add curl and divergence functions to vectors

[Jason Grout]

Make curl work if the vector has 2 or 3 components. Is there a higher-dimensional analogue for curl?

[Jason Grout]

(After some discussion on IRC) It seems that Sage is really lacking in the differential geometry area. If some good work is done there, this request will likely automatically be satisfied.

A short term solution would be to write a vector field class.

[ncalexan]

# a possible implementation of div, for irc user hedgehog

var('x, y, z')
(x, y, z)
f1 = x^2 + 2*y; f2 = x^3 + sin(z); f3 = y*z + 2
F = vector([f1, f2, f3])

print F(x=0, y=2, z=3)

def _variables(F):
    # this is a little funky -- we're finding all the variables that occur
    # in the components of F, and somehow choosing an ordering.  There are
    # other (better ways) but I'm not sure what the correct interface is.
    # For now, the user can specify the variables if they choose, just
    # like the gradient method.
    variables = list(set(flatten([ list(f.variables()) for f in F ])))
    variables.sort()
    return variables

def div(F, variables=None):
    assert len(F) == 3
    if variables is None:
        variables = _variables(F)

    s = 0
    for i in range(len(F)):
        s += F[i].derivative(variables[i])
    return s

print F
print div(F)
print div(F, variables=(y, x, z))

def curl(F, variables=None):
    assert len(F) == 3
    if variables is None:
        variables = _variables(F)
    assert len(variables) == 3
    x, y, z = variables
    Fx, Fy, Fz = F
    i = Fz.derivative(y) - Fy.derivative(z)
    j = Fz.derivative(z) - Fx.derivative(x)
    k = Fy.derivative(x) - Fz.derivative(y)
    return vector([i, j, k])
    
print curl(F)
print curl(F, variables=(y, x, z))

# let's assert that div(curl) == 0
# we need the variables because the ordering is suspect otherwise: for me,
# sage: _variables(F)
# [x, y, z]
# sage: _variables(curl(F))
# [z, x, y]
assert div(curl(F, variables=(x, y, z)), variables=(x, y, z)) == 0

[Jason Grout]

See #5506 for a collection of things to add to a symbolic vectors class

[Robert Bradshaw]

There is a 7-dim curl. Why doesn't the generic vector one work?

[Jason Grout]

What generic curl (I don't think it's in Sage right now)? Or are you saying we should just add curl to generic vectors? I agree; no reason to add this to just the callable symbolic vectors class (what was I thinking?).

[Dan Drake]

Note that in the code above, it should be:

k = Fy.derivative(x) - Fx.derivative(y)

The "Fz" should be "Fx".

[Robert Marik]

Also

 j = Fx.derivative(z) - Fz.derivative(x)

[Robert Bradshaw]

New commits:

​4e34d0d

Add divergence and curl to vectors.

[Eviatar Bach]

I was just going to work on this today!

I haven't tested the code, but looks good. Just a few things:

1. Is there any reason why you're converting the variables to their string representations in ._variables()?
2. The parameter in Expression.gradient is called variables; I think it would be good to switch vars to variables for consistency.
3. I think there should be test(s) for .curl() with the variables parameter.
4. I believe the raise TypeError, "curl only defined for 3 dimensions" syntax is deprecated, and that the string should be passed as an argument.

[Samuel Lelièvre]

Nitpicking on the error message in div: I would change it from "variable list must be equal to the dimension of self" to "number of variables must equal dimension of self".

About point 4 in eviatarbach's comment:

• A reference for the change in exception raising syntax is ​PEP-3109.
• the ValueError in div on follows the new syntax. By contrast, both the TypeError and ValueError in curl follow the old syntax.

[Samuel Lelièvre]

More nitpicking:

• Use "Return" instead of "Returns" in docstrings, see ​PEP 0257:

The docstring is a phrase ending in a period. It prescribes the function or method's effect as a command ("Do this", "Return that"), not as a description; e.g. don't write "Returns the pathname ...".

• Should there be a docstring and tests for the _variables function?
• In examples for div, remove extra space before w in R.<x,y,z, w> = QQ[]

[Eviatar Bach]

Another thought:

The extracting of variables from each element and then sorting by name looks scary, and can give unexpected results (if your vector is (x1, x10, x2), I believe you would get 3 for the divergence instead of 1 as you might expect, due to the alphanumeric sort). Unless there's a better solution, I think we should have it so that the variables list has to always be given explicitly.

[Jason Grout]

If your vectors are callable symbolic vectors, you should be able to get the list of arguments from the base ring (since you've already explicitly given an order to the variables):

sage: f(x,y,z)=(x*y,y*z,z^2)
sage: f
(x, y, z) |--> (x*y, y*z, z^2)
sage: type(f)
<class 'sage.modules.vector_callable_symbolic_dense.Vector_callable_symbolic_dense'>
sage: f.base_ring().arguments()
(x, y, z)

[Eviatar Bach]

That's a great idea. I think we should have that for callable vectors.

For the record, Mathematica and Maple both have default coordinate systems, which is how they deal with this issue.

[git]

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

​f353c94

Missing doctest.

[git]

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

​ae901d1	Merge tag '6.2' into div-curl
​8ab298e	Address reviewer comments.

[Robert Bradshaw]

I've addressed all the comments.

[Karl-Dieter Crisman]

I was going to look at this, but looks like Eviatar and Samuel are on top, so just think of this as a ping :)

[Robert Bradshaw]

Replying to kcrisman:

I was going to look at this, but looks like Eviatar and Samuel are on top, so just think of this as a ping :)

Feel free to look at this yourself :)

[Travis Scrimshaw]

My 2 cents, I'd have curl also take 2-dim inputs and return a vector. It might also be a good idea for a way (likely another method) which takes a 2-dim input and return a scalar as a shorthand (a la Green's theorem).

+1 to getting these (fundamental) methods into Sage.

[Robert Bradshaw]

Are there any enhancements to this with-patch, 7-year-old ticket that simply can't be deferred 'till later so we can finally get this in?

[Travis Scrimshaw]

I added the 2-dim input. If you're happy with my changes, then I think we can set this to a positive review.

---

New commits:

​0de71ac	Merge branch 'u/robertwb/ticket/3021' of trac.sagemath.org:sage into u/tscrim/curl_divergence-3021
​7b49240	Added 2-dim curl which returns a scalar value.