The Mathematica Packages Cartan and MathTensor for Tensor Analysis
Mathematica is a general-purpose software system for mathematical and other applications. You can use Mathematica as a numerical and symbolic calculator, a visualization and sound-generation system, a high-level programming language, a knowledge data base, or as a way to create interactive documents that mix text and animated graphics with active formulae. In this talk I present examples of applications of two packages for tensor analysis which both use Mathematica as a base program. The presentation describes the programs as seen by the users, and I do not dwell upon technical details. Neither do I go into how algorithms are implemented. CARTAN by Soleng is an easy-to-use program for symbolic tensor component calculations in Riemann-Cartan geometries of arbitrary dimensions and signatures. It makes use of the powerful formalism of rigid frames, and it can also do calculations using the Newman-Penrose formalism. The user-friendly high-level commands of CARTAN make it an ideal tool for interactive tensor component calculations. MathTensor by Parker and Christensen also has a component package, but its main purpose and strength is that it provides a framework for indicial tensor manipulation. Using MathTensor and Mathematica you can carry out complicated index gymnastics and be sure to get the right answers. Within seconds you can for example compute the metric variation of the action of the general fourth-order gravitation theory.
With these programs computations can be done in a few seconds or minutes, which would otherwise have taken hours, days, or even weeks. The results can auto-matically be translated to TEX and imported into scientific manuscripts completely without risk of misprints or algebraic errors. In order to use the results as parts of large numerical programs, expressions can also be automatically translated into Fortran or C.
Since both programs are based on Mathematica, they are both guaranteed to have the portability, general availability, and the great graphics and visualization capability of the base program. Via MathLink the Mathematica environment can be further extended to include other programs.
KeywordsTensor Analysis Exterior Derivative Einstein Tensor Mathematica Package Rigid Frame
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