# 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 [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)

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.