What is atlas 2 for Mathematica?
atlas 2 for Mathematica is powerful Mathematica© toolbox for performing calculations in the general area of differential geometry: from formulating and solving 2D/3D problems to working with an N-dimensional manifold as a whole.
atlas package has a lot of unique features and applications such as:
- Visualization of multidimensional differential geometry objects - visualize everything from coordinate systems to surfaces and even more, no matter which dimension it has.
- Library of predefined differential geometry objects - atlas 2 package gives you access to library of predefined differential geometry objects where you can find all from 2D/3D coordinate systems to curves and surfaces.
- atlas Palette - allows visualize differential geometry objects, manipulate its parameters, generate atlas code and much more...
- atlas 2D/3D Wizard - powerful GUI AddOn for atlas package code generation
atlas 2 allows you concentrate on the differential geometry problems not on the programming.
atlas 2 uses standard differential geometry notations
which allows you always get output as you expected.
Visualization of multidimensional
differential geometry objects
makes your work more demonstrative
allows you visualize multidimensional differential objects, projecting them to a lower dimension.
Visualize everything from coordinate systems to surfaces and even more, no matter which dimension it has.
Manipulate your visualizations through graphical user interface (learn more) and use predefined differential geometry objects from the atlas library (learn more).
See differential geometry, not just calculate.
Library of predefined differential geometry objects
Over 580 differential geometry objects makes even powerful
As Platinum Service subscriber, you can get access to online library of predefined differential geometry objects directly from Mathematica using package. In the library you can find hundreds of objects: 2D/3D coordinate systems, plane and space curves, surfaces etc.
Now your work can be enriched by:
Enjoy benefits of Platinum Service for free!
You have unique chance to get absolutely free Platinum Service with any of the atlas Licenses for one year! Get the most out of the atlas tool and feel all benefits and advantages of the service with no limitations.
Platinum Service gives you free personal technical support, free software updates, free access to DG Library, discounts for product upgrades etc.
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Buy atlas 2 right now!
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Documentation
User interface of the library » -
Tutorials
atlas Palette Tutorials » -
Video
Usage of the library »
atlas Palette
atlas Palette is advanced GUI application for atlas package
- extend your keyboard. Now you can forget about hand-writing code and use the palette for typesetting of characters and atlas symbols
- get access to the online library of multidimensional differential geometry objects
- visualize objects and manipulate their parameters through graphical user interface
- generate notebook for any of the library objects that can automatically calculate differential geometry quantities for this entity
- enrich generated notebook with the object visualization and comments, that makes your work more demonstrative
atlas 2D/3D Wizard - GUI AddOn
atlas 2D/3D Wizard is powerful GUI AddOn for atlas package code generation.
This AddOn generates atlas package Mathematica® code to solve typical 2D and 3D differential geometry problems:
- 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 tensor, Gauss curvature for any surface in any 3D coordinate system
- calculation of metric, connection, Laplace operator for any 2D and 3D coordinate system
Just follow the Wizard steps, execute the generated Mathematica® notebook and have your problem solved.
With this AddOn you can solve 2D and 3D differential geometry problems even with a little knowledge in differential geometry!
Modern differential geometry
Modern differential geometry is the basis for the package. Such entities as manifolds, mappings, p-forms, tensor fields, bundles, connections are very important in the modern differential geometry. The package allows to work with these entities without extra efforts, just define an entity with corresponding obvious definition and work with it just as you usually do.
The following declarations are trivial and self explanatory:
- Domain - manifold and domain declaration
- Constants - constants declaration
- Functions - functions declaration
- Tensors - tensors declaration
- Forms - forms declaration
- Vectors - vectors declaration
- Mapping - declaration of a mapping between manifolds or domains
- Coframe - coframe declaration
- Frame - frame declaration
- Metric - metric tensor declaration
No programming just differential geometry
When working on your problem you think in terms of manifolds, mappings, embeddings, submersions, p-forms, tensor fields
etc.
The
package allows you concentrate on the differential geometry problem not on the programming.
You can use predefined declaration operators to declare various differential geometry objects, which are calculated on the fly:
- Projectors - automatic calculation of projectors of a mapping
- Invariants - automatic calculation of invariants of a mapping
- Connection - automatic calculation of connection 1-forms
- Curvature - automatic calculation of curvature 2-forms
- Torsion - automatic calculation of torsion 2-forms
- Riemann - automatic Riemann tensor calculation
- Ricci - automatic Ricci tensor calculation
- RicciScalar - automatic Ricci scalar calculation
No ugly output just standard notations
The
package uses standard differential geometry notations: d
- exterior derivative, 
- Lie derivative, ι
- interior product, 
- exterior product, 
- tensor product, 
- Hodge star, 
- covariant differentiation, δ
- Kronecker's delta symbol etc. You always get output as you expected like the following:
- atlas package output example with Lie derivative calculation:
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- atlas package output example with exterior derivative calculation:
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- atlas package output example with tensor product calculation:
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- atlas package output example with covariant derivative calculation:
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- atlas package output example with interior product and Kronecker's delta symbol calculation:
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- atlas package output example with calculation in a manifold with symbolic dimension:
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Single solving path for almost any problem
With the atlas package you always have one and the same solving path for almost any of your differential geometry problem. You start with definitions of manifolds, vector and tensor fields, p- forms and mappings between the manifolds.
When you get your differential geometry entities defined, you use standard operators to get various quantities of your entities:
- Projectors - automatic calculation of projectors of a mapping
- Invariants - automatic calculation of invariants of a mapping
- Connection - automatic calculation of connection 1-forms
- Curvature - automatic calculation of curvature 2-forms
- Torsion - automatic calculation of torsion 2-forms
- Riemann - automatic Riemann tensor calculation
- Ricci - automatic Ricci tensor calculation
- RicciScalar - automatic Ricci scalar calculation
This is standard procedure which can be automated completely.
All calculations are as coordinate free as possible
In the
package all calculations are performed in terms of tensors, vectors and p-forms (not their components!). For instance, conformally flat metric tensor of sphere 
is presented as
, where 
are coframe 1-forms and symbol - is tensor product operator (see examples below).
To get more information about the main principle of the package structure and to look through some complete examples see Examples.
To look through references list see References.
Some calculations with symbolic dimension are available
The
package allows one to make some useful calculations even if the working dimension is symbolic. For example, if 
is the dimension, 
are coframe 1-forms and 
are frame vectors then decomposition
(of interior product of vector and 1-form ) - is one of the available calculation. Another example is Lie bracket decomposition:
To get more information about this possibility see Dimension
Almost any differential geometry entity can be indexed
In the package any object (constant, tensor, p-form, manifold etc.) can be indexed. This is very flexable feature. For or more information on indexing facilities, see Indexing.
Easy customizable simplification of your results
Because computations with tensors and p-forms usually involve a great number of quantities, it is important to make simplification in each step of the computations. For this reason, the user can customise the simplification routine `atlas/simp` for a particular problem. For more information, see Simplification routine.
