Atlas™ - modern differential
geometry for Maple™
 The
Atlas package is a powerful Maple toolbox which allows you to solve a
variety of differential geometry problems using a simple computational
scheme. The computational process is highly automated which allows
you to save time by concentrating your efforts on the problem not on the
programming.
For low dimensional (2-D and 3-D) problems the package contains atlasWizard
- a Maplet tool which generates corresponding atlas package code and allows
you to formulate and solve problems in seconds. This makes the
package an ideal tool for the teaching and learning of corresponding differential
geometry courses.
The Atlas package allows you to do some differential geometry calculations
with symbolic dimension. All the results are given in standard mathematical
notations and all calculations are as coordinate free as possible.
The unique quality of the package is its ability to work with
manifolds and mappings. With the Atlas package you can define
N-dimensional manifold as a set of charts and mappings (an atlas on the
manifold) and then do calculations on defined charts. You are also able
to transform tensor fields and p-forms from one chart into another, which
allows you to work with a manifold geometry as a whole
(see atlas package Reference Manual, features and examples).
Features
- Automatic formulation and solution of typical 2-D/3-D dimensional
problems (see atlas 2-D/3-D Wizard):
- calculation of metric, connection, Laplace operator for any 2-D
and 3-D coordinate system
- calculation of curvature, torsion, tangent, principal normal and
binormal vectors for plane and space curves in any coordinate system
- calculation of metric, second fundamental form, mean curvature
vectors, Laplace operator, connection, curvature, Riemann and Ricci
tensors, Gauss curvature for any surface in any 3-D coordinate system
- Ability to do differential geometry calculations in any dimension
(some sort of calculations can be done even with symbolic dimension)
- Easy definition of and operation with typical differential geometry
quantities:
- domains (manifold charts), mappings, submersions, immersions etc.
- constants, functions, tensors, forms, vectors, coframes, frames,
metrics, connections, curvatures, torsions etc.
- Calculation of typical differential geometry operators:
- exterior derivative, Lie derivative, interior product, exterior
product, tensor product, natural vectors, Hodge operator, covariant
derivative, dual operator, gradient operator, divergence operator
etc.
- Easy definition of and operation with N-dimensional manifold as a
whole:
- definition of charts and mappings (an atlas on a manifold)
- predefined operators for transformation of differential geometry
quantities from one chart into another
- Automatic calculation of mapping invariants:
- for embedding of a curve: the curve normalised moving frame and
the curve curvatures
- for embedding or immersion: first and second fundamental forms
and mean curvature vector field
- for submersions: A and T tensor fields, mean curvature vector
of corresponding fibres, integrability obstruction of corresponding
horizontal distribution and riemannian obstruction (if the submersion
is not a riemannian one)
- Automatic calculation of mapping projectors:
- for embedding or immersion: normal and tangent projectors
- for submersions: horizontal and vertical projectors
- Technical documentation: reference manual (also available in PDF and
HTML formats), example and template worksheets.
Intended Audience:
- Educators and students using, teaching and learning differential geometry.
- Researchers, scientists and engineers who work with differential geometry
and general relativity.
- Scientists, researchers, mathematicians, physicists, engineers and
educators in a wide range of fields including differential geometry
as a mathematical background.
Technical Requirements:
- Maple 8 or higher, tested against Maple 8, Maple 9 and Maple 10
- 2 MB free disk space
- Download Size: 2 MB
Author Information
DigiArea Group Ltd
DigiArea Group Ltd was founded by Alexander A. Shklyaev. DigiArea Group
develops and markets Maple and Mathematica packages, online and 3-D scientific
games. For further information about DigiArea Group and its products,
please contact info@digi-area.com
Technical Support
MapleConnect products are supported by the developer not Maplesoft. Click
below for support details.
Terms: Standard
Email Address: support@digi-area.com
URL: http://mathshop.digi-area.com/prod/atlas/
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