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What is Atlas 2 for Mathematica?


Atlas 2 is a powerful Mathematica toolbox for calculations in modern differential geometry and its applications.
The tool doesn't stop on common calculations and provides a bunch of flexible options to control every aspect of your project from initial problem to manipulation and visualization of results. Atlas package has a lot of unique features and capabilities described below.

Differential Geometry

Atlas includes a full list of functions for calculating common differential geometry problems, deploys results in standard math notations to maximize your productivity, and automates solving path of your tasks allowing to concentrate on ideas. Even if you're an expert, you solve differential geometry problems faster with Atlas!
Learn more »
Modern differential geometry in Mathematica is the basis for the Atlas 2 package.

Visualization and Manipulation

Atlas offers powerful functionality for visualization of multidimensional differential geometry objects uniquely integrated with Mathematica. Atlas provides access to Differential Geometry Library with hundreds of objects and continuously growing data collection.
Learn more »
Atlas Palette is an advanced Mathematica Palette for Atlas package!

Modern differential geometry

Atlas is the most modern tool for symbolic tensor computations that arise in differential geometry.
Modern differential geometry is the basis for the package. The entities such as manifolds, mappings, p-forms, tensor fields, bundles, connections are very important in the modern differential geometry. The package allows you to work with these entities without extra efforts. Define an entity with the 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 to concentrate on differential geometry problems, but 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- Lie derivative, ι - interior product, Exterior product- exterior product, Tensor product- tensor product, Hodge operator- Hodge star, Covariant derivative- 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:
  • Lie derivative formula
  • atlas package output example with exterior derivative calculation:
  • Exterior derivative formula
  • atlas package output example with tensor product calculation:
  • Tensor product formula
  • atlas package output example with covariant derivative calculation:
  • Covariant derivative formula
  • atlas package output example with interior product and Kronecker's delta symbol calculation:
  • Interior product and Kronecker's delta symbol
  • atlas package output example with calculation in a manifold with symbolic dimension:
  • Manifold with symbolic dimension

This software is absolutely a must have. Very well done. The more I use it, the more I appreciate the flow it has with the Mathematica application I am writing and my work. I am very happy with the cost and the quality of the software. It deserves a lot!

Saeed Assadi, Researcher at Texas A&M University

Single solving path for almost any problem

With the Atlas package you always have one and the same solving path for almost all your differential geometry problems. 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.

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 flexible feature. For or more information on indexing facilities, see Indexing.

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

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 customize the simplification routine `atlas/simp` for a particular problem. For more information, see Simplification routine.

Visualization of multidimensional differential geometry objects

Atlas 2 makes your work more demonstrative

allows you to visualize multidimensional differential geometry objects projecting them to a lower dimension.

Visualize everything from coordinate systems to surfaces and even more, no matter what dimension it has.

Manipulate your visualizations through graphical user interface (learn more) and use predefined differential geometry objects from the Atlas library (learn more).


Calculate. Manipulate. Explore!

Atlas is very user-friendly and doesn't bog down with a lot of programming syntax which is very important for people interested in learning. I use Atlas to explore and learn more about the complex world of differential geometry in physics primarily. I would definitely recommend Atlas as the leading differential geometry package that can work seamlessly with either Maple or Mathematica. Great tool!!!

Dr. John Fraser, Senior Principal & Energy at EnerChemTek, Inc.

Library of predefined
differential geometry objects

Over 580 differential geometry objects make Atlas 2 more powerful

As Platinum Service subscriber, you can get access to the 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:

Atlas Palette

Atlas Palette is an 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 the objects and manipulate their parameters through graphical user interface.

Generate notebook for each of the library objects. The notebook automatically calculates differential geometry quantities for this entity.

Enrich generated notebooks with the objects visualization and comments which make your work more demonstrative.

Atlas 2D/3D Wizard

Atlas 2D/3D Wizard is a powerful GUI Add-On for Atlas package code generation

Atlas 2D/3D Wizard generates Mathematica code to solve typical 2D and 3D differential geometry problems using Atlas:

- 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 the Add-On you can solve 2D and 3D differential geometry problems even with a little knowledge in differential geometry!