Opened 11 months ago

Closed 7 months ago

Last modified 7 months ago

#24623 closed enhancement (fixed)

Euclidean spaces and vector calculus

Reported by: egourgoulhon Owned by:
Priority: major Milestone: sage-8.3
Component: geometry Keywords: Euclidean space, vector calculus, gradient, divergence, curl, Laplacian
Cc: tmonteil Merged in:
Authors: Eric Gourgoulhon Reviewers: Travis Scrimshaw
Report Upstream: N/A Work issues:
Branch: 220726c (Commits) Commit:
Dependencies: #24622, #24792 Stopgaps:

Description (last modified by tscrim)

This ticket implements Euclidean spaces as Riemannian manifolds diffeomorphic to Rn and equipped with a flat metric, which defines the Euclidean dot product. Using the operators introduced in #24622, this provides the standard operators of vector calculus: dot product, norm, cross product, gradient, divergence, curl and Laplacian, along with the standard coordinate systems (Cartesian, spherical, cylindrical, etc.).

See this ask.sagemath question for a motivation, as well as this one.

The implementation is performed via the parent class EuclideanSpace, which inherits from PseudoRiemannianManifold (introduced in #24622). Two subclasses are devoted to specific cases:

  • EuclideanPlane for n=2
  • Euclidean3dimSpace for n=3

The user interface for constructing an Euclidean space relies on the EuclideanSpace.__classcall_private__ to direct to the appropriate subclass.

The implementation through the manifold framework allows for an easy use of various coordinate systems, along with the related transformations. However, the user interface does not assume any knowledge of Riemannian geometry. In particular, no direct manipulation of the metric tensor is required.

A minimal example is

sage: E.<x,y,z> = EuclideanSpace(3)
sage: v = E.vector_field(-y, x, 0)
sage: v.display()
-y e_x + x e_y
sage: v[:]
[-y, x, 0]
sage: w = v.curl()
sage: w.display()
2 e_z
sage: w[:]
[0, 0, 2]

The transformation to spherical coordinates:

sage: spherical.<r,th,ph> = E.spherical_coordinates()
sage: spherical_frame = E.spherical_frame()  # orthonormal frame (e_r, e_th, e_ph)
sage: v.display(spherical_frame)
sqrt(x^2 + y^2) e_ph
sage: v.display(spherical_frame, spherical)
r*sin(th) e_ph
sage: v[spherical_frame, :, spherical]
[0, 0, r*sin(th)]
sage: w.display(spherical_frame, spherical)
2*cos(th) e_r - 2*sin(th) e_th
sage: w[spherical_frame, :, spherical]
[2*cos(th), -2*sin(th), 0]

More detailed examples are in the following Jupyter notebooks:

This work is part of the SageManifolds project, see #18528 for an overview.

Change History (39)

comment:1 Changed 11 months ago by egourgoulhon

  • Description modified (diff)

comment:2 Changed 11 months ago by egourgoulhon

  • Description modified (diff)

comment:3 Changed 11 months ago by tmonteil

  • Cc tmonteil added

comment:4 Changed 10 months ago by egourgoulhon

  • Branch set to public/manifolds/Euclidean_spaces
  • Commit set to 4bf802027ddd8178ab17efa692bda2d877e0f8d6

This is a first sketch (not ready yet).


Last 10 new commits:

55f356cImprove documentation of pseudo-Riemannian manifolds
0fc661aRemove method set_metric and improve doc of pseudo-Riemannian manifolds
b4dddf7Add operators Laplacian, d'Alembertian and curl on pseudo-Riemannian manifolds
040b01cAdd dot product and cross product of vector fields
46b8f66Add norm of vector fields.
135af97Add method volume_form() to class PseudoRiemannianManifold
38e8fd3Add global functions grad, div, curl, etc. for vector/tensor operators on pseudo-Riemannian manifolds
8309fddImprove documentation for operators on pseudo-Riemannian manifolds
29fc5c8Merge branch 'public/manifolds/pseudoRiemannian' of git://trac.sagemath.org/sage into Sage 8.2.beta5
4bf8020First draft of Euclidean spaces as Riemannian manifolds

comment:5 Changed 10 months ago by egourgoulhon

  • Dependencies changed from #24622 to #24622, #24792

#24792 introduces more flexibility in naming the elements of vector frames, for instance using (e_x, e_y, e_z) for a coordinate frame instead of (d/dx, d/dy, d/dz). The current ticket will therefore be build on it for a better user interface.

comment:6 Changed 10 months ago by git

  • Commit changed from 4bf802027ddd8178ab17efa692bda2d877e0f8d6 to f24a5f96a25d12b75c9d98a1b6749513673b4faa

