View source for Lamé coefficients

''of an orthogonal curvilinear coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057370/l0573701.png" /> in space''

The quantities

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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:

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the element of surface area:

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and the volume element:

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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" />:

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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 <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

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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.

====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>