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]]. | Here are details of mathematical tools that have applications in [[classical thermodynamics]], [[statistical mechanics]] and [[computer simulation techniques]]. | ||
==General== | ==General== | ||
*[[Clebsch-Gordan coefficients]] | *[[Clebsch-Gordan coefficients]] | ||
*[[Euler angles]] | *[[Euler angles]] | ||
*[[Gauss theorem]] | |||
*[[Green's theorem]] | *[[Green's theorem]] | ||
*[[Lagrange multipliers]] | |||
*[[Laplace's equation]] | *[[Laplace's equation]] | ||
*[[Lee-Yang circle theorem]] | |||
*[[Linear algebra]] | *[[Linear algebra]] | ||
*[[Liouville's theorem]] | *[[Liouville's theorem]] | ||
| Line 13: | Line 23: | ||
*[[Power series]] | *[[Power series]] | ||
*[[Predictor-corrector methods]] | *[[Predictor-corrector methods]] | ||
*[[Quaternions]] | |||
*[[Random numbers]] | *[[Random numbers]] | ||
*[[Root finding]] | *[[Root finding]] | ||
| Line 18: | Line 29: | ||
*[[Spherical harmonics]] | *[[Spherical harmonics]] | ||
*[[Stirling's approximation]] | *[[Stirling's approximation]] | ||
*[[Substantive derivative]] | |||
*[[Taylor expansion]] | *[[Taylor expansion]] | ||
*[[Trotter formula]] | |||
*[[Wigner D-matrix]] | *[[Wigner D-matrix]] | ||
*[[Wigner D-matrix |Wigner rotation matrices]] | *[[Wigner D-matrix |Wigner rotation matrices]] | ||
| Line 56: | Line 69: | ||
*[[Legendre transform]] | *[[Legendre transform]] | ||
==Computer assisted mathematics== | ==Computer assisted mathematics== | ||
{| class="wikitable sortable" border="1" cellpadding="5" cellspacing="0" | |||
|- | |||
! Packages !! Program or library !! class="unsortable" | Notes !! License | |||
|- | |||
|[[BLAS]] | |||
|routines | |||
| '''B'''asic '''L'''inear '''A'''lgebra '''S'''ubprograms | |||
|free source | |||
|- | |||
|[[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]
- 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 rotation matrices
Functions and distributions[edit]
- Bessel functions
- Dirac delta distribution
- Gamma function
- Gaussian distribution
- Heaviside step distribution
- Kronecker delta
- Poisson distribution
- Ramp function
Geometry[edit]
Integrals and quadrature[edit]
- Standard integrals
- Elliptic integrals
- Newton-Cotes formulas
- Gauss-Chebyshev quadrature
- Gauss-Legendre quadrature
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]
- Eugene P. Wigner "The unreasonable effectiveness of mathematics in the natural sciences", Communications on Pure and Applied Mathematics 13 pp. 1-14 (1960)
- David Ruelle "Is our mathematics natural? The case of equilibrium statistical mechanics", Bulletin of the American Mathematical Society 19 pp. 259-268 (1988)
References[edit]
- ↑ Auguste Comte"Cours de philosophie positive" (Tome 3) 35e Leçon. Considérations philosophiques sur l'ensemble de la chimie (1830)
- ↑ 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)
- ↑ 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.