Hi, A new enhancement, devoted to tensors on generic free modules of finite rank, has been submitted to trac (ticket #15916<http://trac.sagemath.org/ticket/15916>). By *generic*, it is meant *without any distinguished basis*.
*Description* This ticket implements: - tensor products of the type M\otimes ...\otimes M \otimes M^* \otimes...\otimes M^* (k factors of M and l factors of M*, say) where M is a free module of finite rank over a commutative ring R and M^* is its dual - the elements of the above tensor products, considered as tensors of type (k,l) on M, i.e. multilinear forms (M^*)^k \times M^l --> R, thanks to the canonical isomorphism M^** = M (which holds since M is a free module of finite rank) - the following tensor operations: * operations inherent to the module structure (addition, multiplication by a ring element) * tensor product of two tensors * tensor contraction * symmetry / antisymmetry handling (on subset of the tensor arguments or on all arguments) * exterior product of alternating forms No distinguished basis is assumed on the free module M; on the contrary many bases can be introduced. Each tensor has then various representations, via its components in the various bases. *Motivation and context* The ticket has been motivated by tensors on smooth manifolds over \RR, within the SageManifolds <http://sagemanifolds.obspm.fr/> project. In this context, tensors on free modules appear at two places: - tensors on tangent spaces: * commutative ring R = real field \RR * free module M = tangent vector space at a given manifold's point - tensor fields on a manifold: * commutative ring R = the set C^\infty(N) of smooth functions N--> \RR, where N is a parallelizable open set of the manifold * free module M = the set X(N) of smooth vector fields on N (since N is parallelizable, this is a free module; its rank is the manifold's dimension) *Documentation* Apart from the numerous doctests in the code, some pieces of documentation are - the tutorial worksheet posted here<http://sagemanifolds.obspm.fr/examples/html/SM_tensors_modules.html> (the pdf version is here<http://sagemanifolds.obspm.fr/examples/pdf/SM_tensors_modules.pdf> ) - the reference manual<http://sagemanifolds.obspm.fr/doc/tensors_free_module/index.html>(the pdf version is here <http://sagemanifolds.obspm.fr/doc/tensors_free_modules_ref.pdf>); it can also be generated via the command sage -docbuild tensors_free_module html See also this page <http://sagemanifolds.obspm.fr/tensor_modules.html>. *Remarks* 1/ Although developed in the context of SageManifolds, the ticket is self-contained and does not depend on other parts of SageManifolds. It this respect, it can be viewed as some attempt to include a first subset of SageManifolds in Sage, with a moderate size: the ticket comprises 9391 lines of Python code (most of them being doctests), while at present SageManifolds contains 29240 lines of code. 2/ The ticket follows Sage's Parent/Element scheme and the (new) category framework. In particular, the ticket's free module class (FiniteFreeModule) passes the module TestSuite. 3/ It turned out to be necessary to develop a new class to implement free modules of finite rank. Indeed, the category of free modules does not exist yet in Sage: only those of generic modules (Modules) or free modules with a distinguished basis (ModulesWithBasis) are available. Now, the tangent space at a given point of a manifold is a vector space without any distinguished basis (in other words, while the tangent space is isomorphic to \RR^n, there is no *canonical* isomorphism, each isomorphism relying on the choice of some coordinate chart). The new class, FiniteFreeModule<http://sagemanifolds.obspm.fr/doc/tensors_free_module/finite_free_module.html>, does not rely on any distinguished basis. It inherits directly from sage.modules.module.Module. In particular, it does not inherit from sage.modules.module.Module_old.FreeModule_generic since the latter does not conform to the new coercion model and seems to assume a distinguished basis (cf. its method basis()<http://www.sagemath.org/doc/reference/modules/sage/modules/free_module.html#sage.modules.free_module.FreeModule_generic> ). Eric. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
