Tuesday, April 26, 2011

Most useful differential geometry packages for Maple. Overview.

Most useful differential geometry packages for Maple. Overview.

In this article we discuss most useful and famous differential geometry packages for Maple.
DifferentialGeometry Package from Maplesoft, Inc.
Atlas 2 for Maple from DigiArea, Inc.
GRTensor 2 from Queen's University

This is by no means a complete list. If you know of other, similar packages that you find useful, please add a comment to the bottom.



Atlas 2 for Maple from DigiArea

The atlas 2 for Maple package is a powerful Maple toolbox which allows you to do a wide range of modern differential geometry calculations: from formulating and solving 2D/3D problems to working with an N-dimensional manifold as a whole.

What's inside?

No programming just differential geometry - atlas package allows you concentrate on the differential geometry problem not on the programming.
No ugly output just standard notation - atlas package uses standard differential geometry notations: d - exterior derivative, ℒ - Lie derivative, ι - interior product, ⋀ - exterior product, ⊗ - tensor product, ⋆ - Hodge star, ∇ - covariant differentiation, δ - Kronecker's delta symbol etc.
Single solving path for almost any problem - with the atlas package you always have one and the same solving path. It keeps your time and makes the solving really fast.
All calculations are as coordinate free as possible - atlas package all calculations are performed in terms of tensors, vectors and p-forms (not their components!)
Almost any differential geometry entity can be indexed - any object (constant, tensor, p-form, manifold etc.) can be indexed. This is very flexable feature. For or more information on atlas indexing facilities, see atlas[indexing].
Easy customizable simplification of your results - because computations with tensors and p-forms usually involve a great number of quantities, the user can customise the simplification routine `atlas/simp` for a particular problem.

Compatibility

Successfully tested and compatible with all versions no less than the Maple 11.
atlas 2D/3D Wizard - really useful Maplet which generates atlas package Maple code to solve typical 2D and 3D differential geometry problems. You solve differential geometry just clicking the button.
Also see this Examples list and Template Worksheets for fast understanding how it works.

DifferentialGeometry Package from Maplesoft, Inc.

DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds(vector fields, differential forms and transformations), tensor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus.

What's inside?

Package includes subpackages that makes it really powerful:
Tensor package - computations with tensors on the tangent bundle of any manifold or with tensors on any vector bundle, with specialized functionality for advanced general relativity.
LieAlgebras package - for defining Lie algebras from a variety of sources and creating new ones, contains 2 subpackages, LieAlgebraCohomology and LieAlgebraRepresentations.
GroupActions Package - for working with Lie groups and infinitesimal transformations, contains one subpackage, MovingFrames.
JetCalculus package - for symbolic computations on jet spaces and is fully compatible with the other packages and commands in DifferentialGeometry.
Library package - for browsing and searching tables of Lie algebras and differential equations.
Tools package - with utility procedures for developing new differential geometry applications, contains one subpackage, DGmain.

Compatibility

Was included in the Maple package starting with Maple 11
See this Lessons and Tutorials for understanding how it works.
The DifferentialGeometry lessons provide a systematic approach to learning the commands in the DifferentialGeometry, Tensor, LieAlgebras and JetCalculus packages.
Each lesson contains a set of exercises which range in difficulty from simple computational exercise to programming exercises. Solutions are given. The tutorials present specialized applications of the DifferentialGeometry package.



GRTensor 2 from Queen's University

The GRTensor II package is a computer algebra package for performing calculations in the general area of differential geometry: calculations can be carried out in spaces of arbitrary dimension, and in multiple spacetimes simultaneously.

What's inside?

Work in either metric or basis vector formalisms - the purpose of GRTensor II is the calculation of tensor components on curved spacetimes specified in terms of a metric or set of basis vectors.
Perform calculations in any number of dimensions - calculations can be carried out in spaces of arbitrary dimension, and in multiple spacetimes simultaneously.
Calculate standard objects (Riemann, Ricci tensors, etc.) - the package contains a library of standard definitions of a large number of commonly used curvature tensors, as well as the Newman-Penrose formalism. The standard object libraries are easily expandable by a facility for defining new tensors.

Compatibility

Tests show that GRTensorII 1.79 works as expected with Maple 11. Install as with Maple 10, 9.5, 9, 8.
GRTensor II and related software and documentation are distributed free of charge as an aide for both research and teaching. A limited version (GRTensorM) has been ported to Mathematica.
See this Demonstrations and Documentation for understanding how it works.


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