Add method tangent_vector to differentiable manifolds

[egourgoulhon]

Currently, constructing a tangent vector v at some point p of a manifold M requires two steps: first construct the tangent space at p, T_pM, and then construct v as an element of T_pM. For instance:

sage: E.<x,y> = EuclideanSpace()
sage: p = E((1, 2), name='p')
sage: Tp = E.tangent_space(p)   # step 1
sage: v = Tp((-1, 3)); v        # step 2
Vector at Point p on the Euclidean plane E^2

This ticket endows the class DifferentiableManifold with a method tangent_vector, with vector as an alias, to make it a 1-step process. We can now write:

sage: E.<x,y> = EuclideanSpace()
sage: p = E((1, 2), name='p')
sage: v = E.vector(p, -1, 3); v
Vector at Point p on the Euclidean plane E^2

This feature is motivated by this ​ask.sagemath question.

[egourgoulhon]

New commits:

​5b22596

Add method vector() to DifferentiableManifold

[gh-mjungmath]

Sweet! I have just one comment. Since Python 3.6, we have f-strings which are way more convenient and easier to read than format (and also faster). I'd propose to use f-strings more often in the future. For this ticket:

- raise ValueError("{} components must be provided".format(dim))
+ raise ValueError(f"{dim} components must be provided")

[gh-mjungmath]

Here is an explanation also elaborating on more advantages of f-strings over format: ​https://realpython.com/python-f-strings/.

[git]

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

​c4a76e9

Use f-string in DifferentialManifold.vector

[egourgoulhon]

Replying to gh-mjungmath:

Here is an explanation also elaborating on more advantages of f-strings over format: ​https://realpython.com/python-f-strings/.

Thanks for the tip! The change to f-string is performed in the last commit.

[gh-mjungmath]

Is there a reason why you chose vector over tangent_vector for the method's name? The latter would sound more intuitive to me.

[egourgoulhon]

Replying to gh-mjungmath:

Is there a reason why you chose vector over tangent_vector for the method's name? The latter would sound more intuitive to me.

vector is more adapted to elementary use, like in vector calculus in the Euclidean space, when the user might not know what "tangent space" means. Of course, we can make an alias for tangent_vector if you feel it necessary.

[gh-mjungmath]

Mh, I don't know. Manifolds usually don't constitute of vectors. So a vector method wouldn't make much sense there as it sounds to me more like a method that should be reserved for vector spaces only. I'd advocate to rename vector to tangent_vector because mathematically precise.

This is a neat feature though, and I don't want to block it just because of my pedantry. An alias sounds like a good compromise.

[gh-mjungmath]

Alternatively the Euclidean space, as a special case, can be endowed with an alias vector whereas general (differentiable) manifolds only supposed to have tangent_vector. I think that is the best solution to maintain preciseness and having that alias in the elementary case at the same time.

[git]

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

​b5d7a8f

Rename vector to tangent_vector and make vector an alias to it

[egourgoulhon]

Replying to gh-mjungmath:

Mh, I don't know. Manifolds usually don't constitute of vectors. So a vector method wouldn't make much sense there as it sounds to me more like a method that should be reserved for vector spaces only. I'd advocate to rename vector to tangent_vector because mathematically precise.

OK, this is done in the last commit. Note however that saying simply "vector at point p of manifold M" is quite unambiguous and is in line with the terminology "vector field": we do not say "tangent vector field", do we?

This is a neat feature though, and I don't want to block it just because of my pedantry. An alias sounds like a good compromise.

Done. vector is now an alias for tangent_vector.

[egourgoulhon]

Replying to gh-mjungmath:

Alternatively the Euclidean space, as a special case, can be endowed with an alias vector whereas general (differentiable) manifolds only supposed to have tangent_vector. I think that is the best solution to maintain preciseness and having that alias in the elementary case at the same time.

I oppose to this: we shall not introduce on Euclidean spaces method names that break for more general manifolds, while the functionality is exactly the same.

[slelievre]

How about requiring

v = E.vector(p, (-1, 3))

instead of

v = E.vector(p, -1, 3)

I don't mind either way but there's a choice to make.

[egourgoulhon]

Replying to slelievre:

How about requiring

v = E.vector(p, (-1, 3))

instead of

v = E.vector(p, -1, 3)

I don't mind either way but there's a choice to make.

There is no choice to make ;-) Both work in the current implementation (cf. the examples).

[gh-mjungmath]

Replying to egourgoulhon:

OK, this is done in the last commit. Note however that saying simply "vector at point p of manifold M" is quite unambiguous and is in line with the terminology "vector field": we do not say "tangent vector field", do we?

No, we don't say "tangent vector field". But I haven't heard or read the use of "vector" over "tangent vector" either. Historical burden I suppose.

It should be unambiguous though and having both is fine with me. Someone who seeks for (tangent) vectors will find them now.

[gh-mjungmath]

LGTM

[egourgoulhon]

Thanks for the review and suggestions!

[mkoeppe]

Sage development has entered the release candidate phase for 9.3. Setting a new milestone for this ticket based on a cursory review.