Mathematics: Difference between revisions

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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
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>
<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."
'''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>
<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>
Here are details of mathematical tools that have applications in [[classical thermodynamics]], [[statistical mechanics]] and [[computer simulation techniques]].
==General==
*[[Clebsch-Gordan coefficients]]
*[[Euler angles]]
*[[Gauss theorem]]
*[[Green's theorem]]
*[[Lagrange multipliers]]
*[[Laplace's equation]]
*[[Lee-Yang circle theorem]]
*[[Linear algebra]]
*[[Liouville's theorem]]
*[[Lyapunov exponents]]
*[[Padé approximants]]
*[[Perron-Frobenius theorem]]
*[[Poincaré theorem]]
*[[Power series]]
*[[Predictor-corrector methods]]
*[[Quaternions]]
*[[Random numbers]]
*[[Root finding]]
*[[Runge-Kutta method]]
*[[Spherical harmonics]]
*[[Stirling's approximation]]
*[[Substantive derivative]]
*[[Taylor expansion]]
*[[Trotter formula]]
*[[Wigner D-matrix]]
*[[Wigner D-matrix |Wigner rotation matrices]]
==Functions and distributions==
*[[Bessel functions]]
*[[Bessel functions]]
*[[Chebyshev polynomials]]
*[[Dirac delta distribution]]
*[[Dirac delta distribution]]
*[[Elliptic integrals]]
*[[Gamma function]]
*[[Fast Fourier transform]]
*[[Fourier analysis]]
*[[Gaussian distribution]]
*[[Gaussian distribution]]
*[[Green's theorem]]
*[[Heaviside step distribution]]
*[[Heaviside step distribution]]
*[[Kronecker delta]]
*[[Kronecker delta]]
*[[Poisson distribution]]
*[[Ramp function]]
==Geometry==
*[[Alpha shape]]
*[[Delaunay simplexes]]
*[[Dual lattice]]
*[[Lattice Structures]]
*[[Voronoi cells]]
==Integrals and quadrature==
*[[Standard integrals]]
*[[Elliptic integrals]]
*[[Newton-Cotes formulas]]
*[[Gauss-Chebyshev quadrature]]
*[[Gauss-Legendre quadrature]]
==Polynomials==
*[[Chebyshev polynomials]]
*[[Laguerre polynomials]]
*[[Laguerre polynomials]]
*[[Legendre polynomials]]
*[[Legendre polynomials]]
**[[Associated Legendre functions]]
==Transforms==
*[[Fast Fourier transform]]
*[[Fourier analysis]]
*[[Laplace transform]]
*[[Legendre transform]]
*[[Legendre transform]]
*[[Padé approximants]]
==Computer assisted mathematics==
*[[Perron-Frobenius theorem]]
{| class="wikitable sortable" border="1" cellpadding="5" cellspacing="0"
*[[Power series]]
|-
*[[Ramp function]]
! Packages !! Program or library !! class="unsortable" | Notes !! License 
*[[Root finding]]
|-
*[[Spherical harmonics]]
|[[BLAS]]
*[[Standard integrals]]
|routines
*[[Stirling's approximation]]
| '''B'''asic '''L'''inear '''A'''lgebra '''S'''ubprograms
*[[Taylor expansion]]
|free source
*[[Wigner rotation matrices]]
|-
|[[CGAL]]
|library
|'''C'''omputational '''G'''eometry '''A'''lgorithms '''L'''ibrary
|free, Open Source
|-
|[http://freemat.sourceforge.net/ FreeMat]
|program
|
|free (GNU General Public License)
|-
|-
|[[IMSL Numerical Libraries]]
|library
|
|
|-
|[[LAPACK]]
|routines
|'''L'''inear '''A'''lgebra '''PACK'''age
|free source
|-
|[[ScaLAPACK]]
|library
|'''Sca'''lable '''LAPACK'''
|free source
|-
|[[Magma]]
|
|
|
|-
|[[Maple]]
|program
|
|commercial
|-
|[[Mathematica]]
|program
|
|commercial
|-
|[[Math Kernel Library]]
|library
|
|
|-
|[[MATLAB]]
|program
|
|commercial
|-
|[[Maxima]]
|program
|
|free (General Public License)
|-
|[[NAG]]
|routines
|'''N'''umerical '''A'''lgorithms '''G'''roup
|commercial
|-
|[[Numerical Recipes]]
|routines
|
|
|-
|[[Octave]]
|program
|
|free (GNU)
|-
|[[QUADPACK]]
|routines
|
|free
|-
|[[SAGE]]
|program
|
|free (General Public License)
|-
|[[Scilab]]
|program
|
|free, Open Source
|}
 
==Interesting reading==
*[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)]
**[http://dx.doi.org/10.1063/1.2435629 Frank Wilczek "Reasonably Effective: I. Deconstructing a Miracle", Physics Today  '''November''' pp. 8-9 (2006)]
**[http://dx.doi.org/10.1063/1.2743102 Frank Wilczek "Reasonably Effective: II. Devil's Advocate", Physics Today  '''May''' pp. 8-9 (2007)]  
*[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)]
==References==
<references/>
==External links==
*[http://www.mathematics.thetangentbundle.net/wiki/Main_Page TheTangentBundle mathematics wiki]
*[http://mathworld.wolfram.com/ Wolfram MathWorld]
[[category: mathematics]]
[[category: mathematics]]

Latest revision as of 16:52, 24 May 2016

"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 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) [1]

"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." David Hilbert (1900) [2]

"Every mathematician knows it is impossible to understand an elementary course in thermodynamics." Vladimir I. Arnold (1989)[3]


Here are details of mathematical tools that have applications in classical thermodynamics, statistical mechanics and computer simulation techniques.

General[edit]

Functions and distributions[edit]

Geometry[edit]

Integrals and quadrature[edit]

Polynomials[edit]

Transforms[edit]

Computer assisted mathematics[edit]

Packages Program or library Notes License
BLAS routines Basic Linear Algebra Subprograms free source
CGAL library Computational Geometry Algorithms Library free, Open Source
FreeMat program free (GNU General Public License)
IMSL Numerical Libraries library
LAPACK routines Linear Algebra PACKage free source
ScaLAPACK library Scalable LAPACK free source
Magma
Maple program commercial
Mathematica program commercial
Math Kernel Library library
MATLAB program commercial
Maxima program free (General Public License)
NAG routines Numerical Algorithms Group commercial
Numerical Recipes routines
Octave program free (GNU)
QUADPACK routines free
SAGE program free (General Public License)
Scilab program free, Open Source

Interesting reading[edit]

References[edit]

  1. Auguste Comte"Cours de philosophie positive" (Tome 3) 35e Leçon. Considérations philosophiques sur l'ensemble de la chimie (1830)
  2. David Hilbert, excerpt from problem VI of his lecture "Sur les problèmes futurs des Mathématiques", Second International Congress of Mathematicians, pp. 58-114 Paris (1900)
  3. 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.

External links[edit]