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

a527e8eFirst draft adding more flexibility in symbols of free module bases and vector frames
2d43a23More refactoring in vector frames
0840d54More work on vector frames
01681b8Add documentation for new options of vector frames
f24a5f9Merge branch 'public/manifolds/Euclidean_spaces' of git://trac.sagemath.org/sage into public/manifolds/more_basis_flexibility (branch of #24792)

comment:7 Changed 10 months ago by git

  • Commit changed from f24a5f96a25d12b75c9d98a1b6749513673b4faa to d9302b65668e7f62b54306c4e4c916c73c72dc31

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

d9302b6Add documentation and doctests for Euclidean spaces

comment:8 follow-ups: Changed 10 months ago by kcrisman

I wonder whether some of this (in terms of the div/grad/curl) could be exposed in the symbolic calculus places, at least with some well-placed and well-formed examples that would be understandable to someone teaching Calc III to engineers (in the US framework) who may not be a differential geometer.

comment:9 in reply to: ↑ 8 Changed 10 months ago by egourgoulhon

Replying to kcrisman:

I wonder whether some of this (in terms of the div/grad/curl) could be exposed in the symbolic calculus places, at least with some well-placed and well-formed examples that would be understandable to someone teaching Calc III to engineers (in the US framework) who may not be a differential geometer.

Indeed, the aim here is to provide some interface which does not require any knowledge of differential geometry on manifolds. The class EuclideanSpaceGeneric does inherit from PseudoRiemannianManifold, but the end user has not to know it if he is not interested by this aspect and prefers to focus on vector calculus. Regarding examples, I plan to prepare some Jupyter notebooks and will expose them here, when the ticket is ready for review. Already, you can have a glimpse of the div/grad/curl capabilities introduced in the dependency #24622 in this notebook.

comment:10 follow-up: Changed 10 months ago by kcrisman

That's great! The reason I brought it up is because folders like sage/calculus and sage/symbolic might benefit from some good examples that are easy to find.

Is there a way to use something like (say) vector([x^2+y,y^2+z,z^2+x]) as "vector field" immediately, without much (or any) of the formalism necessary in that notebook? I think that is how most end users I've talked to about this issue would want it. See e.g. https://ask.sagemath.org/question/10104/gradient-divergence-curl-and-vector-products/ and ticket:3021 for one version already in Sage - maybe these should be combined somehow, if possible.

comment:11 in reply to: ↑ 10 Changed 10 months ago by egourgoulhon

Replying to kcrisman:

Is there a way to use something like (say) vector([x^2+y,y^2+z,z^2+x]) as "vector field" immediately, without much (or any) of the formalism necessary in that notebook?

Yes, this is precisely the aim of the current ticket. The above notebook is still at the level of semi-Riemannian manifolds (i.e. it illustrates only ticket #24622), while the user interface of the current ticket will be much simpler. It should also provide an easy way to change from Cartesian coordinates to e.g. polar or cylindrical ones.

See e.g. https://ask.sagemath.org/question/10104/gradient-divergence-curl-and-vector-products/ and ticket:3021 for one version already in Sage - maybe these should be combined somehow, if possible.

Thanks for pointing this version; I was not aware of it.

comment:12 Changed 10 months ago by git

  • Commit changed from d9302b65668e7f62b54306c4e4c916c73c72dc31 to 56a1a4a37f60f8f61ba8da18ece1e281c6c07aa9

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

9ddc467Add method set_name to classes VectorFrame and CoFrame
82c37b4Cut long lines in src/sage/manifolds/differentiable/vectorframe.py
95b2736Merge branch 'public/manifolds/more_basis_flexibility' of git://trac.sagemath.org/sage into Euclidean (to get latest version of the dependency #24792)
56a1a4aAdd method vector_field to Euclidean spaces + internal changes regarding charts and frames

comment:13 Changed 10 months ago by git

  • Commit changed from 56a1a4a37f60f8f61ba8da18ece1e281c6c07aa9 to 70a2a749c2d5c07a85182c0d51fdc9f16d050b01

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

799066bList entries of VectorFrame processed by __classcall_private__
da06d12Small improvement in treatment of symbol attributes of free module bases
244c1f4Merge branch 'public/manifolds/more_basis_flexibility' into 'public/manifolds/Euclidean_spaces' to get the latest version of dependency #24792 in #24623.
70a2a74Vector fields on Euclidean spaces can be initialized from a vector of symbolic expressions

comment:14 Changed 10 months ago by egourgoulhon

  • Description modified (diff)

comment:15 Changed 10 months ago by egourgoulhon

Here is some status update: functionalities regarding the 2-dimensional case are almost complete, as you can see in this demo worksheet (any feedback is of course appreciated). The 3-dimensional case is under preparation. The generic n-dimensional case, with n=1 or n>=4 is ready, since only Cartesian coordinates will be introduced (by default) in this case.

