Differential Geometry: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=137
en-us2019 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 12 Dec 2019 20:16:49 GMTThu, 12 Dec 2019 20:16:49 GMTNew applications in the Differential Geometry categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgDifferential Geometry: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=137
Gödel's Universe
https://www.maplesoft.com/applications/view.aspx?SID=154430&ref=Feed
In 1949, Kurt Gödel proposed a solution to Einstein's field equations that exhibited a rotation of matter (with a nonvanishing cosmological constant). This solution allows the existence of closed timelike curves, which implies the possibility of time travel. In this worksheet, we demonstrate the use of DifferentialGeometry and Tensor packages to calculate the Ricci tensor, Ricci scalar, and Einstein tensor. We also apply the calculus of variations to calculate the null geodesics (trajectories of light rays) in Gödel's universe. The notation is based on the presentation in the monograph of Stephen Hawking and George Ellis.<img src="https://www.maplesoft.com/view.aspx?si=154430/godelnullgeodesics.png" alt="Gödel's Universe" style="max-width: 25%;" align="left"/>In 1949, Kurt Gödel proposed a solution to Einstein's field equations that exhibited a rotation of matter (with a nonvanishing cosmological constant). This solution allows the existence of closed timelike curves, which implies the possibility of time travel. In this worksheet, we demonstrate the use of DifferentialGeometry and Tensor packages to calculate the Ricci tensor, Ricci scalar, and Einstein tensor. We also apply the calculus of variations to calculate the null geodesics (trajectories of light rays) in Gödel's universe. The notation is based on the presentation in the monograph of Stephen Hawking and George Ellis.https://www.maplesoft.com/applications/view.aspx?SID=154430&ref=FeedMon, 02 Apr 2018 04:00:00 ZDr. Frank WangDr. Frank WangCalculating Gaussian Curvature Using Differential Forms
https://www.maplesoft.com/applications/view.aspx?SID=153720&ref=Feed
<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="https://www.maplesoft.com/view.aspx?si=153720/c119c404932805fdc4af274016b48a13.gif" alt="Calculating Gaussian Curvature Using Differential Forms" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=153720&ref=FeedTue, 09 Dec 2014 05:00:00 ZDr. Frank WangDr. Frank WangDifferential Geometry in Maple 16
https://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="https://www.maplesoft.com/view.aspx?si=132224/thumb.jpg" alt="Differential Geometry in Maple 16" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=132224&ref=FeedTue, 27 Mar 2012 04:00:00 ZMaplesoftMaplesoftParameterizing Motion along a Curve
https://www.maplesoft.com/applications/view.aspx?SID=130465&ref=Feed
<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="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Parameterizing Motion along a Curve" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=130465&ref=FeedWed, 08 Feb 2012 05:00:00 ZShawn HedmanShawn HedmanClassroom Tips and Techniques: Directional Derivatives in Maple
https://www.maplesoft.com/applications/view.aspx?SID=126623&ref=Feed
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="https://www.maplesoft.com/view.aspx?si=126623/thumb.jpg" alt="Classroom Tips and Techniques: Directional Derivatives in Maple" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=126623&ref=FeedFri, 14 Oct 2011 04:00:00 ZDr. Robert LopezDr. Robert LopezDifferential Geometry in Maple
https://www.maplesoft.com/applications/view.aspx?SID=103787&ref=Feed
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="https://www.maplesoft.com/view.aspx?si=103787/thumb.jpg" alt="Differential Geometry in Maple" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=103787&ref=FeedWed, 06 Apr 2011 04:00:00 ZMaplesoftMaplesoftDifforms2
https://www.maplesoft.com/applications/view.aspx?SID=99700&ref=Feed
<p>An extension of the package difforms</p><img src="https://www.maplesoft.com/view.aspx?si=99700/maple_icon.jpg" alt="Difforms2" style="max-width: 25%;" align="left"/><p>An extension of the package difforms</p>https://www.maplesoft.com/applications/view.aspx?SID=99700&ref=FeedWed, 01 Dec 2010 05:00:00 ZFabian Schulte-HengesbachFabian Schulte-HengesbachVisualizing 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 LopezInvolute of an Ellipse
https://www.maplesoft.com/applications/view.aspx?SID=5195&ref=Feed
<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="https://www.maplesoft.com/view.aspx?si=5195/involute_30.jpg" alt="Involute of an Ellipse" style="max-width: 25%;" 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>https://www.maplesoft.com/applications/view.aspx?SID=5195&ref=FeedThu, 17 Dec 2009 05:00:00 ZProf. Josef BettenProf. Josef BettenClassroom 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 LopezPlane Reflection Caustics
