Tensors: New Applications
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en-us2020 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 27 Feb 2020 20:00:57 GMTThu, 27 Feb 2020 20:00:57 GMTNew applications in the Tensors categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgTensors: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=151
Mathematics for Chemistry
https://www.maplesoft.com/applications/view.aspx?SID=154267&ref=Feed
This interactive electronic textbook in the form of Maple worksheets comprises two parts.
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Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
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Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
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Other chapters are in preparation and will be released in due course.
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Last updated on March 19, 2019<img src="https://www.maplesoft.com/view.aspx?si=154267/molecule.PNG" alt="Mathematics for Chemistry" style="max-width: 25%;" align="left"/>This interactive electronic textbook in the form of Maple worksheets comprises two parts.
<BR><BR>
Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
<BR><BR>
Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
<BR><BR>
Other chapters are in preparation and will be released in due course.
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Last updated on March 19, 2019https://www.maplesoft.com/applications/view.aspx?SID=154267&ref=FeedTue, 30 May 2017 04:00:00 ZProf. John OgilvieProf. John OgilvieVisualizing a Parallel Field in a Curved Manifold
https://www.maplesoft.com/applications/view.aspx?SID=35113&ref=Feed
<p>My PhD thesis was in relativistic cosmology, a study that took me into differential geometry, continuous group theory, and tensor calculus. One of the most difficult concepts in all this was the notion of parallel transport of a vector from one tangent space to another. Of course, the image I had in my head was a basketball for a manifold, and a vector in a tangent plane on this (unit) sphere. The manifold sat in an enveloping R<sup>3</sup>, and I struggled mightily to visualize the difference between the transported field appearing parallel to the surface observer and Euclidean parallelism as seen by an external observer. The Kantian imperative is true - it's natural to imagine the vectors in R<sup>3</sup>, but devilishly difficult to visualize the difference between Euclidean parallelism and parallel transport in an intrinsically curved space.</p><img src="https://www.maplesoft.com/view.aspx?si=35113/thumb.jpg" alt="Visualizing a Parallel Field in a Curved Manifold" style="max-width: 25%;" align="left"/><p>My PhD thesis was in relativistic cosmology, a study that took me into differential geometry, continuous group theory, and tensor calculus. One of the most difficult concepts in all this was the notion of parallel transport of a vector from one tangent space to another. Of course, the image I had in my head was a basketball for a manifold, and a vector in a tangent plane on this (unit) sphere. The manifold sat in an enveloping R<sup>3</sup>, and I struggled mightily to visualize the difference between the transported field appearing parallel to the surface observer and Euclidean parallelism as seen by an external observer. The Kantian imperative is true - it's natural to imagine the vectors in R<sup>3</sup>, but devilishly difficult to visualize the difference between Euclidean parallelism and parallel transport in an intrinsically curved space.</p>https://www.maplesoft.com/applications/view.aspx?SID=35113&ref=FeedThu, 28 Jan 2010 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Geodesics on a Surface
https://www.maplesoft.com/applications/view.aspx?SID=34940&ref=Feed
<p>Several months ago we provided the article Tensor Calculus with the Differential Geometry Package in which we found geodesics in the plane when the plane was referred to polar coordinates. In this month's article we find geodesics on a surface embedded in R<sup>3</sup>. We illustrate three approaches: numeric approximation, the calculus of variations, and differential geometry.</p><img src="https://www.maplesoft.com/view.aspx?si=34940/thumb.jpg" alt="Classroom Tips and Techniques: Geodesics on a Surface" style="max-width: 25%;" align="left"/><p>Several months ago we provided the article Tensor Calculus with the Differential Geometry Package in which we found geodesics in the plane when the plane was referred to polar coordinates. In this month's article we find geodesics on a surface embedded in R<sup>3</sup>. We illustrate three approaches: numeric approximation, the calculus of variations, and differential geometry.</p>https://www.maplesoft.com/applications/view.aspx?SID=34940&ref=FeedTue, 08 Dec 2009 05:00:00 ZDr. Robert LopezDr. Robert LopezGravitation
