Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning. Unlike the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold. in spacetime).ĭeveloped by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his general theory of relativity. In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields ( tensors that may vary over a manifold, e.g. ![]() ![]() GR calculations: adding support for GR calculations to Sage.It has been suggested that this article be merged into Ricci calculus. 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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