add curl and divergence functions to vectors

[jasongrout]

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

CC: @eviatarbach

Component: calculus

Author: Robert Bradshaw

Branch/Commit: 7b49240

Reviewer: Eviatar Bach, Samuel Lelièvre, Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/3021

[jasongrout]

comment:1

(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]

comment:2

# 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

[jasongrout]

comment:3

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

[robertwb]

comment:4

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

[jasongrout]

comment:5

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?).

[dandrake]

comment:6

Note that in the code above, it should be:

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

The "Fz" should be "Fx".

[robert-marik]

comment:7

Also

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