https://www.maplesoft.com/applications/view.aspx?SID=4891&ref=Feed
An object-oriented package for gravitational tensor calculation, comprising calculation of;
- the Levi-Civita connection, the Riemann tensor, the Ricci tensor, the Ricci scalar, the Einstein tensor, the Weyl tensor, and the Kretschmann scalar
- the spin connection, and the spin curvature tensor
The package treats all the following three cases;
- the purely metric case
- the purely tetrad case
- the mixed case, where both metric and tetrad are present (may, or may not, be independent)
The package is able to treat as independent variables the metric, the Levi-Civita connection, the tetrad, and the spin connection<img src="https://www.maplesoft.com/view.aspx?si=4891//applications/images/app_image_blank_lg.jpg" alt="Gravitation" style="max-width: 25%;" align="left"/>An object-oriented package for gravitational tensor calculation, comprising calculation of;
- the Levi-Civita connection, the Riemann tensor, the Ricci tensor, the Ricci scalar, the Einstein tensor, the Weyl tensor, and the Kretschmann scalar
- the spin connection, and the spin curvature tensor
The package treats all the following three cases;
- the purely metric case
- the purely tetrad case
- the mixed case, where both metric and tetrad are present (may, or may not, be independent)
The package is able to treat as independent variables the metric, the Levi-Civita connection, the tetrad, and the spin connectionhttps://www.maplesoft.com/applications/view.aspx?SID=4891&ref=FeedThu, 05 Apr 2007 00:00:00 ZJohn FredstedJohn FredstedStokes' Theorem
https://www.maplesoft.com/applications/view.aspx?SID=1755&ref=Feed
There are some examples for Stokes' integral Theorem in the worksheet. One can check the Theorem by examples, in arbitrary dimensional vector space, for abitrary dimensional submanifolds, for differentable functions.<img src="https://www.maplesoft.com/view.aspx?si=1755/stokesend_175.gif" alt="Stokes' Theorem" style="max-width: 25%;" align="left"/>There are some examples for Stokes' integral Theorem in the worksheet. One can check the Theorem by examples, in arbitrary dimensional vector space, for abitrary dimensional submanifolds, for differentable functions.https://www.maplesoft.com/applications/view.aspx?SID=1755&ref=FeedMon, 26 Jun 2006 00:00:00 ZAttila AndaiAttila AndaiDynamics on Our Rotating Earth
https://www.maplesoft.com/applications/view.aspx?SID=1650&ref=Feed
The general relationship between two rotating coordinate systems has been established and the behaviour of moving objects in the vicinity of the rotating earth's surface has been analysed. Amongst others, a mathematical analysis of the historical demonstration of the earth's rotation with his pendulum by Foucault, 1850 in the pantheon in Paris is presented.<img src="https://www.maplesoft.com/view.aspx?si=1650/Freier_Fall_88.gif" alt="Dynamics on Our Rotating Earth" style="max-width: 25%;" align="left"/>The general relationship between two rotating coordinate systems has been established and the behaviour of moving objects in the vicinity of the rotating earth's surface has been analysed. Amongst others, a mathematical analysis of the historical demonstration of the earth's rotation with his pendulum by Foucault, 1850 in the pantheon in Paris is presented.https://www.maplesoft.com/applications/view.aspx?SID=1650&ref=FeedMon, 08 Aug 2005 04:00:00 ZDr. Friedrich FUTSCHIKDr. Friedrich FUTSCHIKTrajectory Near a Black Hole: an application of Lagrangian mechanics
https://www.maplesoft.com/applications/view.aspx?SID=4240&ref=Feed
The lagrangian formulation of mechanics has great advantages in practical use. Calculations of this type require finding the derivative of a function with respect to another function. Although our method is a pedagogic approach that might involve longer steps, it is a straightforward attack on this problem, and practically all problems in classical mechanics can be solved once the lagrangian is found. In most real physics problems, there are no analytic solutions to differential equations. We particularly emphasize forming plots numerically. We introduce an example in general relativity, to find the trajectory of a particle near a black hole, which corresponds to the shortest path between two points in a curved space.