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''of a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m0631602.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m0631603.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m0631604.png" />''
+
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Half of the sum of the principal curvatures (cf. [[Principal curvature|Principal curvature]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m0631605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m0631606.png" />, calculated at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m0631607.png" /> of this surface:
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/m/m063/m063160/m0631608.png" /></td> </tr></table>
+
''of a surface  $  \Phi  ^ {2} $
 +
in  $  3 $-
 +
dimensional Euclidean space  $  \mathbf R  ^ {3} $''
  
For a hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m0631609.png" /> in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316010.png" />, this formula is generalized in the following way:
+
Half of the sum of the principal curvatures (cf. [[Principal curvature|Principal curvature]])  $  k _ {1} $
 +
and  $  k _ {2} $,  
 +
calculated at a point  $  A $
 +
of this surface:
  
<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/m/m063/m063160/m06316011.png" /></td> </tr></table>
+
$$
 +
H( A)  =
 +
\frac{k _ {1} + k _ {2} }{2}
 +
.
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316013.png" />, are the principal curvatures of the hypersurface, calculated at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316014.png" />.
+
For a hypersurface  $  \Phi  ^ {n} $
 +
in the Euclidean space  $  \mathbf R  ^ {n+} 1 $,  
 +
this formula is generalized in the following way:
  
The mean curvature of a surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316015.png" /> can be expressed by means of the coefficients of the first and second fundamental forms of this surface:
+
$$
 +
H( A)  =
 +
\frac{k _ {1} + \dots + k _ {n} }{n}
 +
,
 +
$$
  
<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/m/m063/m063160/m06316016.png" /></td> </tr></table>
+
where  $  k _ {i} $,
 +
$  i = 1 \dots n $,
 +
are the principal curvatures of the hypersurface, calculated at a point  $  A \in \Phi  ^ {n} $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316017.png" /> are the coefficients of the [[First fundamental form|first fundamental form]], and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316018.png" /> are the coefficients of the [[Second fundamental form|second fundamental form]], calculated at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316019.png" />. In the particular case where the surface is defined by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316020.png" />, the mean curvature is calculated using the formula:
+
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:
  
<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/m/m063/m063160/m06316021.png" /></td> </tr></table>
+
$$
 +
H( A)  =
 +
\frac{1}{2}
 +
 +
\frac{LG - 2MF + NE }{EG - F ^ { 2 } }
 +
,
 +
$$
  
<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/m/m063/m063160/m06316022.png" /></td> </tr></table>
+
where  $  E, F, G $
 +
are the coefficients of the [[First fundamental form|first fundamental form]], and  $  L, M, N $
 +
are the coefficients of the [[Second fundamental form|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:
  
which is generalized for a hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316024.png" />, defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316025.png" />, as follows:
+
$$
 +
H ( A) =
 +
$$
  
<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/m/m063/m063160/m06316026.png" /></td> </tr></table>
+
$$
 +
= \
  
where
+
\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:
  
<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/m/m063/m063160/m06316027.png" /></td> </tr></table>
+
$$
 +
H ( A)  =
 +
\frac{\sum _ { i= } 1 ^ { n }  \left ( 1 + p  ^ {2} - \left (
 +
\frac{\partial
 +
f }{\partial  x _ {i} }
 +
\right )  ^ {2} \right )
 +
\frac{\partial  ^ {2} f }{\partial  x _ {i}  ^ {2} }
 +
- \sum _ { i,j= } 1 ^ { n } 
 +
\frac{\partial  f
 +
}{\partial  x _ {i} }
 +
 +
\frac{\partial  f }{\partial  x _ {j} }
 +
 +
\frac{
 +
\partial  ^ {2} f }{\partial  x _ {i} \partial  x _ {j} }
 +
}{( 1 + p  ^ {2} )  ^ {3/2} }
 +
,
 +
$$
  
 +
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====
 
====Comments====
For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316028.png" />-dimensional submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316029.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316030.png" />-dimensional Euclidean space of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316031.png" />, the mean curvature generalizes to the notion of the mean curvature normal
+
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
  
<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/m/m063/m063160/m06316032.png" /></td> </tr></table>
+
$$
 +
\nu _ {p}  =
 +
\frac{1}{m}
 +
\sum _ { j= } 1 ^ { n- m [  \mathop{\rm Tr}  A
 +
( e _ {j} ) ] e _ {j} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316033.png" /> is an orthonormal frame of the normal space (cf. [[Normal space (to a surface)|Normal space (to a surface)]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316034.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316036.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316037.png" /> denotes the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316038.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316039.png" />) is the shape operator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316040.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316041.png" /> in the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316042.png" />, which is related to the second fundamental tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316043.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316044.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316045.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063160/m06316046.png" />.
+
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 $
 +
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====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Blaschke,  K. Leichtweiss,  "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B.-Y. Chen,  "Geometry of submanifolds" , M. Dekker  (1973)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Blaschke,  K. Leichtweiss,  "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B.-Y. Chen,  "Geometry of submanifolds" , M. Dekker  (1973)</TD></TR></table>

Latest revision as of 08:00, 6 June 2020


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=12526
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article