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scalarfield_algebra.py
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scalarfield_algebra.py
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r"""
Algebra of Scalar Fields
The class :class:`ScalarFieldAlgebra` implements the commutative algebra
`C^0(M)` of scalar fields on a topological manifold `M` over a topological
field `K`. By *scalar field*, it
is meant a continuous function `M \to K`. The set
`C^0(M)` is an algebra over `K`, whose ring product is the pointwise
multiplication of `K`-valued functions, which is clearly commutative.
AUTHORS:
- Eric Gourgoulhon, Michal Bejger (2014-2015): initial version
- Travis Scrimshaw (2016): review tweaks
REFERENCES:
- [Lee2011]_
- [KN1963]_
"""
#******************************************************************************
# Copyright (C) 2015 Eric Gourgoulhon <eric.gourgoulhon@obspm.fr>
# Copyright (C) 2015 Michal Bejger <bejger@camk.edu.pl>
# Copyright (C) 2016 Travis Scrimshaw <tscrimsh@umn.edu>
#
# Distributed under the terms of the GNU General Public License (GPL)
# as published by the Free Software Foundation; either version 2 of
# the License, or (at your option) any later version.
# http://www.gnu.org/licenses/
#******************************************************************************
from sage.structure.parent import Parent
from sage.structure.unique_representation import UniqueRepresentation
from sage.misc.cachefunc import cached_method
from sage.categories.commutative_algebras import CommutativeAlgebras
from sage.symbolic.ring import SR
from sage.manifolds.scalarfield import ScalarField
class ScalarFieldAlgebra(UniqueRepresentation, Parent):
r"""
Commutative algebra of scalar fields on a topological manifold.
If `M` is a topological manifold over a topological field `K`, the
commutative algebra of scalar fields on `M` is the set `C^0(M)` of all
continuous maps `M \to K`. The set `C^0(M)` is an algebra over `K`,
whose ring product is the pointwise multiplication of `K`-valued
functions, which is clearly commutative.
If `K = \RR` or `K = \CC`, the field `K` over which the
algebra `C^0(M)` is constructed is represented by the :class:`Symbolic
Ring <sage.symbolic.ring.SymbolicRing>` ``SR``, since there is no exact
representation of `\RR` nor `\CC`.
INPUT:
- ``domain`` -- the topological manifold `M` on which the scalar fields
are defined
EXAMPLES:
Algebras of scalar fields on the sphere `S^2` and on some open
subsets of it::
sage: M = Manifold(2, 'M', structure='topological') # the 2-dimensional sphere S^2
sage: U = M.open_subset('U') # complement of the North pole
sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: V = M.open_subset('V') # complement of the South pole
sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: M.declare_union(U,V) # S^2 is the union of U and V
sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)),
....: intersection_name='W',
....: restrictions1= x^2+y^2!=0,
....: restrictions2= u^2+v^2!=0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: CM = M.scalar_field_algebra(); CM
Algebra of scalar fields on the 2-dimensional topological manifold M
sage: W = U.intersection(V) # S^2 minus the two poles
sage: CW = W.scalar_field_algebra(); CW
Algebra of scalar fields on the Open subset W of the
2-dimensional topological manifold M
`C^0(M)` and `C^0(W)` belong to the category of commutative
algebras over `\RR` (represented here by
:class:`~sage.symbolic.ring.SymbolicRing`)::
sage: CM.category()
Category of commutative algebras over Symbolic Ring
sage: CM.base_ring()
Symbolic Ring
sage: CW.category()