https://www.maplesoft.com/applications/view.aspx?SID=4867&ref=Feed
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="https://www.maplesoft.com/view.aspx?si=4867/ReflectionCaustics_31.jpg" alt="Plane Reflection Caustics" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=4867&ref=FeedMon, 05 Feb 2007 00:00:00 ZHakan TiftikciHakan TiftikciStokes' 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 AndaiA Maple Package for Computation with Differential Forms
https://www.maplesoft.com/applications/view.aspx?SID=1734&ref=Feed
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="https://www.maplesoft.com/view.aspx?si=1734//applications/images/app_image_blank_lg.jpg" alt="A Maple Package for Computation with Differential Forms" style="max-width: 25%;" 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 packagehttps://www.maplesoft.com/applications/view.aspx?SID=1734&ref=FeedMon, 01 May 2006 00:00:00 ZJohn FredstedJohn FredstedA Package for Drawing Geometric Curves (documentation in Spanish)
https://www.maplesoft.com/applications/view.aspx?SID=4402&ref=Feed
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="https://www.maplesoft.com/view.aspx?si=4402//applications/images/app_image_blank_lg.jpg" alt="A Package for Drawing Geometric Curves (documentation in Spanish)" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=4402&ref=FeedThu, 10 Jul 2003 15:25:31 ZDante MontenegroDante MontenegroNon trivial Lie homomorphism
https://www.maplesoft.com/applications/view.aspx?SID=4359&ref=Feed
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="https://www.maplesoft.com/view.aspx?si=4359//applications/images/app_image_blank_lg.jpg" alt="Non trivial Lie homomorphism" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=4359&ref=FeedMon, 03 Feb 2003 14:32:03 ZYuri GribovYuri GribovThe VectorCalculus Package
https://www.maplesoft.com/applications/view.aspx?SID=1382&ref=Feed
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="https://www.maplesoft.com/view.aspx?si=1382/veccalc.gif" alt="The VectorCalculus Package" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=1382&ref=FeedMon, 15 Apr 2002 16:13:38 ZMaplesoftMaplesoftFrenet frame of a 3D curve
https://www.maplesoft.com/applications/view.aspx?SID=4019&ref=Feed
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="https://www.maplesoft.com/view.aspx?si=4019//applications/images/app_image_blank_lg.jpg" alt="Frenet frame of a 3D curve" style="max-width: 25%;" 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.https://www.maplesoft.com/applications/view.aspx?SID=4019&ref=FeedThu, 02 Aug 2001 14:03:08 ZArthur BelmonteArthur BelmonteParametrizing surfaces
https://www.maplesoft.com/applications/view.aspx?SID=4012&ref=Feed
How to parametrize surfaces. Special attention is given to surfaces of revolution.
<img src="https://www.maplesoft.com/view.aspx?si=4012//applications/images/app_image_blank_lg.jpg" alt="Parametrizing surfaces " style="max-width: 25%;" align="left"/>How to parametrize surfaces. Special attention is given to surfaces of revolution.
https://www.maplesoft.com/applications/view.aspx?SID=4012&ref=FeedThu, 02 Aug 2001 13:36:02 ZProf. Michael MayProf. Michael MayParametric curves in R2 and R3
https://www.maplesoft.com/applications/view.aspx?SID=4011&ref=Feed
Parameterizing curves in R2 and R3
<img src="https://www.maplesoft.com/view.aspx?si=4011//applications/images/app_image_blank_lg.jpg" alt="Parametric curves in R2 and R3" style="max-width: 25%;" align="left"/>Parameterizing curves in R2 and R3
https://www.maplesoft.com/applications/view.aspx?SID=4011&ref=FeedThu, 02 Aug 2001 11:51:48 ZProf. Michael MayProf. Michael MayMaptools package
https://www.maplesoft.com/applications/view.aspx?SID=3925&ref=Feed
This package enables the user to produce 2-D and 3-D world map projections of the earth, including a full database of coastal points, on the plane or a parametrizable surface using whatever equations one wants! The maps it produces are high quality, useable in books or journal articles<img src="https://www.maplesoft.com/view.aspx?si=3925/map.gif" alt="Maptools package" style="max-width: 25%;" align="left"/>This package enables the user to produce 2-D and 3-D world map projections of the earth, including a full database of coastal points, on the plane or a parametrizable surface using whatever equations one wants! The maps it produces are high quality, useable in books or journal articleshttps://www.maplesoft.com/applications/view.aspx?SID=3925&ref=FeedTue, 10 Jul 2001 10:55:05 ZVincent CostanzoVincent Costanzo