Differentiable manifolds: tangent spaces

[egourgoulhon]

This ticket implements tangent vectors and tangent spaces on differentiable manifolds. This is a follow-up of #18843 within the ​SageManifolds project (see the metaticket #18528 for an overview). As in #18843, the non-discrete topological field K over which the differentiable manifold is defined is generic, although in most applications, K=R or K=C.

This ticket implements the following Python classes:

• the parent class TangentSpace for tangent spaces to a differentiable manifold
• the element class TangentVector for tangent vectors to a differentiable manifold

It also implements

• the method at() in tensor field classes (introduced in #18843) to take the value of a given tensor field (in particular a vector field) at a given point; the method at() is implemented for vector frames and coframes as well.
• the method plot() in class VectorField, to get some graphical representations of vector fields
• the methods differential(), differential_functions() and jacobian_matrix() to the class DiffMap introduced in #18783 (differential of a map between differentiable manifolds at a given point).

Documentation: The reference manual is produced by sage -docbuild reference/manifolds html It can also be accessed online at ​http://sagemanifolds.obspm.fr/doc/19092/reference/manifolds/ More documentation (e.g. example worksheets) can be found ​here.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​6a1fc97

Add method differential() to class DiffMap

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​31d6789

Add method plot() to class VectorField; full doctest coverage.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​04360b0

Major improvements in the documentation of diff. manifolds (tangent spaces part)

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​513f5ba

Improve TensorField.__eq__ (case with no open cover known)

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​00c327d	Slight reorganization of the reference manual of topological manifolds (morphisms part)
​f8d3f27	Merge #18725 into #18640
​f2fef7b	Small improvements in the documentation of differentiable manifolds
​8ab80d8	Improvement in simplify_sqrt_real(); minor modif. in documentation
​2f231b6	Major improvements in the documentation of diff. manifolds (basic part)
​f0ca2de	Merge #18783 into #18725
​307e655	Major improvements in the documentation of diff. manifolds (tensor field part)
​dc7f7a1	Improve TensorField.__eq__ (case with no open cover known)
​4d6f21c	Merge #18843 into #18783
​e43e03b	Merge #19092 into #18843

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​041a5d1	Adding p-adics to metric spaces and some cleanup.
​bfa0cdf	One last doc tweak.
​d13c368	Fixing doc of metric spaces.
​2605c0b	Merge #18529 (Topological manifolds: basics) into #18175 (Implement categories for topological...)
​6dec6d5	Implement topological manifolds (basics, #18529) on the new categories for manifolds (#18175)
​3403978	Implement top. manifolds (scalar fields, #18640) on the new manifold categories (#18175)
​b0521ef	Implement top. manifolds (morphisms, #18725) on the new manifold categories (#18175)
​f643097	Implement diff. manifolds (basics, #18783) on the new manifold categories (#18175)
​a527726	Implement diff. manifolds (tensor fields, #18843) on the new manifold categories (#18175)
​22923ab	Implement diff. manifolds (tangent spaces, #19092) on the new manifold categories (#18175).

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​668bc26	Remove UniqueRepresentation, leaving only WithEqualityById, for topological manifold homsets
​6518699	Introduce the attribute _field_type in class TopologicalManifold to check for real and complex manifolds.
​22383e6	Check for real/complex manifold performed on base_field_type() instead of RR/CC
​66f2c5a	Change function('f', x) to function('f')(x) to accomodate the deprecation warning introduced in #17447
​a28ed04	Morphisms of topological manifolds with the use of base_field_type()
​f31bed1	Remove UniqueRepresentation from differentiable manifolds
​0ee4c41	Tensor fields on differentiable manifolds without unique representation
​e8f11ff	Fix pickling test in tensor field modules.
​a967787	Suppress direct call to _element_constructor_ in tensor field parent classes
​1bf77fd	Tangent spaces on differentiable manifolds without unique representation

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​d8397c1	Merge branch 'public/manifolds/top_manif_basics' of trac.sagemath.org:sage into public/manifolds/top_manif_basics
​0b08b11	Some small tweaks as part of the review.
​d3e5d4d	Revert to UniqueRepresentation for topological manifolds
​2481359	Revert to UniqueRepresentation for ScalarFieldAlgebra; better ScalarField constructor
​0cc06e8	Revert to UniqueRepresentation for topological manifold homsets
​322d5bd	Revert to UniqueRepresentation for differentiable manifolds
​45beae5	Revert to UniqueRepresentation for differentiable manifolds: tensor fields
​98c6036	Revert to UniqueRepresentation for differentiable manifolds: tangent spaces

