Differential Geometry: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=137
en-us2020 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 13 Aug 2020 18:20:16 GMTThu, 13 Aug 2020 18:20:16 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
Applications of Differential Forms in Thermodynamics
https://www.maplesoft.com/applications/view.aspx?SID=154631&ref=Feed
We use Maple's DifferentialGeometry package to derive Maxwell and other mathematical identities in thermodynamics.<img src="https://www.maplesoft.com/view.aspx?si=154631/gradS.png" alt="Applications of Differential Forms in Thermodynamics" style="max-width: 25%;" align="left"/>We use Maple's DifferentialGeometry package to derive Maxwell and other mathematical identities in thermodynamics.https://www.maplesoft.com/applications/view.aspx?SID=154631&ref=FeedSat, 11 Apr 2020 04:00:00 ZDr. Frank WangDr. Frank WangBee-Cell Structure.mw
https://www.maplesoft.com/applications/view.aspx?SID=154603&ref=Feed
Nature, in general, affords many examples of economy of space and time for anyone who is curious enough to stop for a while and think.
Pythagoras (6th Century BC) was reported to have known the fact that the circle is the figure that has the greatest surface area among all plane figures having the same perimeter which is obviously an example of economy of space.
Heron (2d Century BC) deduced the fact that light after reflection follows the shortest path, hence an example of economy of space.
Fermat (1601-1665) was seeking a way to put the law of light refraction under a form similar to that given by Heron for the light reflection but this time he was looking for an economy of time rather than space.
Huygens (1629-1695), building on Fermat finding, considered the path of light not as a straight line but rather as a curve when passing through mediums where its velocity is variable from one point to an other. Hence economy of time.
Jean Bernoulli (1667-1748) he too was building on Huygens concept when he solved the problem of the brachystochron which is based on economy of time.
This is not to say that nature economy is concerned with only physical phenomena but examples taken from living creatures abound around us. To take one example that many researches have examined very carefully and in many details in the past is that of the honeycomb building in a bee hive.
It turns out, as we shall soon prove, that the bottom of any bee-cell has the form of a trihedron with 3 equal rhombi (rhombotrihedron) which, once added to the hexagonal right prism, will make the total surface area smaller resulting in economy on the precious wax that is secreted and used by worker bee in the construction of the entire cell.
Our plan in this article has a double purpose:
1- to prove the minimal surface we referred to above using a classical proof.
2- To start with no preconceived idea about the bee-cell then
A- to consider a trihedron having 90 degrees dihedral angle between all 3 planes.
B- To get an equation relating dihedral angle with the larger angle in a rhombus that, once
solved, gives exactly the dihedral angle of 120 degrees along with the larger angle in each
rhombus as 109.47122 degrees which are the exact data one can find at the bottom end of a
honey-cell.
This configuration which is that of a minimal surface of the cell is the only one that our
equation can give for all 3 planes to have in common these two angles . All others have
different dihedral and larger angles.
I believe that the neat and simple equation I arrived at is somehow original and so far I have no idea if anyone else has found it before me.<img src="https://www.maplesoft.com/view.aspx?si=154603/3b4238cc9e7aeb803a42872bde31a350.gif" alt="Bee-Cell Structure.mw" style="max-width: 25%;" align="left"/>Nature, in general, affords many examples of economy of space and time for anyone who is curious enough to stop for a while and think.
Pythagoras (6th Century BC) was reported to have known the fact that the circle is the figure that has the greatest surface area among all plane figures having the same perimeter which is obviously an example of economy of space.
Heron (2d Century BC) deduced the fact that light after reflection follows the shortest path, hence an example of economy of space.
Fermat (1601-1665) was seeking a way to put the law of light refraction under a form similar to that given by Heron for the light reflection but this time he was looking for an economy of time rather than space.
Huygens (1629-1695), building on Fermat finding, considered the path of light not as a straight line but rather as a curve when passing through mediums where its velocity is variable from one point to an other. Hence economy of time.
Jean Bernoulli (1667-1748) he too was building on Huygens concept when he solved the problem of the brachystochron which is based on economy of time.
This is not to say that nature economy is concerned with only physical phenomena but examples taken from living creatures abound around us. To take one example that many researches have examined very carefully and in many details in the past is that of the honeycomb building in a bee hive.
It turns out, as we shall soon prove, that the bottom of any bee-cell has the form of a trihedron with 3 equal rhombi (rhombotrihedron) which, once added to the hexagonal right prism, will make the total surface area smaller resulting in economy on the precious wax that is secreted and used by worker bee in the construction of the entire cell.
Our plan in this article has a double purpose:
1- to prove the minimal surface we referred to above using a classical proof.
2- To start with no preconceived idea about the bee-cell then
A- to consider a trihedron having 90 degrees dihedral angle between all 3 planes.
B- To get an equation relating dihedral angle with the larger angle in a rhombus that, once
solved, gives exactly the dihedral angle of 120 degrees along with the larger angle in each
rhombus as 109.47122 degrees which are the exact data one can find at the bottom end of a
honey-cell.
This configuration which is that of a minimal surface of the cell is the only one that our
equation can give for all 3 planes to have in common these two angles . All others have
different dihedral and larger angles.
I believe that the neat and simple equation I arrived at is somehow original and so far I have no idea if anyone else has found it before me.https://www.maplesoft.com/applications/view.aspx?SID=154603&ref=FeedThu, 13 Feb 2020 17:34:36 ZAhmed BaroudyAhmed BaroudyGödel'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 May