Namespaces
Variants
Actions

Difference between revisions of "Conormal"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (typo bold \nu = \pmb\nu)
 
Line 11: Line 11:
 
{{TEX|done}}
 
{{TEX|done}}
  
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} ) $
+
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 $ \pmb\nu = ( \nu _ {1} \dots \nu _ {n} ) $
 
be the outward normal at a point  $  x $
 
be the outward normal at a point  $  x $
 
to a smooth surface  $  S $
 
to a smooth surface  $  S $
Line 28: Line 28:
 
where  $  \nu  ^ {i} = g  ^ {ik} \nu _ {k} $.  
 
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} $)  
 
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 $
+
of the normal covariant vector  $  \pmb\nu $
to  $  S $(
+
to  $  S $ (in the space with Euclidean metric).
in the space with Euclidean metric).
 
  
 
====References====
 
====References====

Latest revision as of 17:17, 5 June 2020


A term used in the theory of boundary value problems for partial differential equations (cf. Boundary value problem, partial differential equations). Let $ \pmb\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 $ \pmb\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=46517
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article