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m (tex encoded by computer)
m (fixing superscripts)
 
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{{TEX|done}}
 
{{TEX|done}}
  
''of a surface  $  \Phi  ^ {2} $
+
''of a surface  $  \Phi  ^ {2} $ in  $  3 $-dimensional Euclidean space  $  \mathbf R  ^ {3} $''
in  $  3 $-
 
dimensional Euclidean space  $  \mathbf R  ^ {3} $''
 
  
 
Half of the sum of the principal curvatures (cf. [[Principal curvature|Principal curvature]])  $  k _ {1} $
 
Half of the sum of the principal curvatures (cf. [[Principal curvature|Principal curvature]])  $  k _ {1} $
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For a hypersurface  $  \Phi  ^ {n} $
 
For a hypersurface  $  \Phi  ^ {n} $
in the Euclidean space  $  \mathbf R  ^ {n+} 1 $,  
+
in the Euclidean space  $  \mathbf R  ^ {n+1} $,  
 
this formula is generalized in the following way:
 
this formula is generalized in the following way:
  
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which is generalized for a hypersurface  $  \Phi  ^ {n} $
 
which is generalized for a hypersurface  $  \Phi  ^ {n} $
in  $  \mathbf R  ^ {n+} 1 $,  
+
in  $  \mathbf R  ^ {n+1} $,  
defined by the equation  $  x _ {n+} 1 = f( x _ {1} \dots x _ {n} ) $,  
+
defined by the equation  $  x _ {n+1} = f( x _ {1} \dots x _ {n} ) $,  
 
as follows:
 
as follows:
  
 
$$  
 
$$  
 
H ( A)  =   
 
H ( A)  =   
\frac{\sum _ { i= } 1 ^ { n }  \left ( 1 + p  ^ {2} - \left (  
+
\frac{\sum _ { i= 1} ^ { n }  \left ( 1 + p  ^ {2} - \left (  
 
\frac{\partial  
 
\frac{\partial  
 
f }{\partial  x _ {i} }
 
f }{\partial  x _ {i} }
 
  \right )  ^ {2} \right )  
 
  \right )  ^ {2} \right )  
 
\frac{\partial  ^ {2} f }{\partial  x _ {i}  ^ {2} }
 
\frac{\partial  ^ {2} f }{\partial  x _ {i}  ^ {2} }
  - \sum _ { i,j= } 1 ^ { n }   
+
  - \sum _ { i,j= 1} ^ { n }   
 
\frac{\partial  f
 
\frac{\partial  f
 
  }{\partial  x _ {i} }
 
  }{\partial  x _ {i} }
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====Comments====
 
====Comments====
For an  $  m $-
+
For an  $  m $-dimensional submanifold  $  M $
dimensional submanifold  $  M $
+
of an  $  n $-dimensional Euclidean space of codimension  $  n - m > 1 $,  
of an  $  n $-
 
dimensional Euclidean space of codimension  $  n - m > 1 $,  
 
 
the mean curvature generalizes to the notion of the mean curvature normal
 
the mean curvature generalizes to the notion of the mean curvature normal
  
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\nu _ {p}  =   
 
\nu _ {p}  =   
 
\frac{1}{m}
 
\frac{1}{m}
  \sum _ { j= } 1 ^ { n- } m [  \mathop{\rm Tr}  A
+
  \sum _ { j= 1} ^ { n-  m} [  \mathop{\rm Tr}  A
 
( e _ {j} ) ] e _ {j} ,
 
( e _ {j} ) ] e _ {j} ,
 
$$
 
$$
  
where  $  e _ {1} \dots e _ {n-} m $
+
where  $  e _ {1} \dots e _ {n-m} $
 
is an orthonormal frame of the normal space (cf. [[Normal space (to a surface)|Normal space (to a surface)]]) of  $  M $
 
is an orthonormal frame of the normal space (cf. [[Normal space (to a surface)|Normal space (to a surface)]]) of  $  M $
 
at  $  p $
 
at  $  p $
and  $  A ( e _ {j} ) :  T _ {p} M \rightarrow T _ {p} M $(
+
and  $  A ( e _ {j} ) :  T _ {p} M \rightarrow T _ {p} M $ ($  T _ {p} M $
$  T _ {p} M $
 
 
denotes the tangent space to  $  M $
 
denotes the tangent space to  $  M $
 
at  $  p $)  
 
at  $  p $)  