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​85d03dc	Change the argument type to structure in Manifold
​5251ef0	Remove method _test_pickling from class TopologicalManifoldPoint
​f69c9ee	Fix doctest error in coord_func_symb.py due to #19312 (update to pynac-0.5.2)
​7889a5d	Change in simplify_sqrt_real to cope with the change of != operator induced by #19312 (Sage 6.10.beta7)
​e8d2ba6	Differentiable manifolds: basics with the change in symbolic expression logic induced by #19312 (Sage 6.10.beta7)
​0d68f86	Diff. manifolds: tensor fields, with the change in symbolic expression logic induced by #19312 (Sage 6.10.beta7)
​347a9cc	Diff. manifolds: tangent spaces, with the change in symbolic expression logic induced by #19312 (Sage 6.10.beta7)

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​1c1fccc	Parallel plot for Manifold: only 2D. Flat parallelization.
​9b87ff1	Vectorfields : plot parallelization improvement and bug corrction.
​db99641	Parallel plotting vectorfield: added example

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​ce03503	Scalar fields on topological manifolds: slight improvements in the documentation
​9d25aa0	Scalar field algebras with AbstractNamedObject without argument full_name
​ed4aa58	ScalarFieldAlgebra does not longer inherit from AbstractNamedObject
​bc77059	Morphisms on the refactored topoological manifolds
​f6c6a34	Topological manifold morphisms with AbstractNamedObject without full_name
​2df7af7	Refactorization of differentiable manifolds, with the mixin class DifferentiableMixin
​7204572	Basics of diff. manifolds with AbstractNamedObject without full_name
​ecfbf2c	Tensor fields on refactored differentiable manifolds
​8b720a5	Tensor fields with AbstractNamedObject without full_name
​b07ffd9	Tangent spaces on refactored differentiable manifolds

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​3cd03a4	Add methods lift() and retract() to ManifoldSubset; add method __eq__() in CoordChange
​984c3c2	Revert to simple hierarchy for manifold classes
​c01048f	Scalar fields with the simplified hierarchy for manifold classes
​c866d6c	Morphisms of topological manifolds with the simplified hierarchy for manifold classes
​0e04631	Basics of diff. manifolds with the simplified hierarchy for manifold classes
​1f226bd	Tensor fields with the simplified hierarchy for manifold classes
​9fa77c4	Tangent spaces with the simplified hierarchy for manifold classes

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​8e17d54	Merge into the latest version of #18529; improve treatment of composite functions in ExpressionNice
​f00be00	Topological manifold morphisms: solved merge conflict with Sage 7.1.beta1
​21b3968	Basics of diff. manifolds: solve merge conflict with Sage 7.1.beta1
​8ba4b91	Tensor fields: solve merge conflict with Sage 7.1.beta1
​2fad9db	Correct doctest in class DiffFormParal
​b47f55f	Tangent spaces: solve merge conflict with Sage 7.1.beta1

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​f11df50	Set CoordFunctionSymbRing in the category of commutative algebras over SR; minor documentation changes
​f3f5470	Fixing a (essentially trivial) doctest failure.
​7b3dab3	Merge branch 'public/manifolds/top_manif_scalar_fields' of git://trac.sagemath.org/sage into Sage 7.2.rc0
​9ec7d3e	Python 3 format for print in manifolds
​d190b37	Morphisms of topological manifolds with coordinate functions as algebra elements
​2577fc5	Fixing a (essentially trivial) doctest failure.
​7125e29	Python3 format for print in morphims of topological manifolds
​bd2f35a	Diff. manifolds with coordinate functions as algebra elements
​73b4a03	Tensor fields with with coordinate functions as algebra elements
​37521ae	Tangent spaces with coordinate functions as algebra elements

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​4fad094	Merge branch 'public/manifolds/top_manif_morphisms' of trac.sagemath.org:sage into public/manifolds/top_manif_morphisms
​cfecb18	Reviewer changes and tweaks for continuous maps ticket.
​efcb618	Modify authorship for continuous maps.
​84051b0	Merge branch 'public/manifolds/diff_manif_basics' of trac.sagemath.org:sage into public/manifolds/diff_manif_basics
​97172dd	Basics of differentiable manifolds with changes in morphisms of topological manifolds
​29832ec	Use @cached_method for VectorFieldModule.identity_map() and AutomorphismFieldGroup.one()
​159afb4	Tangent spaces upon latest version of tensor fields

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​2150a43	Correct doctests in vector fields after the merge of #20770 in sage 7.3.beta3
​3b1b794	Fix doctests in tangent spaces after the merge of #20770 in sage 7.3.beta3