Regarding a question raised in comment:10, it is now possible to construct a vector field on an Euclidean space from a vector of symbolic expressions like this:

v = E.vector_field(vector([-y,x]))

where E is the underlying Euclidean space.

comment:16 Changed 9 months ago by git

  • Commit changed from 70a2a749c2d5c07a85182c0d51fdc9f16d050b01 to 6e9fce6b191d1d1fcf84153ea90478c444687703

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

18b63a6Differential operators are no longer imported in the global namespace by PseudoRiemannianManifold.__init__
8367232Merge branch 'public/manifolds/pseudoRiemannian' of git://trac.sagemath.org/sage into 'public/manifolds/Euclidean_spaces' to get the latest version of the dependency #24622 in #24623.
6e9fce6First draft of Euclidean 3-spaces, with the Cartesian, spherical and cylindrical coordinate systems

comment:17 Changed 9 months ago by git

  • Commit changed from 6e9fce6b191d1d1fcf84153ea90478c444687703 to 825ba7831fc3774455fd23b4c3b78dee3ee1fa0c

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

825ba78Add doctests to Euclidean spaces

comment:18 Changed 9 months ago by git

  • Commit changed from 825ba7831fc3774455fd23b4c3b78dee3ee1fa0c to 88f12ed85714bfa068c0a4e8f911681771ca4f20

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

185e438Improve documentation of pseudo-Riemannian manifolds
52b10eeMerge branch 'public/manifolds/pseudoRiemannian' of git://trac.sagemath.org/sage into 'public/manifolds/Euclidean_spaces' to get the final version of the dependency #24622 in #24623.
88f12edImprove documentation of Euclidean spaces

comment:19 Changed 9 months ago by git

  • Commit changed from 88f12ed85714bfa068c0a4e8f911681771ca4f20 to 371e05ee9f2d118c8f4b1d13376668a85b6fa639

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

371e05eImprove treatment of changes of frames in Euclidean spaces + documentation added

comment:20 Changed 9 months ago by git

  • Commit changed from 371e05ee9f2d118c8f4b1d13376668a85b6fa639 to 8a87f70a8efbf5e5f57f682459fe984a8df92338

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

8a87f70Add scalar triple product in Euclidean spaces

comment:21 Changed 9 months ago by git

  • Commit changed from 8a87f70a8efbf5e5f57f682459fe984a8df92338 to dd1951f87df19bdab818ded912e38579ad8f2fc5

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

dd1951fImprove documentation of Euclidean spaces, especially of vector operators

comment:22 Changed 9 months ago by git

  • Commit changed from dd1951f87df19bdab818ded912e38579ad8f2fc5 to d8b748baf218baf6d35b767d358553619a9045cb

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

7c4a9a9Merge branch 'public/manifolds/Euclidean_spaces' of git://trac.sagemath.org/sage into Sage 8.2.beta8.
d8b748bAdd references to Euclidean spaces and vector calculus in the Reference Manual

comment:23 Changed 9 months ago by git

  • Commit changed from d8b748baf218baf6d35b767d358553619a9045cb to de74bb734f9f531bb6d991b14af20e58e6ebd683

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

de74bb7Slight reorganization of the documentation of Euclidean spaces

comment:24 Changed 9 months ago by egourgoulhon

  • Description modified (diff)

comment:25 Changed 9 months ago by egourgoulhon

  • Status changed from new to needs_review

The documentation has been completed in the above commits and new Jupyter notebook examples have been added. This is now ready for review.

comment:26 in reply to: ↑ 8 Changed 9 months ago by egourgoulhon

Replying to kcrisman:

I wonder whether some of this (in terms of the div/grad/curl) could be exposed in the symbolic calculus places, at least with some well-placed and well-formed examples that would be understandable to someone teaching Calc III to engineers (in the US framework) who may not be a differential geometer.