Latest revision as of 03:50, 21 March 2022


of a surface $ \Phi ^ {2} $ in $ 3 $-dimensional Euclidean space $ \mathbf R ^ {3} $

Half of the sum of the principal curvatures (cf. Principal curvature) $ k _ {1} $ and $ k _ {2} $, calculated at a point $ A $ of this surface:

$$ H( A) = \frac{k _ {1} + k _ {2} }{2} . $$

For a hypersurface $ \Phi ^ {n} $ in the Euclidean space $ \mathbf R ^ {n+1} $, this formula is generalized in the following way:

$$ H( A) = \frac{k _ {1} + \dots + k _ {n} }{n} , $$

where $ k _ {i} $, $ i = 1 \dots n $, are the principal curvatures of the hypersurface, calculated at a point $ A \in \Phi ^ {n} $.

The mean curvature of a surface in $ \mathbf R ^ {3} $ can be expressed by means of the coefficients of the first and second fundamental forms of this surface:

$$ H( A) = \frac{1}{2} \frac{LG - 2MF + NE }{EG - F ^ { 2 } } , $$

where $ E, F, G $ are the coefficients of the first fundamental form, and $ L, M, N $ are the coefficients of the second fundamental form, calculated at a point $ A \in \Phi ^ {2} $. In the particular case where the surface is defined by an equation $ z = f( x, y) $, the mean curvature is calculated using the formula:

$$ H ( A) = $$

$$ = \ \frac{\left ( 1 + \left ( \frac{\partial f }{\partial y } \right ) ^ {2} \right ) \frac{\partial ^ {2} f }{\partial x ^ {2} } - 2 \frac{\partial f }{\partial x } \frac{\partial f }{\partial y } \frac{\partial ^ {2} f }{\partial x \partial y } + \left ( 1 + \left ( \frac{\partial f }{ \partial x } \right ) ^ {2} \right ) \frac{\partial ^ {2} f }{\partial y ^ {2} } }{\left ( 1 + \left ( \frac{\partial f }{\partial x } \right ) ^ {2} + \left ( \frac{\partial f }{\partial y } \right ) ^ {2} \right ) ^ {3/2} } , $$

which is generalized for a hypersurface $ \Phi ^ {n} $ in $ \mathbf R ^ {n+1} $, defined by the equation $ x _ {n+1} = f( x _ {1} \dots x _ {n} ) $, as follows:

$$ H ( A) = \frac{\sum _ { i= 1} ^ { n } \left ( 1 + p ^ {2} - \left ( \frac{\partial f }{\partial x _ {i} } \right ) ^ {2} \right ) \frac{\partial ^ {2} f }{\partial x _ {i} ^ {2} } - \sum _ { i,j= 1} ^ { n } \frac{\partial f }{\partial x _ {i} } \frac{\partial f }{\partial x _ {j} } \frac{ \partial ^ {2} f }{\partial x _ {i} \partial x _ {j} } }{( 1 + p ^ {2} ) ^ {3/2} } , $$

where

$$ p ^ {2} = | \mathop{\rm grad} f | ^ {2} = \ \left ( \frac{\partial f }{\partial x _ {1} } \right ) ^ {2} + \dots + \left ( \frac{\partial f }{\partial x _ {n} } \right ) ^ {2} . $$

Comments

For an $ m $-dimensional submanifold $ M $ of an $ n $-dimensional Euclidean space of codimension $ n - m > 1 $, the mean curvature generalizes to the notion of the mean curvature normal

$$ \nu _ {p} = \frac{1}{m} \sum _ { j= 1} ^ { n- m} [ \mathop{\rm Tr} A ( e _ {j} ) ] e _ {j} , $$

where $ e _ {1} \dots e _ {n-m} $ is an orthonormal frame of the normal space (cf. Normal space (to a surface)) of $ M $ at $ p $ and $ A ( e _ {j} ) : T _ {p} M \rightarrow T _ {p} M $ ($ T _ {p} M $ denotes the tangent space to $ M $ at $ p $) is the shape operator of $ M $ at $ p $ in the direction $ e _ {j} $, which is related to the second fundamental tensor $ V $ of $ M $ at $ p $ by $ \langle A ( e _ {j} ) ( X) , Y \rangle = \langle V ( X , Y ) , e _ {j} \rangle $.

References

[a1] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2] M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976)
[a3] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
[a4] B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973)
How to Cite This Entry:
Mean curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mean_curvature&oldid=52238
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article