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''of an orthogonal curvilinear coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l0573701.png" /> in space''
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''of an orthogonal curvilinear coordinate system  $  u , v , w $
 +
in space''
  
 
The quantities
 
The quantities
  
<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/l/l057/l057370/l0573702.png" /></td> </tr></table>
+
$$
 +
L _ {u}  = \
 +
\sqrt {
 +
\left (
 +
\frac{\partial  x }{\partial  u }
 +
\right )  ^ {2} +
 +
\left (
 +
\frac{\partial  y }{\partial  u }
 +
\right )  ^ {2} +
 +
\left (
 +
\frac{\partial  z }{\partial  u }
 +
\right )  ^ {2} } ,
 +
$$
  
<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/l/l057/l057370/l0573703.png" /></td> </tr></table>
+
$$
 +
L _ {v}  = \sqrt {\left (
 +
\frac{\partial  x }{\partial  v }
 +
\right )
 +
^ {2} + \left (
 +
\frac{\partial  y }{\partial  v }
 +
\right )  ^ {2}
 +
+ \left (
 +
\frac{\partial  z }{\partial  v }
 +
\right )  ^ {2} } ,
 +
$$
  
<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/l/l057/l057370/l0573704.png" /></td> </tr></table>
+
$$
 +
L _ {w}  = \sqrt {\left (
 +
\frac{\partial  x }{\partial  w }
 +
\right )
 +
^ {2} + \left (
 +
\frac{\partial  y }{\partial  w }
 +
\right )  ^ {2}
 +
+ \left (
 +
\frac{\partial  z }{\partial  w }
 +
\right )  ^ {2} } .
 +
$$
  
The Lamé coefficients in the plane are defined similarly. In terms of the Lamé coefficients in the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l0573705.png" /> one can express the element of arc length:
+
The Lamé coefficients in the plane are defined similarly. In terms of the Lamé coefficients in the coordinates $  u , v , w $
 +
one can express the element of arc length:
  
<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/l/l057/l057370/l0573706.png" /></td> </tr></table>
+
$$
 +
d l  = \
 +
\sqrt {
 +
L _ {u}  ^ {2}  d u  ^ {2} +
 +
L _ {v}  ^ {2}  d v  ^ {2} +
 +
L _ {w}  ^ {2}  d w  ^ {2} } ;
 +
$$
  
 
the element of surface area:
 
the element of surface area:
  
<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/l/l057/l057370/l0573707.png" /></td> </tr></table>
+
$$
 +
d \sigma  = \
 +
\sqrt {
 +
( L _ {u} L _ {v}  d u  d v )  ^ {2} +
 +
( L _ {u} L _ {w}  d u  d w )  ^ {2} +
 +
( L _ {v} L _ {w}  d v  d w )  ^ {2} } ;
 +
$$
  
 
and the volume element:
 
and the volume element:
  
<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/l/l057/l057370/l0573708.png" /></td> </tr></table>
+
$$
 +
d V  = L _ {u} L _ {v} L _ {w}  d u  d v  d w .
 +
$$
 +
 
 +
The Lamé coefficients occur in the expressions for the operations of vector analysis in the coordinates  $  u , v , w $:
 +
 
 +
$$
 +
\mathop{\rm grad} _ {u}  \psi  = \
 +
 
 +
\frac{1}{L _ {u} }
 +
 
 +
\frac{\partial  \psi }{\partial  u }
 +
,\ \
 +
\mathop{\rm grad} _ {v}  \psi  = \
 +
 
 +
\frac{1}{L _ {v} }
 +
 
 +
\frac{\partial  \psi }{\partial  v }
 +
,\ \
 +
\mathop{\rm grad} _ {w}  \psi  = \
 +
 
 +
\frac{1}{L _ {w} }
 +
 
 +
\frac{\partial  \psi }{\partial  w }
 +
;
 +
$$
 +
 
 +
$$
 +
\mathop{\rm div}  a  = \
 +
 
 +
\frac{1}{L _ {u} L _ {v} L _ {w} }
 +
 
 +
\left [
 +
 
 +
\frac \partial {\partial  u }
 +
 
 +
( a _ {u} L _ {v} L _ {w} ) +
 +
 
 +
\frac \partial {\partial  v }
 +
 
 +
( a _ {v} L _ {u} L _ {w} ) \right . +
 +
$$
 +
 
 +
$$
 +
+ \left .
 +
 
 +
\frac \partial {\partial  w }
 +
( a _ {w} L _ {u} L _ {v} ) \right ] ;
 +
$$
 +
 
