# Lamé coefficients

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 .

How to Cite This Entry:
Lamé coefficients. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lam%C3%A9_coefficients&oldid=47570
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article