![]() ![]() In addition, a tensor with rank may be of mixed type, consisting of so-called "contravariant" (upper) indices The notation for a tensor is similar to that of a matrix (i.e., ), except that a tensor, ,, etc., may have an arbitrary number of indices. Tensors provide a natural and concise mathematical framework for formulating and solving problems in areas of physics such as elasticity, fluid mechanics, and general relativity. Have exactly two indices) to an arbitrary number of indices. (that have exactly one index), and matrices (that Tensors are generalizations of scalars (that have no indices), vectors (with the notable exception of the contracted Kroneckerĭelta). However, the dimension of the space is largely irrelevant in most tensor equations Of a tensor ranges over the number of dimensions of space. GR calculations: adding support for GR calculations to Sage.Tensor in -dimensional space is a mathematicalĬomponents and obeys certain transformation rules. There are a few Sage projects in the works that might be interesting in the context of differential forms and tensor calculus. GluCat implements a model of each Clifford algebra corresponding to each non-degenerate quadratic form up to a maximum number of dimensions. 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. JET: Axiom code to deal with jet bundles, geometric ODEs/PDEs, Cartan-Kuranishi prolongations, etc. Mathematica also has two more packages for doing differential forms: and (the last one has a nice Integral command, for example). Scmutils has lots of code to deal with forms, Riemannian geometry, etc. Maxima seems to have a differential forms package.įriCAS has support for a De Rham complex, which (among others) apparently allows you to represent differential forms. xAct implements state-of-the-art algorithms for fast manipulations of indices and has been modelled on the current geometric approach to General Relativity. XAct is a suite of free packages for tensor computer algebra in Mathematica. It looks very powerful and versatile, but the syntax is very terse. The Ricci package in Mathematica looks terrific, but I don't have Mathematica so I can't experiment with it.Ĭadabra is a tensor package designed for computations in field theory (HEP, GR). Its purpose is the calculation of tensor components on curved spacetimes specified in terms of a metric or set of basis vectors. ![]() From the web page: GRTensor II is a computer algebra package for performing calculations in the general area of differential geometry. GRTensor is a package for Maple (with a port to Mathematica) for geometry computations in general relativity. See this paper for some real-world applications. This list is by no means complete, so please feel free to edit.Īs 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. ![]() 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 page arose out of a thread at sage-devel on the use of differential forms in Sage. ![]() Examples of use are here see also the tutorial. NB: this page is obsolete: tensor calculus is now fully implemented in Sage, see the Manifolds section of the reference manual. ![]()
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