Differential Geometry: New Applications
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en-us2016 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSat, 13 Feb 2016 15:25:05 GMTSat, 13 Feb 2016 15:25:05 GMTNew applications in the Differential Geometry categoryhttp://www.mapleprimes.com/images/mapleapps.gifDifferential Geometry: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=137
Calculating Gaussian Curvature Using Differential Forms
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<p>Riemannian geometry is customarily developed by tensor methods, which is not necessarily the most computationally efficient approach. Using the language of differential forms, Elie Cartan's formulation of the Riemannian geometry can be elegantly summarized in two structural equations. Essentially, the local curvature of the manifold is a measure of how the connection varies from point to point. This Maple worksheet uses the <strong>DifferentialGeometry</strong> package to solves three problems in Harley Flanders' book on differential forms to demonstrate the implementation of Cartan's method. </p><img src="/view.aspx?si=153720/c119c404932805fdc4af274016b48a13.gif" alt="Calculating Gaussian Curvature Using Differential Forms" align="left"/><p>Riemannian geometry is customarily developed by tensor methods, which is not necessarily the most computationally efficient approach. Using the language of differential forms, Elie Cartan's formulation of the Riemannian geometry can be elegantly summarized in two structural equations. Essentially, the local curvature of the manifold is a measure of how the connection varies from point to point. This Maple worksheet uses the <strong>DifferentialGeometry</strong> package to solves three problems in Harley Flanders' book on differential forms to demonstrate the implementation of Cartan's method. </p>153720Tue, 09 Dec 2014 05:00:00 ZDr. Frank WangDr. Frank WangDerivation of Schwarzschild Metric Using Newman-Penrose Formalism
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<p>This document is an attempt to use the DifferentialGeometry tool to derive a standard metric. The Schwarzschild metric is the simpilist, so will provide a straightforward example of the DifferentialGeometry commands. Instead of plugging the general metric directly into Einstein's equations, we use the Newman-Penrose formalism.<br /><br /></p><img src="/applications/images/app_image_blank_lg.jpg" alt="Derivation of Schwarzschild Metric Using Newman-Penrose Formalism" align="left"/><p>This document is an attempt to use the DifferentialGeometry tool to derive a standard metric. The Schwarzschild metric is the simpilist, so will provide a straightforward example of the DifferentialGeometry commands. Instead of plugging the general metric directly into Einstein's equations, we use the Newman-Penrose formalism.<br /><br /></p>146772Sun, 05 May 2013 04:00:00 ZDr. Michael WatsonDr. Michael WatsonAlexander Friedmann's Cosmic Scenarios
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<p>The Russian mathematician and physicist Alexander Friedmann (1888-1925) is well known among relativists, but his contributions to cosmology are largely misunderstood. Even the Royal Swedish Academy of Sciences misrepresented Friedmann's work in the 2011 Nobel Prize scientific background essay. Friedmann was the first physicist who demonstrated that Albert Einstein's general relativity admits non-static solutions, and the universe can expand, oscillate, and be born in a singularity. Friedmann's conclusion was based on his analysis of an elliptic integral; this worksheet employs Maple's utility of handling elliptic integrals to present Friedmann's results graphically. Friedmann's differential equation governing the evolution of the universe based on Einstein's general theory of relativity is also derived using Maple's tensor package. </p><img src="/view.aspx?si=142459/friedmannscenario.jpg" alt="Alexander Friedmann's Cosmic Scenarios" align="left"/><p>The Russian mathematician and physicist Alexander Friedmann (1888-1925) is well known among relativists, but his contributions to cosmology are largely misunderstood. Even the Royal Swedish Academy of Sciences misrepresented Friedmann's work in the 2011 Nobel Prize scientific background essay. Friedmann was the first physicist who demonstrated that Albert Einstein's general relativity admits non-static solutions, and the universe can expand, oscillate, and be born in a singularity. Friedmann's conclusion was based on his analysis of an elliptic integral; this worksheet employs Maple's utility of handling elliptic integrals to present Friedmann's results graphically. Friedmann's differential equation governing the evolution of the universe based on Einstein's general theory of relativity is also derived using Maple's tensor package. </p>142459Sun, 20 Jan 2013 05:00:00 ZDr. Frank WangDr. Frank WangDifferential Geometry in Maple 16
http://www.maplesoft.com/applications/view.aspx?SID=132224&ref=Feed
With over 250 commands, the DifferentialGeometry package allows sophisticated computations from basic jet calculus to the realm of the mathematics behind general relativity. In addition, 19 differential geometry lessons, from beginner to advanced level, and 6 tutorials illustrate the use of the package in applications. This applications demonstrates some of the new functionality in Maple 16 for working with abstractly defined differential forms, general relativity, and Lie algebras.<img src="/view.aspx?si=132224/thumb.jpg" alt="Differential Geometry in Maple 16" align="left"/>With over 250 commands, the DifferentialGeometry package allows sophisticated computations from basic jet calculus to the realm of the mathematics behind general relativity. In addition, 19 differential geometry lessons, from beginner to advanced level, and 6 tutorials illustrate the use of the package in applications. This applications demonstrates some of the new functionality in Maple 16 for working with abstractly defined differential forms, general relativity, and Lie algebras.132224Tue, 27 Mar 2012 04:00:00 ZMaplesoftMaplesoftParameterizing Motion along a Curve
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<p>We use the Euler-Lagrange equation to parameterize the motion of a bead on a parabola, a helix, and a piecewise defined combination of the two.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Parameterizing Motion along a Curve" align="left"/><p>We use the Euler-Lagrange equation to parameterize the motion of a bead on a parabola, a helix, and a piecewise defined combination of the two.</p>130465Wed, 08 Feb 2012 05:00:00 ZShawn HedmanShawn HedmanClassroom Tips and Techniques: Directional Derivatives in Maple
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Several identities in vector calculus involve the operator A . (VectorCalculus[Nabla]) acting on a vector B. The resulting expression (A . (VectorCalculus[Nabla]))B is interpreted as the directional derivative of the vector B in the direction of the vector A. This is not easy to implement in Maple's VectorCalculus packages. However, this functionality exists in the Physics:-Vectors package, and in the DifferentialGeometry package where it is properly called the DirectionalCovariantDerivative.
