Opened 15 years ago

Closed 8 years ago

# add curl and divergence functions to vectors

Reported by: Owned by: Jason Grout William Stein major sage-6.4 calculus Eviatar Bach Robert Bradshaw Eviatar Bach, Samuel Lelièvre, Travis Scrimshaw N/A 7b49240 7b49240972f369a441432b2e2736b9dbc65d1337

### Description

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

### comment:1 Changed 15 years ago by 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.

### comment:2 Changed 13 years ago by 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
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
```

### comment:3 Changed 13 years ago by Jason Grout

Report Upstream: → N/A

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

### comment:4 Changed 13 years ago by Robert Bradshaw

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

### comment:5 Changed 13 years ago by Jason Grout

Summary: add curl and divergence functions to symbolic vectors → add curl and divergence functions to vectors

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?).

### comment:6 Changed 11 years ago by Dan Drake

Note that in the code above, it should be:

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

The "`Fz`" should be "`Fx`".

### comment:7 Changed 11 years ago by Robert Marik

Also

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

### comment:8 Changed 9 years ago by Jeroen Demeyer

Milestone: sage-5.11 → sage-5.12

### comment:9 Changed 9 years ago by For batch modifications

Milestone: sage-6.1 → sage-6.2

### comment:11 Changed 9 years ago by Robert Bradshaw

Branch: → u/robertwb/ticket/3021 Mar 11, 2014, 7:34:59 AM → Mar 11, 2014, 7:34:59 AM

### comment:12 follow-up:  14 Changed 9 years ago by Robert Bradshaw

Commit: → 4e34d0d90624c54e233be003c460c3bbaf6e3dcf new → needs_review

New commits:

 ​4e34d0d `Add divergence and curl to vectors.`

### comment:13 Changed 9 years ago by 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.

### comment:14 in reply to:  12 Changed 9 years ago by 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.

### comment:15 Changed 9 years ago by 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[]`

### comment:16 Changed 9 years ago by 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, since `'10'` comes before `'2'` in 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.

Last edited 9 years ago by Eviatar Bach (previous) (diff)

### comment:17 follow-up:  18 Changed 9 years ago by 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)
```

### comment:18 in reply to:  17 Changed 9 years ago by 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.

### comment:19 Changed 9 years ago by For batch modifications

Milestone: sage-6.2 → sage-6.3

### comment:20 Changed 8 years ago by git

Commit: 4e34d0d90624c54e233be003c460c3bbaf6e3dcf → f353c94832b7ddd0b76d18eba32a1dd5440c3a8e

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

 ​f353c94 `Missing doctest.`

### comment:21 Changed 8 years ago by git

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

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

### comment:23 Changed 8 years ago by For batch modifications

Milestone: sage-6.3 → sage-6.4

### comment:24 follow-up:  25 Changed 8 years ago by 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 :)

### comment:25 in reply to:  24 Changed 8 years ago by Robert Bradshaw

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 :)

### comment:26 Changed 8 years ago by 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.

### comment:27 Changed 8 years ago by 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?

### comment:28 Changed 8 years ago by Travis Scrimshaw

Authors: → Robert Bradshaw u/robertwb/ticket/3021 → u/tscrim/curl_divergence-3021 8ab298e3617a2a803db07f27ad6515a1a8551821 → 7b49240972f369a441432b2e2736b9dbc65d1337 → Eviatar Bach, Samuel Lelièvre, 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.`

### comment:29 Changed 8 years ago by Robert Bradshaw

Status: needs_review → positive_review

### comment:30 Changed 8 years ago by Volker Braun

Branch: u/tscrim/curl_divergence-3021 → 7b49240972f369a441432b2e2736b9dbc65d1337 → fixed positive_review → closed
Note: See TracTickets for help on using tickets.