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Difference between revisions of "D'Alembert operator"

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====Comments====
 
====Comments====
In the last equation above, the Einstein (summation) convention applies to the right-hand side (i.e. there is a summation involved over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030080/d03008011.png" />).
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In the last equation above, the Einstein [[summation convention]] applies to the right-hand side (i.e. there is a summation involved over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030080/d03008011.png" />).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. John,  "Partial differential equations" , Springer  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. John,  "Partial differential equations" , Springer  (1968)</TD></TR></table>

Revision as of 19:13, 21 January 2016

wave operator, d'Alembertian

The second-order differential operator which in Cartesian coordinates assumes the following form:

where is the Laplace operator and is a constant. Its form in spherical coordinates is:

in cylindrical coordinates:

in general curvilinear coordinates:

where is the determinant of the matrix formed from the coefficients of the metric tensor .

Named after J. d'Alembert (1747), who considered its simplest form when solving the one-dimensional wave equation.


Comments

In the last equation above, the Einstein summation convention applies to the right-hand side (i.e. there is a summation involved over all ).

References

[a1] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[a2] F. John, "Partial differential equations" , Springer (1968)
How to Cite This Entry:
D'Alembert operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=D%27Alembert_operator&oldid=11498
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article