Visit the Atlas 2 for Maple™ site:


Buy atlas 2 for Maple »

What is Atlas 2 for Maple?


Atlas 2 for Maple is a powerful Maple 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 2 allows you to concentrate on differential geometry problems, but not on the programming.
Atlas 2 uses standard differential geometry notations which allow you to always get output as you expected.

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

DG Library Browser - Maple maplet for Atlas package to access DG Library directly from Maple.

Atlas for Maple is upgraded for Maple 18. Atlas for Maple is available for Windows, Mac and Unix.

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 Maple 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:

DG Library Browser

DG Library Browser is a Maple maplet for Atlas package

DG Library Browser allows you to access the online library of multidimensional differential geometry objects.

Now you can generate a Maple worksheet for each of the library objects. The worksheet automatically calculates differential geometry quantities for this entity.



Atlas 2D/3D Wizard - GUI Add-On

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

The Add-On generates Maple™ 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!


I am a teacher who is enrolled at the University of Padua for a second degree in Mathematics. What struck me most in Atlas tool is simple and intuitive approach, taking into account that the subject is not easy. I would definitely recommend Atlas for those learning differential geometry using traditional textbooks and as an alternative way to expand topics you already know. I am absolutely satisfied!

Diego Zampiva, Math Teacher

Modern 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

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.

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