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​3ab0af0	Initial reviewer changes.
​d7ad8e1	Merge branch 'public/manifolds/diff_manif_tensor_fields' of git://trac.sagemath.org/sage into sage 7.3.beta5
​dcb08fc	Decrease doctest times in tensor fields; remove xder from the global namespace
​c70792e	Update tangent spaces to the latest version of #18843 (vector fields)

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​88295c6	Merge branch 'develop' into public/manifolds/diff_manif_tensor_fields
​58f7fb9	Merge branch 'develop' into public/manifolds/diff_manif_tensor_fields
​3e73095	Merge branch 'develop' into public/manifolds/diff_manif_tensor_fields
​4955c3c	Final review of everything.
​7cf4ff4	Merge branch 'public/manifolds/diff_manif_tensor_fields' of trac.sagemath.org:sage into public/manifolds/diff_manif_tensor_fields
​58ae506	Merge branch 'public/manifolds/diff_manif_tensor_fields' of git://trac.sagemath.org/sage into Sage 7.4.beta6
​8219403	Fix documentation error; change in TensorField comparison to zero
​a841f79	Python 3 compatible syntax in tensor fields
​5a5f400	Slight change in TensorField.__eq__ (comparison to zero)
​98efa66	Update tangent spaces to the final version of the dependency ticket #18843

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​8860387	Add missing reference Lee13
​8205dca	Update tangent spaces to the last change (adding a missing reference) in the dependency ticket #18843

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​fb7f4dd	Bibliographic references for tensor fields moved to the master file
​785300b	Bibliographic references for tangent spaces moved to the master file created by #21454

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​4ce1118	Merge branch 'public/manifolds/diff_manif_tangent_spaces' of git://trac.sagemath.org/sage into public/manifolds/diff_manif_tangent_spaces
​567b892	Doing some reviewer changes for #19092.

[tscrim]

Okay, I've made it through this one. Mostly it is documentation format changes, but the exception:

For plotting, I used the @options for scale and max_range as these seemed like something a user might want to not have to pass every single time (you change change their default with v.plot.options). From the user perspective, nothing really changes unless you explicitly set said default value.

So if you agree with my changes, then you can set a positive review.

[tscrim]

I'm handling rebasing #19124 as I'm starting my review of that now.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​05f8cc7

A little bit of extra cleanup.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​23eb9d8	Correct doctest and documentation in tangent spaces
​b4a4530	Use sphinx_plot for 3D plots in the documentation of charts and vector fields

[egourgoulhon]

Thank you so much for the review!

In the first commit above (23eb9d8), I've corrected a doctest error and fixed some documentation issue revealed by the patchbot.

In the second commit (b4a4530), I've added sphinx directives to render the 3D plots in the documentation of VectorField.plot (as well as in RealChart.plot), on the same footing as what is now done in the "3D Graphics" section of the reference manual.

Thanks for the addition of @options to VectorField.plot; I've added color to these options, to be consistent with what is done in RealChart.plot. I've also added some documentation about the use of v.plot.options.

A question: in the keyword arguments of VectorField.plot, you have changed nb_values to number_values, but it is still nb_values in RealChart.plot. For consistency, both plot methods should use the same name. It seems that nb is not a standard English abbreviation for number, is it? (I find number_values a little bit long, especially that we don't have the TAB completion to type the options).

[tscrim]

Replying to egourgoulhon:

Thank you so much for the review!

Not a problem; sorry it took so long.

A question: in the keyword arguments of VectorField.plot, you have changed nb_values to number_values, but it is still nb_values in RealChart.plot. For consistency, both plot methods should use the same name. It seems that nb is not a standard English abbreviation for number, is it? (I find number_values a little bit long, especially that we don't have the TAB completion to type the options).

Not to my knowledge. The standard abbreviation is "no." (with a period) but "num" has become standard (at least in the programming world), but IMO the verbosity is better. However, you can do tab completion for explicit arguments.

[egourgoulhon]

Replying to tscrim:

Not to my knowledge. The standard abbreviation is "no." (with a period) but "num" has become standard (at least in the programming world), but IMO the verbosity is better. However, you can do tab completion for explicit arguments.

OK, so let use number_values everywhere. I am going to change it in RealChart.plot.

[tscrim]

Thanks.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​ff6eb09

Name of argument 'nb_values' in RealChart.plot changed to 'number_values'.

[egourgoulhon]

Change nb_values --> number_values done.

[tscrim]

I'm sorry I missed that on the previous ticket. LGTM. Thank you.

[egourgoulhon]

Thanks for the review!