 +
$$
 +
\mathop{\rm rot} _ {u}  \mathbf a  = \
 +
 
 +
\frac{1}{L _ {v} L _ {w} }
 +
 
 +
\left [
 +
 
 +
\frac \partial {\partial  v }
 +
 
 +
( a _ {w} L _ {w} ) -
 +
 
 +
\frac \partial {\partial  w }
  
The Lamé coefficients occur in the expressions for the operations of vector analysis in the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l0573709.png" />:
+
( a _ {v} L _ {v} )
 +
\right ] ,
 +
$$
  
<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/l/l057/l057370/l05737010.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm rot} _ {v}  \mathbf a  =
 +
\frac{1}{L _ {u} L _ {w} }
 +
\left [
  
<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/l/l057/l057370/l05737011.png" /></td> </tr></table>
+
\frac \partial {\partial  w }
  
<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/l/l057/l057370/l05737012.png" /></td> </tr></table>
+
( a _ {u} L _ {u} ) -
  
<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/l/l057/l057370/l05737013.png" /></td> </tr></table>
+
\frac \partial {\partial  u }
  
<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/l/l057/l057370/l05737014.png" /></td> </tr></table>
+
( a _ {w} L _ {w} ) \right ] ,
 +
$$
  
<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/l/l057/l057370/l05737015.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm rot} _ {w}  \mathbf a  =
 +
\frac{1}{L _ {u} L _ {v} }
 +
\left [
  
<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/l/l057/l057370/l05737016.png" /></td> </tr></table>
+
\frac \partial {\partial  u }
  
<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/l/l057/l057370/l05737017.png" /></td> </tr></table>
+
( a _ {v} L _ {v} ) -
 +
 
 +
\frac \partial {\partial  v }
 +
 
 +
( a _ {u} L _ {u} ) \right ] ;
 +
$$
 +
 
 +
$$
 +
\Delta \psi  =   \mathop{\rm div}  (  \mathop{\rm grad}  \psi )  =
 +
\frac{1}{L _ {u} L _ {v} L _ {w} }
 +
\left [
 +
 
 +
\frac \partial {\partial  u }
 +
 
 +
\left (
 +
\frac{L _ {v} L _ {w} }{L _ {u} }
 +
 +
\frac{\partial  \psi }{\partial  u }
 +
\right ) \right . +
 +
$$
 +
 
 +
$$
 +
+ \left .
 +
 
 +
\frac \partial {\partial  v }
 +
 
 +
\left (
 +
\frac{L _ {u} L _ {w} }{L _ {v} }
 +
 +
\frac{\partial  \psi }{\partial  v }
 +
\right ) +
 +
 
 +
\frac \partial {\partial  w }
 +
 
 +
\left (
 +
\frac{L _ {u} L _ {v} }{L _ {w} }
 +
 
 +
\frac{\partial  \psi }{\partial  w }
 +
\right )  \right ] .
 +
$$
  
 
For the Lamé coefficients of various orthogonal curvilinear coordinates see the corresponding articles on those coordinates.
 
For the Lamé coefficients of various orthogonal curvilinear coordinates see the corresponding articles on those coordinates.
Line 45: Line 212:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Lamé,  "Leçons sur les coordonnées curvilignes et leurs diverses applications" , Paris  (1859)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.F. Laptev,  "Elements of vector calculus" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.M. Morse,  H. Feshbach,  "Methods of theoretical physics" , '''1''' , McGraw-Hill  (1953)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Lamé,  "Leçons sur les coordonnées curvilignes et leurs diverses applications" , Paris  (1859)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.F. Laptev,  "Elements of vector calculus" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.M. Morse,  H. Feshbach,  "Methods of theoretical physics" , '''1''' , McGraw-Hill  (1953)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The name  "Lamé coefficients"  for the quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l05737018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l05737019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l05737020.png" /> is not often used in the Western literature. Instead one finds  "scale factors of an orthogonal curvilinear coordinate systemscale factors"  [[#References|[3]]] or  "metric coefficients of an orthogonal curvilinear coordinate systemmetric coefficients"  [[#References|[a1]]]. The latter terminology of course derives from the fact that the original Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l05737021.png" /> with respect to the new orthogonal curvilinear system of coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l05737022.png" /> takes the form
+
The name  "Lamé coefficients"  for the quantities $  L _ {u} $,  
 +
$  L _ {v} $,  
 +
$  L _ {w} $
 +
is not often used in the Western literature. Instead one finds  "scale factors of an orthogonal curvilinear coordinate systemscale factors"  [[#References|[3]]] or  "metric coefficients of an orthogonal curvilinear coordinate systemmetric coefficients"  [[#References|[a1]]]. The latter terminology of course derives from the fact that the original Riemannian metric $  d s  ^ {2} $
 +
with respect to the new orthogonal curvilinear system of coordinates $  u , v , w $
 +
takes the form
  