<img src="https://www.maplesoft.com/view.aspx?si=4240//applications/images/app_image_blank_lg.jpg" alt="Trajectory Near a Black Hole: an application of Lagrangian mechanics " style="max-width: 25%;" align="left"/>The lagrangian formulation of mechanics has great advantages in practical use. Calculations of this type require finding the derivative of a function with respect to another function. Although our method is a pedagogic approach that might involve longer steps, it is a straightforward attack on this problem, and practically all problems in classical mechanics can be solved once the lagrangian is found. In most real physics problems, there are no analytic solutions to differential equations. We particularly emphasize forming plots numerically. We introduce an example in general relativity, to find the trajectory of a particle near a black hole, which corresponds to the shortest path between two points in a curved space.https://www.maplesoft.com/applications/view.aspx?SID=4240&ref=FeedTue, 12 Mar 2002 11:24:18 ZProf. J. OgilvieProf. J. OgilvieRelativistic Astrophysics and Cosmology with Maple
https://www.maplesoft.com/applications/view.aspx?SID=3554&ref=Feed
The basics of the relativistic astrophysics including the celestial mechanics in weak field, black holes and cosmological models are illustrated and analyzed by means of Maple 6. <img src="https://www.maplesoft.com/view.aspx?si=3554//applications/images/app_image_blank_lg.jpg" alt="Relativistic Astrophysics and Cosmology with Maple " style="max-width: 25%;" align="left"/>The basics of the relativistic astrophysics including the celestial mechanics in weak field, black holes and cosmological models are illustrated and analyzed by means of Maple 6. https://www.maplesoft.com/applications/view.aspx?SID=3554&ref=FeedMon, 18 Jun 2001 00:00:00 ZVladimir KalashnikovVladimir KalashnikovTensor Analysis of Deformation
https://www.maplesoft.com/applications/view.aspx?SID=3609&ref=Feed
This is a complement to text books about large deformations. Different coordinate systems and motions can be defined and deformation tensors used for describing the deformation can be computed. Thus it can be used for problem solution in courses but also for verifying computational implementations in for example finite element codes. The worksheet gives the mathematical definitions and also contains some derivations.
<img src="https://www.maplesoft.com/view.aspx?si=3609//applications/images/app_image_blank_lg.jpg" alt="Tensor Analysis of Deformation" style="max-width: 25%;" align="left"/>This is a complement to text books about large deformations. Different coordinate systems and motions can be defined and deformation tensors used for describing the deformation can be computed. Thus it can be used for problem solution in courses but also for verifying computational implementations in for example finite element codes. The worksheet gives the mathematical definitions and also contains some derivations.
https://www.maplesoft.com/applications/view.aspx?SID=3609&ref=FeedMon, 18 Jun 2001 00:00:00 ZLars-Erik LindgrenLars-Erik LindgrenCommutator package for Maple 6
https://www.maplesoft.com/applications/view.aspx?SID=123869&ref=Feed
This package provides for manipulation and simplification of commutators, expanding commutators in terms of &* Maple's non-commutatorive multiplication operator, and converting an expression in terms of &* to commutator form. <img src="https://www.maplesoft.com/view.aspx?si=3452//applications/images/app_image_blank_lg.jpg" alt="Commutator package for Maple 6" style="max-width: 25%;" align="left"/>This package provides for manipulation and simplification of commutators, expanding commutators in terms of &* Maple's non-commutatorive multiplication operator, and converting an expression in terms of &* to commutator form. https://www.maplesoft.com/applications/view.aspx?SID=123869&ref=FeedMon, 18 Jun 2001 00:00:00 ZMichael MonaganMichael MonaganTensor product of two irreducible finite dimensional fundamental representations of a simple Lie algebra
https://www.maplesoft.com/applications/view.aspx?SID=3454&ref=Feed
Worksheet explains how to use 'crystal' to compute "3 tensor 3 tensor 3"<img src="https://www.maplesoft.com/view.aspx?si=3454//applications/images/app_image_blank_lg.jpg" alt="Tensor product of two irreducible finite dimensional fundamental representations of a simple Lie algebra" style="max-width: 25%;" align="left"/>Worksheet explains how to use 'crystal' to compute "3 tensor 3 tensor 3"https://www.maplesoft.com/applications/view.aspx?SID=3454&ref=FeedMon, 18 Jun 2001 00:00:00 ZMichael FourteMichael Fourte