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| <blockquote>"Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry. If
| | *[[Stirling's approximation]] |
| mathematical analysis should ever hold a prominent place in chemistry – an aberration which is happily almost impossible – it would occasion a rapid and widespread degeneration of that science." '''Auguste Comte''' (1830) <ref>[http://gallica.bnf.fr/ark:/12148/bpt6k109113f/f4.item.r=Cours%20de%20philosophie%20positive%20Tome%203 Auguste Comte"Cours de philosophie positive" (Tome 3) 35e Leçon. Considérations philosophiques sur l'ensemble de la chimie (1830)]</ref></blockquote>
| | *[[Root finding]] |
| <blockquote>"Important investigations by physicists on the foundations of mechanics are at hand; I refer to the writings of Mach, Hertz, Boltzmann and Volkmann. It is therefore very desirable that the discussion of the foundations of mechanics be taken up by mathematicians also."
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| '''David Hilbert''' (1900) <ref>David Hilbert, excerpt from problem VI of his lecture [http://www.mathunion.org/ICM/ICM1900/Main/icm1900.0058.0114.ocr.pdf "Sur les problèmes futurs des Mathématiques", Second International Congress of Mathematicians, pp. 58-114 Paris (1900)]</ref></blockquote>
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| <blockquote>"Every mathematician knows it is impossible to understand an elementary course in thermodynamics." '''Vladimir I. Arnold''' (1989)<ref> Vladimir I. Arnold "Contact geometry: The geometrical method of Gibbs' thermodynamics," in ''Proceedings of the Gibbs Symposium, Yale University, May 15-17, 1989'' Ed. D. G. Caldi and G. D. Mostow, American Mathematical Society (1990) ISBN 978-0-8218-0157-4 p. 163. </ref></blockquote>
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| Here are details of mathematical tools that have applications in [[classical thermodynamics]], [[statistical mechanics]] and [[computer simulation techniques]].
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| ==General==
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| *[[Clebsch-Gordan coefficients]]
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| *[[Euler angles]]
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| *[[Gauss theorem]]
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| *[[Green's theorem]]
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| *[[Lagrange multipliers]]
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| *[[Laplace's equation]]
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| *[[Lee-Yang circle theorem]]
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| *[[Linear algebra]]
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| *[[Liouville's theorem]] | |
| *[[Lyapunov exponents]] | |
| *[[Padé approximants]] | | *[[Padé approximants]] |
| *[[Perron-Frobenius theorem]] | | *[[Fast Fourier transform]] |
| *[[Poincaré theorem]] | | *[[Standard integrals]] |
| *[[Power series]]
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| *[[Predictor-corrector methods]]
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| *[[Quaternions]]
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| *[[Random numbers]]
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| *[[Root finding]]
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| *[[Runge-Kutta method]]
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| *[[Spherical harmonics]] | | *[[Spherical harmonics]] |
| *[[Stirling's approximation]] | | *[[Legendre polynomials]] |
| *[[Substantive derivative]] | | *[[Legendre transform]] |
| *[[Taylor expansion]]
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| *[[Trotter formula]]
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| *[[Wigner D-matrix]]
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| *[[Wigner D-matrix |Wigner rotation matrices]]
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| ==Functions and distributions==
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| *[[Bessel functions]] | | *[[Bessel functions]] |
| | *[[Laguerre polynomials]] |
| | *[[Chebyshev polynomials]] |
| | *[[Heaviside step distribution]] |
| *[[Dirac delta distribution]] | | *[[Dirac delta distribution]] |
| *[[Gamma function]]
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| *[[Gaussian distribution]]
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| *[[Heaviside step distribution]]
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| *[[Kronecker delta]] | | *[[Kronecker delta]] |
| *[[Poisson distribution]] | | *[[Taylor expansion]] |
| *[[Ramp function]] | | *[[Wigner rotation matrices]] |
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| ==Geometry==
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| *[[Alpha shape]]
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| *[[Delaunay simplexes]]
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| *[[Dual lattice]]
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| *[[Lattice Structures]]
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| *[[Voronoi cells]]
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| ==Integrals and quadrature==
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| *[[Standard integrals]]
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| *[[Elliptic integrals]]
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| *[[Newton-Cotes formulas]]
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| *[[Gauss-Chebyshev quadrature]]
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| *[[Gauss-Legendre quadrature]]
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| ==Polynomials==
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| *[[Chebyshev polynomials]]
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| *[[Laguerre polynomials]]
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| *[[Legendre polynomials]]
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| **[[Associated Legendre functions]]
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| ==Transforms==
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| *[[Fast Fourier transform]]
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| *[[Fourier analysis]]
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| *[[Laplace transform]]
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| *[[Legendre transform]]
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| ==Computer assisted mathematics==
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| {| class="wikitable sortable" border="1" cellpadding="5" cellspacing="0"
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| |-
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| ! Packages !! Program or library !! class="unsortable" | Notes !! License