I have added an entry "Vector Calculus" to src/doc/en/reference/calculus/index.rst. I have also added some SEEALSO sections in the definitions of curl and div in src/sage/modules/free_module_element.pyx, as well as in the definition of gradient in src/sage/symbolic/expression.pyx.

comment:27 Changed 9 months ago by egourgoulhon

  • Description modified (diff)

comment:28 Changed 8 months ago by egourgoulhon

  • Description modified (diff)

comment:29 Changed 8 months ago by git

  • Commit changed from de74bb734f9f531bb6d991b14af20e58e6ebd683 to 3578e5f49d0f8ef92fca4b7cb5e1d818fa49a633

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

b821dacMerge branch 'public/manifolds/Euclidean_spaces' of git://trac.sagemath.org/sage into public/manifolds/Euclidean_spaces
3578e5fFusing EuclideanSpace function with EuclideanSpaceGeneric plus other misc reviewer changes.

comment:30 Changed 8 months ago by git

  • Commit changed from 3578e5f49d0f8ef92fca4b7cb5e1d818fa49a633 to ab249df0d9dc01064ef68dd366941fe38e5a8e13

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

ab249dfFusing EuclideanSpace function with EuclideanSpaceGeneric plus other misc reviewer changes.

comment:31 follow-up: Changed 8 months ago by tscrim

  • Description modified (diff)
  • Reviewers set to Travis Scrimshaw

Okay, I have made my way through the code. Overall, it looks good.

I remove the function EuclideanSpace and just used the __classcall_private__ mechanism (provided by UniqueRepresentation) to make the class EuclideanSpace (formerly EuclideanSpaceGeneric) the main entry point.

There is something very subtle happening as some of the doctest output for display names of arctan2(y, x) needed to be arctan(y/x). I do not understand why nor could I see what was causing this. It is not a blocker issue, but it is an indication that something might not be working properly.

The rest of my changes were mostly cosmetic.

comment:32 in reply to: ↑ 31 Changed 8 months ago by egourgoulhon

Replying to tscrim:

Okay, I have made my way through the code. Overall, it looks good.

Thanks a lot for looking into this.

I remove the function EuclideanSpace and just used the __classcall_private__ mechanism (provided by UniqueRepresentation) to make the class EuclideanSpace (formerly EuclideanSpaceGeneric) the main entry point.

This is a nice improvement, thanks!

There is something very subtle happening as some of the doctest output for display names of arctan2(y, x) needed to be arctan(y/x). I do not understand why nor could I see what was causing this. It is not a blocker issue, but it is an indication that something might not be working properly.

Indeed; I will look into this...

The rest of my changes were mostly cosmetic.

Thanks for all of them.

comment:33 Changed 8 months ago by tscrim

I should also say that if you approve of my changes, then you can set a positive review.

(Unfortunately, a number of the doctests are just slow because of feeding calls off to Maxima and the simplifications. Although I don't think there is anything we can do about that.)

comment:34 Changed 8 months ago by git

  • Commit changed from ab249df0d9dc01064ef68dd366941fe38e5a8e13 to 220726c07bc00ba69b12a5cc7dc9499d930f1e8c

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

654b0aaMerge branch 'public/manifolds/Euclidean_spaces' of git://trac.sagemath.org/sage into Sage 8.2.rc4
220726cA few minor changes in doctests of Euclidean spaces

comment:35 Changed 8 months ago by egourgoulhon

The above commit fixes the arctan issue mentioned in comment:31: it was due to the declaration

sage: E3.<x,t,p> = EuclideanSpace(coordinates='spherical')

in line 591 of euclidean.py, which had the side effect of adding x>0 to Sage's assumptions. Then subsequent doctests involving arctan2(y, x) automatically replaced it by arctan(y/x). I changed the variable x to r in the declaration of E3 (which btw fits better with standard notation for spherical coordinates) and everything was OK, i.e. all arctan2(y, x) could be restored.

Moreover, since it is now possible to skip the dimension as first argument of EuclideanSpace in constructions of the type E.<x,y> = EuclideanSpace() (thank you for this improvement!), I systematically removed the dimension argument and added a sentence in the documentation about this.

I also took the opportunity of this commit to add a single-line fix of a (small) bug in src/sage/manifolds/differentiable/vectorfield_module.py revealed after the dependency ticket #24792 was merged. I hope you do not mind.

comment:36 follow-up: Changed 8 months ago by tscrim

  • Status changed from needs_review to positive_review

Yep, LGTM. Thank you for tracking that down.

comment:37 in reply to: ↑ 36 Changed 8 months ago by egourgoulhon

Replying to tscrim:

Yep, LGTM. Thank you for tracking that down.

Thank you very much for the review, and the associated improvements in the code!

comment:38 Changed 7 months ago by vbraun

  • Branch changed from public/manifolds/Euclidean_spaces to 220726c07bc00ba69b12a5cc7dc9499d930f1e8c
  • Resolution set to fixed
  • Status changed from positive_review to closed

comment:39 Changed 7 months ago by slelievre

  • Commit 220726c07bc00ba69b12a5cc7dc9499d930f1e8c deleted
  • Milestone changed from sage-8.2 to sage-8.3

Merged in Sage 8.3.beta0.

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