This article examines how to obtain (A . (VectorCalculus[Nabla]))B in Maple.<img src="/view.aspx?si=126623/thumb.jpg" alt="Classroom Tips and Techniques: Directional Derivatives in Maple" align="left"/>Several identities in vector calculus involve the operator A . (VectorCalculus[Nabla]) acting on a vector B. The resulting expression (A . (VectorCalculus[Nabla]))B is interpreted as the directional derivative of the vector B in the direction of the vector A. This is not easy to implement in Maple's VectorCalculus packages. However, this functionality exists in the Physics:-Vectors package, and in the DifferentialGeometry package where it is properly called the DirectionalCovariantDerivative.
This article examines how to obtain (A . (VectorCalculus[Nabla]))B in Maple.126623Fri, 14 Oct 2011 04:00:00 ZDr. Robert LopezDr. Robert LopezDifferential Geometry in Maple
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The Maple 15 DifferentialGeometry package is the most comprehensive mathematical software available in the area of differential geometry, with 224 commands covering a wide range of topics from basic jet calculus to the realm of the mathematics behind general relativity. It includes thorough documentation including extensive examples for all these commands, 19 differential geometry lessons covering both beginner and advanced topics, and 5 tutorials illustrating the use of package in applications.
Key features include being able to perform computations in user-specified frames, inclusion of a variety of homotopy operators for the de Rham and variational bicomplexes, algorithms for the decomposition of Lie algebras, and functionality for the construction of a solvable Lie group from its Lie algebra. Also included are extensive tables of Lie algebras, Lie algebras of vectors and differential equations taken from the mathematics and mathematical physics literature.<img src="/view.aspx?si=103787/thumb.jpg" alt="Differential Geometry in Maple" align="left"/>The Maple 15 DifferentialGeometry package is the most comprehensive mathematical software available in the area of differential geometry, with 224 commands covering a wide range of topics from basic jet calculus to the realm of the mathematics behind general relativity. It includes thorough documentation including extensive examples for all these commands, 19 differential geometry lessons covering both beginner and advanced topics, and 5 tutorials illustrating the use of package in applications.
Key features include being able to perform computations in user-specified frames, inclusion of a variety of homotopy operators for the de Rham and variational bicomplexes, algorithms for the decomposition of Lie algebras, and functionality for the construction of a solvable Lie group from its Lie algebra. Also included are extensive tables of Lie algebras, Lie algebras of vectors and differential equations taken from the mathematics and mathematical physics literature.103787Wed, 06 Apr 2011 04:00:00 ZMaplesoftMaplesoftDifforms2
http://www.maplesoft.com/applications/view.aspx?SID=99700&ref=Feed
<p>An extension of the package difforms</p><img src="/view.aspx?si=99700/maple_icon.jpg" alt="Difforms2" align="left"/><p>An extension of the package difforms</p>99700Wed, 01 Dec 2010 05:00:00 ZFabian Schulte-HengesbachFabian Schulte-HengesbachVisualizing a Parallel Field in a Curved Manifold
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<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 LopezInvolute of an Ellipse
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<p>Using Maple 11 Ellipse-Evolvents have been constructed. This can be done by solving elliptic integrals with Maple. Furthermore, the author proposes a simple approximation, which is nearly identical to the elliptic-integral-solution.</p><img src="/view.aspx?si=5195/involute_30.jpg" alt="Involute of an Ellipse" align="left"/><p>Using Maple 11 Ellipse-Evolvents have been constructed. This can be done by solving elliptic integrals with Maple. Furthermore, the author proposes a simple approximation, which is nearly identical to the elliptic-integral-solution.</p>5195Thu, 17 Dec 2009 05:00:00 ZProf. Josef BettenProf. Josef BettenClassroom Tips and Techniques: Geodesics on a Surface
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<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 LopezPlane Reflection Caustics
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A procedure for computation plane reflection caustics is presented. Procedure is applied to circle and wavy circle geometries. Caustic front and field intensities are rendered.<img src="/view.aspx?si=4867/ReflectionCaustics_31.jpg" alt="Plane Reflection Caustics" align="left"/>A procedure for computation plane reflection caustics is presented. Procedure is applied to circle and wavy circle geometries. Caustic front and field intensities are rendered.4867Mon, 05 Feb 2007 00:00:00 ZHakan TiftikciHakan TiftikciStokes' Theorem
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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 AndaiA Maple Package for Computation with Differential Forms