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| |-
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| |[[BLAS]]
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| |routines
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| | '''B'''asic '''L'''inear '''A'''lgebra '''S'''ubprograms
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| |free source
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| |-
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| |[[CGAL]]
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| |library
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| |'''C'''omputational '''G'''eometry '''A'''lgorithms '''L'''ibrary
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| |free, Open Source
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| |-
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| |[http://freemat.sourceforge.net/ FreeMat]
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| |program
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| |free (GNU General Public License)
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| |-
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| |-
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| |[[IMSL Numerical Libraries]]
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| |library
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| |-
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| |[[LAPACK]]
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| |routines
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| |'''L'''inear '''A'''lgebra '''PACK'''age
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| |free source
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| |-
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| |[[ScaLAPACK]]
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| |library
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| |'''Sca'''lable '''LAPACK'''
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| |free source
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| |-
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| |[[Magma]]
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| |-
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| |[[Maple]]
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| |program
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| |commercial
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| |-
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| |[[Mathematica]]
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| |program
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| |commercial
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| |-
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| |[[Math Kernel Library]]
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| |library
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| |-
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| |[[MATLAB]]
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| |program
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| |commercial
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| |-
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| |[[Maxima]]
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| |program
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| |free (General Public License)
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| |-
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| |[[NAG]]
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| |routines
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| |'''N'''umerical '''A'''lgorithms '''G'''roup
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| |commercial
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| |-
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| |[[Numerical Recipes]]
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| |routines
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| |-
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| |[[Octave]]
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| |program
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| |free (GNU)
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| |-
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| |[[QUADPACK]]
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| |routines
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| |free
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| |-
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| |[[SAGE]]
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| |program
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| |free (General Public License)
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| |-
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| |[[Scilab]]
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| |program
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| |free, Open Source
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| |}
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| ==Interesting reading==
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| *[http://dx.doi.org/10.1002/cpa.3160130102 Eugene P. Wigner "The unreasonable effectiveness of mathematics in the natural sciences", Communications on Pure and Applied Mathematics '''13''' pp. 1-14 (1960)]
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| **[http://dx.doi.org/10.1063/1.2435629 Frank Wilczek "Reasonably Effective: I. Deconstructing a Miracle", Physics Today '''November''' pp. 8-9 (2006)]
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| **[http://dx.doi.org/10.1063/1.2743102 Frank Wilczek "Reasonably Effective: II. Devil's Advocate", Physics Today '''May''' pp. 8-9 (2007)]
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| *[http://www.ams.org/bull/1988-19-01/S0273-0979-1988-15634-0/S0273-0979-1988-15634-0.pdf David Ruelle "Is our mathematics natural? The case of equilibrium statistical mechanics", Bulletin of the American Mathematical Society '''19''' pp. 259-268 (1988)]
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| ==References==
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| <references/>
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| ==External links==
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| *[http://www.mathematics.thetangentbundle.net/wiki/Main_Page TheTangentBundle mathematics wiki]
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| *[http://mathworld.wolfram.com/ Wolfram MathWorld]
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| [[category: mathematics]]
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