<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/l/l057/l057370/l05737023.png" /></td> </tr></table>
+
$$
 +
d s  ^ {2}  = L _ {u}  ^ {2} \
 +
d u  ^ {2} + L _ {v}  ^ {2} \
 +
d v  ^ {2} + L _ {w}  ^ {2} \
 +
d w  ^ {2} .
 +
$$
  
Thus the squares of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l05737024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l05737025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l05737026.png" /> are the diagonal components of the standard metric tensor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l05737027.png" /> expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l05737028.png" />. The other components are zero because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l05737029.png" /> is an orthogonal curvilinear coordinate system.
+
Thus the squares of the $  L _ {u} $,  
 +
$  L _ {v} $,  
 +
$  L _ {w} $
 +
are the diagonal components of the standard metric tensor on $  \mathbf R  ^ {3} $
 +
expressed in terms of $  u , v, w $.  
 +
The other components are zero because $  u , v , w $
 +
is an orthogonal curvilinear coordinate system.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.S. [I.S. Sokolnikov] Sokolnikoff,  R.M. Redheffer,  "Mathematics of physics and engineering" , McGraw-Hill  (1958)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.F. Davis,  A.D. Snider,  "Introduction to vector analysis" , Allyn &amp; Bacon  (1979)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.S. [I.S. Sokolnikov] Sokolnikoff,  R.M. Redheffer,  "Mathematics of physics and engineering" , McGraw-Hill  (1958)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.F. Davis,  A.D. Snider,  "Introduction to vector analysis" , Allyn &amp; Bacon  (1979)</TD></TR></table>

Latest revision as of 22:15, 5 June 2020


of an orthogonal curvilinear coordinate system $ u , v , w $ in space

The quantities

$$ L _ {u} = \ \sqrt { \left ( \frac{\partial x }{\partial u } \right ) ^ {2} + \left ( \frac{\partial y }{\partial u } \right ) ^ {2} + \left ( \frac{\partial z }{\partial u } \right ) ^ {2} } , $$

$$ L _ {v} = \sqrt {\left ( \frac{\partial x }{\partial v } \right ) ^ {2} + \left ( \frac{\partial y }{\partial v } \right ) ^ {2} + \left ( \frac{\partial z }{\partial v } \right ) ^ {2} } , $$

$$ L _ {w} = \sqrt {\left ( \frac{\partial x }{\partial w } \right ) ^ {2} + \left ( \frac{\partial y }{\partial w } \right ) ^ {2} + \left ( \frac{\partial z }{\partial w } \right ) ^ {2} } . $$

The Lamé coefficients in the plane are defined similarly. In terms of the Lamé coefficients in the coordinates $ u , v , w $ one can express the element of arc length:

$$ d l = \ \sqrt { L _ {u} ^ {2} d u ^ {2} + L _ {v} ^ {2} d v ^ {2} + L _ {w} ^ {2} d w ^ {2} } ; $$

the element of surface area:

$$ d \sigma = \ \sqrt { ( L _ {u} L _ {v} d u d v ) ^ {2} + ( L _ {u} L _ {w} d u d w ) ^ {2} + ( L _ {v} L _ {w} d v d w ) ^ {2} } ; $$

and the volume element:

$$ d V = L _ {u} L _ {v} L _ {w} d u d v d w . $$

The Lamé coefficients occur in the expressions for the operations of vector analysis in the coordinates $ u , v , w $:

$$ \mathop{\rm grad} _ {u} \psi = \ \frac{1}{L _ {u} } \frac{\partial \psi }{\partial u } ,\ \ \mathop{\rm grad} _ {v} \psi = \ \frac{1}{L _ {v} } \frac{\partial \psi }{\partial v } ,\ \ \mathop{\rm grad} _ {w} \psi = \ \frac{1}{L _ {w} } \frac{\partial \psi }{\partial w } ; $$

$$ \mathop{\rm div} a = \ \frac{1}{L _ {u} L _ {v} L _ {w} } \left [ \frac \partial {\partial u } ( a _ {u} L _ {v} L _ {w} ) + \frac \partial {\partial v } ( a _ {v} L _ {u} L _ {w} ) \right . + $$

$$ + \left . \frac \partial {\partial w } ( a _ {w} L _ {u} L _ {v} ) \right ] ; $$

$$ \mathop{\rm rot} _ {u} \mathbf a = \ \frac{1}{L _ {v} L _ {w} } \left [ \frac \partial {\partial v } ( a _ {w} L _ {w} ) - \frac \partial {\partial w } ( a _ {v} L _ {v} ) \right ] , $$

$$ \mathop{\rm rot} _ {v} \mathbf a = \frac{1}{L _ {u} L _ {w} } \left [ \frac \partial {\partial w } ( a _ {u} L _ {u} ) - \frac \partial {\partial u } ( a _ {w} L _ {w} ) \right ] , $$

$$ \mathop{\rm rot} _ {w} \mathbf a = \frac{1}{L _ {u} L _ {v} } \left [ \frac \partial {\partial u } ( a _ {v} L _ {v} ) - \frac \partial {\partial v } ( a _ {u} L _ {u} ) \right ] ; $$

$$ \Delta \psi = \mathop{\rm div} ( \mathop{\rm grad} \psi ) = \frac{1}{L _ {u} L _ {v} L _ {w} } \left [ \frac \partial {\partial u } \left ( \frac{L _ {v} L _ {w} }{L _ {u} } \frac{\partial \psi }{\partial u } \right ) \right . + $$

$$ + \left . \frac \partial {\partial v } \left ( \frac{L _ {u} L _ {w} }{L _ {v} } \frac{\partial \psi }{\partial v } \right ) + \frac \partial {\partial w } \left ( \frac{L _ {u} L _ {v} }{L _ {w} } \frac{\partial \psi }{\partial w } \right ) \right ] . $$

For the Lamé coefficients of various orthogonal curvilinear coordinates see the corresponding articles on those coordinates.

Lamé coefficients were introduced by G. Lamé [1].

References

[1] G. Lamé, "Leçons sur les coordonnées curvilignes et leurs diverses applications" , Paris (1859)
[2] G.F. Laptev, "Elements of vector calculus" , Moscow (1975) (In Russian)
[3] P.M. Morse, H. Feshbach, "Methods of theoretical physics" , 1 , McGraw-Hill (1953)

Comments

The name "Lamé coefficients" for the quantities $ L _ {u} $, $ L _ {v} $, $ L _ {w} $ is not often used in the Western literature. Instead one finds "scale factors of an orthogonal curvilinear coordinate systemscale factors" [3] or "metric coefficients of an orthogonal curvilinear coordinate systemmetric coefficients" [a1]. The latter terminology of course derives from the fact that the original Riemannian metric $ d s ^ {2} $ with respect to the new orthogonal curvilinear system of coordinates $ u , v , w $ takes the form

$$ d s ^ {2} = L _ {u} ^ {2} \ d u ^ {2} + L _ {v} ^ {2} \ d v ^ {2} + L _ {w} ^ {2} \ d w ^ {2} . $$

Thus the squares of the $ L _ {u} $, $ L _ {v} $, $ L _ {w} $ are the diagonal components of the standard metric tensor on $ \mathbf R ^ {3} $ expressed in terms of $ u , v, w $. The other components are zero because $ u , v , w $ is an orthogonal curvilinear coordinate system.

References

[a1] I.S. [I.S. Sokolnikov] Sokolnikoff, R.M. Redheffer, "Mathematics of physics and engineering" , McGraw-Hill (1958)
[a2] H.F. Davis, A.D. Snider, "Introduction to vector analysis" , Allyn & Bacon (1979)
How to Cite This Entry:
Lamé coefficients. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lam%C3%A9_coefficients&oldid=23360
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article