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A package for computation with differential forms, using neither neutral- nor rebound operators
The package treats both p-forms and p-vectors (p-vector-units being the duals to p-form-units), and their mutual interaction through interior multiplication
Powerful commands like multiplication, duality, and differentiation are included in the package<img src="/view.aspx?si=1734//applications/images/app_image_blank_lg.jpg" alt="A Maple Package for Computation with Differential Forms" align="left"/>A package for computation with differential forms, using neither neutral- nor rebound operators
The package treats both p-forms and p-vectors (p-vector-units being the duals to p-form-units), and their mutual interaction through interior multiplication
Powerful commands like multiplication, duality, and differentiation are included in the package1734Mon, 01 May 2006 00:00:00 ZJohn FredstedJohn FredstedA Package for Drawing Geometric Curves (documentation in Spanish)
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A comprehensive package for drawing and analyzing a large variety of curves and surfaces. The documentation and the names of the routines are in Spanish.<img src="/view.aspx?si=4402//applications/images/app_image_blank_lg.jpg" alt="A Package for Drawing Geometric Curves (documentation in Spanish)" align="left"/>A comprehensive package for drawing and analyzing a large variety of curves and surfaces. The documentation and the names of the routines are in Spanish.4402Thu, 10 Jul 2003 15:25:31 ZDante MontenegroDante MontenegroNon trivial Lie homomorphism
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This worksheet demonstrates the use of Maple for costructing a non-trivial vector field from a given matrix G and it's representation in canonical local coords.<img src="/view.aspx?si=4359//applications/images/app_image_blank_lg.jpg" alt="Non trivial Lie homomorphism" align="left"/>This worksheet demonstrates the use of Maple for costructing a non-trivial vector field from a given matrix G and it's representation in canonical local coords.4359Mon, 03 Feb 2003 14:32:03 ZYuri GribovYuri GribovThe VectorCalculus Package
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Maple 8 provides a new package called VectorCalculus for computing with vectors, vector fields, multivariate functions and parametric curves. Computations include vector arithmetic using basis vectors, "del"-operations, multiple integrals over regions and solids, line and surface integrals, differential-geometric properties of curves, and many others. You can easily perform these computations in any coordinate system and convert results between coordinate systems. The package is fully compatible with the Maple LinearAlgebra package. You can also extend the VectorCalculus package by defining your own coordinate systems.<img src="/view.aspx?si=1382/veccalc.gif" alt="The VectorCalculus Package" align="left"/>Maple 8 provides a new package called VectorCalculus for computing with vectors, vector fields, multivariate functions and parametric curves. Computations include vector arithmetic using basis vectors, "del"-operations, multiple integrals over regions and solids, line and surface integrals, differential-geometric properties of curves, and many others. You can easily perform these computations in any coordinate system and convert results between coordinate systems. The package is fully compatible with the Maple LinearAlgebra package. You can also extend the VectorCalculus package by defining your own coordinate systems.1382Mon, 15 Apr 2002 16:13:38 ZMaplesoftMaplesoftFrenet frame of a 3D curve
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In this worksheet we will see how Maple and the vec_calc package can be used to analyse a parametrized curve. Examples include the winding line on a torus and the frenet frame of a curve.<img src="/view.aspx?si=4019//applications/images/app_image_blank_lg.jpg" alt="Frenet frame of a 3D curve" align="left"/>In this worksheet we will see how Maple and the vec_calc package can be used to analyse a parametrized curve. Examples include the winding line on a torus and the frenet frame of a curve.4019Thu, 02 Aug 2001 14:03:08 ZArthur BelmonteArthur BelmonteParametrizing surfaces
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How to parametrize surfaces. Special attention is given to surfaces of revolution.
<img src="/view.aspx?si=4012//applications/images/app_image_blank_lg.jpg" alt="Parametrizing surfaces " align="left"/>How to parametrize surfaces. Special attention is given to surfaces of revolution.
4012Thu, 02 Aug 2001 13:36:02 ZProf. Michael MayProf. Michael MayParametric curves in R2 and R3
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Parameterizing curves in R2 and R3
<img src="/view.aspx?si=4011//applications/images/app_image_blank_lg.jpg" alt="Parametric curves in R2 and R3" align="left"/>Parameterizing curves in R2 and R3
4011Thu, 02 Aug 2001 11:51:48 ZProf. Michael MayProf. Michael May