## 18 Tensor Calculus & Differential Geometry packages

This is the most complete list of packages for modern differential geometry and tensor calculus.

To make it even easier for you to choose, I also added information about cost, version of computational software program and short description under every package!

If you know of other, similar packages, please add a comment to the bottom.

### For Mathematica:

Atlas for Mathematica (commercial, run at Mathematica 8 and Mathematica 9)

The Atlas 2 for Mathematica package is the most powerful Mathematica add-on for doing tensor calculations in modern differential geometry. It has the following features:

- Standard differential geometry notations

- Visualization of multidimensional differential geometry objects

- Library of predefined differential geometry objects

and much more...

xAct (free, compatible with Mathematica 8)

xAct is a suite of free packages for tensor computer algebra in Mathematica. xAct implements state-of-the-art algorithms for fast manipulations of indices and has been modelled on the current geometric approach to General Relativity.

MathTensor (commercial, run at Mathematica 2.x to 8)

MathTensor for Mathematica provides more than 250 new functions and objects for elementary and advanced tensor analysis.

Ricci (free, run at Mathematica 2.0 or greater)

Ricci is a Mathematica package for doing symbolic tensor computations that arise in differential geometry. It has the following features and capabilities:

- Manipulation of tensor expressions with and without indices

- Implicit use of the Einstein summation convention

- Correct manipulation of dummy indices

- Display of results in mathematical notation, with upper and lower indices

- Riemannian metrics and curvatures

and much more...

EinS (free)

EinS is a Mathematica package allowing one to perform operations with indexed objects, which may or may not be tensors. The main application field of EinS is computations with indexed objects involving implicit (Einstein) summations (EinS stands for "Einstein Summation handler").

Tools of Tensor Calculus (TTC) (free, run at Mathematica 2.x to 7.x)

TTC is a Mathematica package for doing tensor and exterior calculus on differentiable manifolds. It can be of interest to the scientist working on any field where tensor calculus is needed. TTC can also be interesting for teaching purposes.

Tensorial (commercial, for Mathematica 5.0 or later)

Tensorial is a general purpose tensor calculus package for Mathematica 5.0 or later. Easy to learn and convenient for students and researchers.Some of the features of Tensorial 4.0 are:

- Complete freedom in choosing tensor labels, indices and base indices.

- Flavored (colored or annotated) indices for various coordinate systems.

- Differently flavor indices may have different dimensions and base index sets.

- Minimum keystroke tensor input and formatted output that can be copied and pasted.

- Detailed set of index manipulation routines.

and more other...

RGTC (free, for Mathematica 3.0 or later)

This package introduces definitions for tensor calculations in Riemannian Geometry.

CARTAN (commercial, run at Mathematica 2.x to 6.x)

Cartan is an easy-to-use tensor component package for interactive tensor calculations in general Riemann-Cartan spaces of arbitrary dimensions and signatures. The program employs the powerful formalism of rigid frames (e.g., orthonormal frames or vielbeins) and can return results both in the rigid frame or in the coordinate basis. Tensors such as Riemann, Ricci, Weyl, Einstein, Lanczos, and Cotton-York are predefined. It is also possible to extend the program by adding your own functions and variables.

### For Maple:

atlas for Maple (commercial, run at Maple 11 and later)

The atlas 2 for Maple package is 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.

DifferentialGeometry (included in the Maple package)

The new DifferentialGeometry package is a comprehensive suite of commands and subpackages featuring a collection of tightly integrated tools for computations in the areas of : calculus on manifolds (vector fields, differential forms and transformations); tensor analysis; calculus on jet spaces; Lie algebras and Lie and transformation groups.

GRTensor || (commercial, run at Maple 6 to 11)

GRTensor II is a computer algebra package for performing calculations in the general area of differential geometry. Its purpose is the calculation of tensor components on curved spacetimes specified in terms of a metric or set of basis vectors.

Riemann (free, run at Maple 3 to 11)

Riemann is a Maple package for calculating components of user-defined or built-in tensors used in General Relativity. One must define the metric or the tetrad in order to calculate tensor components. Riemann tensor package can calculate scalars or components of tensors generated by the Canon tensor package.

### For Reduce:

GRG (free, based on the computer algebra system Reduce)

The computer algebra system GRG is designed to make calculation in differential geometry and field theory as simple and natural as possible. GRG is based on the computer algebra system Reduce but GRG has its own simple input language whose commands resemble short English phrases.

EXCALC (free)

EXCALC is designed for easy use by all who are familiar with the calculus of Modern Differential Geometry. Its syntax is kept as close as possible to standard textbook notations. Therefore, no great experience in writing computer algebra programs is required. It is almost possible to input to the computer the same as what would have been written down for a hand-calculation.

RicciR (free)

Ricci calculus is performed in the framework of the computer algebra system REDUCE by the package being presented. The program called RICCIR is specialized in the requirements of general relativity.

### For Maxima:

itensor (free)

The itensor package can carry out differentiation with respect to an indexed variable, which allows one to use the package when dealing with Lagrangian and Hamiltonian formalisms. As it is possible to differentiate a field Lagrangian with respect to an (indexed) field variable, one can use Maxima to derive the corresponding Euler-Lagrange equations in indicial form.

### You also want to see...

Most useful differential geometry packages for Maple

This is the overview article about three most useful and famous differential geometry packages.

- DifferentialGeometry Package from Maplesoft, Inc.

- atlas 2 for Maple from DigiArea, Inc.

- GRTensor 2 from Queen's University