Tensors: New Applications
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en-us2017 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSun, 26 Feb 2017 03:20:44 GMTSun, 26 Feb 2017 03:20:44 GMTNew applications in the Tensors categoryhttp://www.mapleprimes.com/images/mapleapps.gifTensors: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=151
Visualizing a Parallel Field in a Curved Manifold
http://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="/view.aspx?si=35113/thumb.jpg" alt="Visualizing a Parallel Field in a Curved Manifold" 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>35113Thu, 28 Jan 2010 05:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Geodesics on a Surface
http://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="/view.aspx?si=34940/thumb.jpg" alt="Classroom Tips and Techniques: Geodesics on a Surface" 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>34940Tue, 08 Dec 2009 05:00:00 ZDr. Robert LopezDr. Robert LopezGravitation
http://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="/view.aspx?si=4891//applications/images/app_image_blank_lg.jpg" alt="Gravitation" 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 connection4891Thu, 05 Apr 2007 00:00:00 ZJohn FredstedJohn FredstedStokes' Theorem
http://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="/view.aspx?si=1755/stokesend_175.gif" alt="Stokes' Theorem" 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.1755Mon, 26 Jun 2006 00:00:00 ZAttila AndaiAttila AndaiDynamics on Our Rotating Earth
http://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="/view.aspx?si=1650/Freier_Fall_88.gif" alt="Dynamics on Our Rotating Earth" 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.1650Tue, 09 Aug 2005 00:00:00 ZDr. Friedrich FUTSCHIKDr. Friedrich FUTSCHIKTrajectory Near a Black Hole: an application of Lagrangian mechanics
http://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="/view.aspx?si=4240//applications/images/app_image_blank_lg.jpg" alt="Trajectory Near a Black Hole: an application of Lagrangian mechanics " 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.4240Tue, 12 Mar 2002 11:24:18 ZProf. J. OgilvieProf. J. OgilvieRelativistic Astrophysics and Cosmology with Maple
http://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="/view.aspx?si=3554//applications/images/app_image_blank_lg.jpg" alt="Relativistic Astrophysics and Cosmology with Maple " 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. 3554Mon, 18 Jun 2001 00:00:00 ZVladimir KalashnikovVladimir KalashnikovTensor Analysis of Deformation
http://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="/view.aspx?si=3609//applications/images/app_image_blank_lg.jpg" alt="Tensor Analysis of Deformation" 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.
3609Mon, 18 Jun 2001 00:00:00 ZLars-Erik LindgrenLars-Erik LindgrenCommutator package for Maple 6
http://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="/view.aspx?si=3452//applications/images/app_image_blank_lg.jpg" alt="Commutator package for Maple 6" 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. 123869Mon, 18 Jun 2001 00:00:00 ZMichael MonaganMichael MonaganTensor product of two irreducible finite dimensional fundamental representations of a simple Lie algebra
http://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="/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" align="left"/>Worksheet explains how to use 'crystal' to compute "3 tensor 3 tensor 3"3454Mon, 18 Jun 2001 00:00:00 ZMichael FourteMichael Fourte