Differentiable manifolds: tangent spaces

egourgoulhon commented 9 years ago

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.

Depends on #15916 Depends on #18843

CC: @sagetrac-mbejger

Component: geometry

Keywords: differentiable manifold, tangent vector, tangent space

Author: Eric Gourgoulhon, Michal Bejger, Marco Mancini

Branch/Commit: ff6eb09

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/19092

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

Changed commit from d5562a2 to 6a1fc97

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

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

6a1fc97 Add method differential() to class DiffMap

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

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

31d6789 Add method plot() to class VectorField; full doctest coverage.

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

Changed commit from 6a1fc97 to 31d6789

egourgoulhon commented 9 years ago

Changed dependencies from #15916, #18100, #18843 to #15916, #18843

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

Changed commit from 31d6789 to 04360b0

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

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

04360b0 Major improvements in the documentation of diff. manifolds (tangent spaces part)

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

Changed commit from 04360b0 to 513f5ba

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

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

513f5ba Improve TensorField.__eq__ (case with no open cover known)

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

Changed commit from 513f5ba to e43e03b

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

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`

egourgoulhon commented 9 years ago

Description changed:

--- 
+++ 
@@ -11,4 +11,11 @@
   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). 
+  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](http://sagemanifolds.obspm.fr/documentation.html).
+

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

Changed commit from e43e03b to 22923ab

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

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).`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

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`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from 22923ab to 1bf77fd

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

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`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from 1bf77fd to 98c6036

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from 98c6036 to 347a9cc

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

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)`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from 347a9cc to db99641

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

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`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

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`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from db99641 to b07ffd9

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from b07ffd9 to 9fa77c4

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

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`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from 9fa77c4 to b47f55f

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

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`

egourgoulhon commented 8 years ago

Changed author from Eric Gourgoulhon, Michal Bejger to Eric Gourgoulhon, Michal Bejger, Marco Mancini

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from b47f55f to 37521ae

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

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`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from 37521ae to 159afb4

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

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`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

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`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from 159afb4 to 3b1b794

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

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)`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from 3b1b794 to c70792e

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

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`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from c70792e to 98efa66

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

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`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from 98efa66 to 8205dca

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 7 years ago

Changed commit from 8205dca to 785300b

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 7 years ago

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`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 7 years ago

Changed commit from 785300b to 567b892

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 7 years ago

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 commented 7 years ago

Reviewer: Travis Scrimshaw

tscrim commented 7 years ago

comment:29

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 commented 7 years ago

comment:30

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

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 7 years ago

Changed commit from 567b892 to 05f8cc7

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 7 years ago

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

05f8cc7 A little bit of extra cleanup.

sagemath / sage

Differentiable manifolds: tangent spaces #19092