots \nu ^ {n} ), $$

where $ \nu ^ {i} = g ^ {ik} \nu _ {k} $. In other words, the conormal is the contravariant description (in the space with metric defined by the tensor inverse to $ g ^ {ij} $) of the normal covariant vector $ bold \nu $ to $ S $( in the space with Euclidean metric).

References

[1] A.V. Bitsadze, "Equations of mathematical physics" , MIR (1980) (Translated from Russian)
[2] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)

Comments

References

[a1] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)
How to Cite This Entry:
Conormal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conormal&oldid=46480
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article