另外一个Mathematica张量计算工具Ricci

http://www.math.washington.edu/~lee/Ricci/

Ricci

A Mathematica package for doing tensor calculations in differential geometry

Version 1.53

Last Updated September 16, 2011

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

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• 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
• Automatic calculation of covariant derivatives
• Automatic application of tensor symmetries
• Riemannian metrics and curvatures
• Differential forms
• Any number of vector bundles with user-defined characteristics
• Names of indices indicate which bundles they refer to
• Complex bundles and tensors
• Conjugation indicated by barred indices
• Connections with and without torsion

Limitations: Ricci currently does not support computation of explicit values for tensor components in coordinates, or derivatives of tensors depending on parameters (as in geometric evolution equations or calculus of variations), although support for these is planned for a future release. Ricci also has no explicit support for general relativity, or for other mathematical physics or engineering applications, and none is planned. If you are interested in such support, I recommend that you consider the commercial package MathTensor, which is far more extensive than Ricci, and provides all these capabilities and more. MathTensor is available fromMathTensor, Inc. <[e-mail]>.

Disclaimer: Be warned that I make no claims that this is a professional-quality software package. I have tried to make it as general and error-free as possible, and I think it is reasonably robust. However, I do most of the work on this package in my "spare time", with only very limited programming assistance, so I don't have time to check everything. I will try to fix any bugs that you encounter.

If you use this package at all, I would appreciate it if you would send me a message at [e-mail] describing your experience, and telling me whether you found the package useful or not. I'd especially like to hear about any bugs, anomalous behavior, things that look like they should simplify but don't, suggestions for improvement, things that seem to take longer than they should, etc. If I get e-mail from you, I'll put you on my mailing list to be informed whenever I release a new production version. 

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How to get Ricci

Before you obtain or use Ricci, please read the copyright notice. Then download the following files:

• Manual.pdf: User's Manual (380K - about 90 pages when printed)
• Ricci.m: the source file for the latest version of Ricci (291K)
• Example.txt: an example of Ricci usage (ASCII, 20K).
• Changes.txt: A list of all significant changes to Ricci since the first beta release (ASCII, 6K).
• Ricci.tex: TeX macros needed for Ricci's TeXForm output (ASCII, 2K)

Once you have downloaded the files, put the source file Ricci.m in the directory in which you place Mathematica input files. If you plan to use TeXForm output from Ricci, put Ricci.tex in the directory in which TeX looks for its input files. 

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Using Ricci

Ricci requires Mathematica version 2.0 or greater. The source takes approximately 283K bytes of disk storage, including about 49K bytes of on-line documentation. I have tested the current version of the package with Mathematica 5.0 under Windows. Although I haven't tested it in other situations, I know of no reason why it should not run on any platform using Mathematica 2.0 or later.

To use Ricci, put the Ricci.m source file into a directory of your own that is accessible to Mathematica. (You may need to change the value of Mathematica's $Pathvariable in your initialization file--see the documentation for the version of Mathematica that you're using.) Then load the Ricci package by typing the following Mathematica command:

<<Ricci.m

Once you've loaded Ricci into Mathematica, you can type ?name for information about any Ricci function or command.

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Changes in Version 1.53

• Fixed protection error caused by new system symbol PermutationOrder.

Here's a log of changes in prior versions of Ricci.

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Plans for future releases

What follows is an (incomplete) list of capabilities that Ricci does not currently have, but that I plan to add to some future version. I have no idea how soon these will be ready, but they're listed in approximately the order in which I plan to add them.

• WYSIWIG input and output under Mathematica 3.0 or higher.
• Tensors depending on parameters, such as g[t] or u[s,t], together with support for computing derivatives of tensor expressions with respect to the parameters. This will be useful for studying evolution equations in geometry, and for computing variational equations of geometric functionals.
• Tensors whose rank is a symbolic constant, such as n-forms or k-forms on an n-manifold. Along with this, I will implement Riemannian volume elements (what physicists call the Levi-Civita tensor or epsilon tensor) and the Hodge star operator. The limitation will be that you cannot insert indices into a tensor expression unless its rank is an explicit nonnegative integer.
• Computation of explicit values for the components of tensor expressions in terms of local coordinates.
• Kähler metrics, and the holomorphic and antiholomorphic parts of the exterior derivative (Cauchy-Riemann operators).
• Vector-valued differential forms, including wedge products and covariant exterior derivatives.

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John M. Lee
University of Washington 
Department of Mathematics 
Box 354350 
Seattle, WA 98195-4350 
USA

Phone: [phone] 
Fax: [phone] 
E-mail: [e-mail]