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Euler's identity; Fibonacci's identity see Brahmagupta–Fibonacci identity or Cassini and Catalan identities; Heine's identity; Hermite's identity; Lagrange's identity; Lagrange's trigonometric identities; List of logarithmic identities; MacWilliams identity; Matrix determinant lemma; Newton's identity; Parseval's identity; Pfister's sixteen ...
In the study of ordinary differential equations and their associated boundary value problems in mathematics, Lagrange's identity, named after Joseph Louis Lagrange, gives the boundary terms arising from integration by parts of a self-adjoint linear differential operator. Lagrange's identity is fundamental in Sturm–Liouville theory.
where , and are the wavenumbers in their respective coordinate axes: = + +. The expansion is named after Hermann Weyl, who published it in 1919. [3] The Weyl identity is largely used to characterize the reflection and transmission of spherical waves at planar interfaces; it is often used to derive the Green's functions for Helmholtz equation in layered media.
Green's functions are also useful tools in solving wave equations and diffusion equations. In quantum mechanics, Green's function of the Hamiltonian is a key concept with important links to the concept of density of states. The Green's function as used in physics is usually defined with the opposite sign, instead.
This technique was first introduced into isotropic turbulence by Howard P. Robertson in 1940 where he was able to derive Kármán–Howarth equation from the invariant principle. [7] George Batchelor and Subrahmanyan Chandrasekhar exploited this technique and developed an extended treatment for axisymmetric turbulence. [8] [9] [10]
All about the Third House in astrology, including the meaning and themes of this part of your birth chart.
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The first two identities abstract the skew symmetry and Jacobi identity for the triple commutator, while the third identity means that the linear map L u,v: V → V, defined by L u,v (w) = [u, v, w], is a derivation of the triple product.