View source for Lamé coefficients

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''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 [[#References|[1]]].

====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>

====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"  [[#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

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

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====
<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>