Tensor Calculus in Sage
This page arose out of a thread at sage-devel on the use of differential forms in Sage. Differential forms have been mentioned on the mailing list a few times before, and in the current discussion a number of interesting packages for tensor calculus were mentioned, which are listed here.
This list is by no means complete, so please feel free to edit.
As tensor calculus is a vast subject, at some stage we will want to have a roadmap of which tasks to handle first, benchmarks, and useful applications. See this paper for some real-world applications.
GRTensor is a package for Maple (with a port to Mathematica) for geometry computations in general relativity. From the web page: 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.
The Ricci package in Mathematica looks terrific, but I don't have Mathematica so I can't experiment with it.
Cadabra is a tensor package designed for computations in field theory (HEP, GR). It looks very powerful and versatile, but the syntax is very terse.
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.
Maxima seems to have a differential forms package.
FriCAS has support for a De Rham complex, which (among others) apparently allows you to represent differential forms.
Scmutils has lots of code to deal with forms, Riemannian geometry, etc. plus lots of cool applications.
Mathematica also has two more packages for doing differential forms: http://library.wolfram.com/infocenter/MathSource/683 and http://library.wolfram.com/infocenter/MathSource/482/ (the last one has a nice Integral command, for example).
GluCat: GluCat is a library of template classes which model the universal Clifford algebras over the field of real numbers, with arbitrary dimension and arbitrary signature. GluCat implements a model of each Clifford algebra corresponding to each non-degenerate quadratic form up to a maximum number of dimensions.
Related Sage discussions
There are a few Sage projects in the works that might be interesting in the context of differential forms and tensor calculus. A quick search brings up the following.
GR calculations: adding support for GR calculations to Sage.
Topology: Sage support for topology.