Category of commutative algebras over Symbolic Ring
sage: CW.base_ring()
Symbolic Ring
The elements of `C^0(M)` are scalar fields on `M`::
sage: CM.an_element()
Scalar field on the 2-dimensional topological manifold M
sage: CM.an_element().display() # this sample element is a constant field
M --> R
on U: (x, y) |--> 2
on V: (u, v) |--> 2
Those of `C^0(W)` are scalar fields on `W`::
sage: CW.an_element()
Scalar field on the Open subset W of the 2-dimensional topological
manifold M
sage: CW.an_element().display() # this sample element is a constant field
W --> R
(x, y) |--> 2
(u, v) |--> 2
The zero element::
sage: CM.zero()
Scalar field zero on the 2-dimensional topological manifold M
sage: CM.zero().display()
zero: M --> R
on U: (x, y) |--> 0
on V: (u, v) |--> 0
::
sage: CW.zero()
Scalar field zero on the Open subset W of the 2-dimensional
topological manifold M
sage: CW.zero().display()
zero: W --> R
(x, y) |--> 0
(u, v) |--> 0
The unit element::
sage: CM.one()
Scalar field 1 on the 2-dimensional topological manifold M
sage: CM.one().display()
1: M --> R
on U: (x, y) |--> 1
on V: (u, v) |--> 1
::
sage: CW.one()
Scalar field 1 on the Open subset W of the 2-dimensional topological
manifold M
sage: CW.one().display()
1: W --> R
(x, y) |--> 1
(u, v) |--> 1
A generic element can be constructed by using a dictionary of
the coordinate expressions defining the scalar field::
sage: f = CM({c_xy: atan(x^2+y^2), c_uv: pi/2 - atan(u^2+v^2)}); f
Scalar field on the 2-dimensional topological manifold M
sage: f.display()
M --> R
on U: (x, y) |--> arctan(x^2 + y^2)
on V: (u, v) |--> 1/2*pi - arctan(u^2 + v^2)
sage: f.parent()
Algebra of scalar fields on the 2-dimensional topological manifold M
Specific elements can also be constructed in this way::
sage: CM(0) == CM.zero()
True
sage: CM(1) == CM.one()
True
Note that the zero scalar field is cached::
sage: CM(0) is CM.zero()
True
Elements can also be constructed by means of the method
:meth:`~sage.manifolds.manifold.TopologicalManifold.scalar_field` acting
on the domain (this allows one to set the name of the scalar field at the
construction)::
sage: f1 = M.scalar_field({c_xy: atan(x^2+y^2), c_uv: pi/2 - atan(u^2+v^2)},
....: name='f')
sage: f1.parent()
Algebra of scalar fields on the 2-dimensional topological manifold M
sage: f1 == f
True
sage: M.scalar_field(0, chart='all') == CM.zero()
True
The algebra `C^0(M)` coerces to `C^0(W)` since `W` is an open
subset of `M`::
sage: CW.has_coerce_map_from(CM)
True
The reverse is of course false::
sage: CM.has_coerce_map_from(CW)
False
The coercion map is nothing but the restriction to `W` of scalar fields
on `M`::
sage: fW = CW(f) ; fW
Scalar field on the Open subset W of the
2-dimensional topological manifold M
sage: fW.display()
W --> R
(x, y) |--> arctan(x^2 + y^2)
(u, v) |--> 1/2*pi - arctan(u^2 + v^2)
::
sage: CW(CM.one()) == CW.one()
True
The coercion map allows for the addition of elements of `C^0(W)`
with elements of `C^0(M)`, the result being an element of
`C^0(W)`::
sage: s = fW + f
sage: s.parent()
Algebra of scalar fields on the Open subset W of the
2-dimensional topological manifold M
sage: s.display()
W --> R
(x, y) |--> 2*arctan(x^2 + y^2)
(u, v) |--> pi - 2*arctan(u^2 + v^2)
Another coercion is that from the Symbolic Ring.
Since the Symbolic Ring is the base ring for the algebra ``CM``, the
coercion of a symbolic expression ``s`` is performed by the operation
``s*CM.one()``, which invokes the (reflected) multiplication operator.
If the symbolic expression does not involve any chart coordinate,
the outcome is a constant scalar field::
sage: h = CM(pi*sqrt(2)) ; h
Scalar field on the 2-dimensional topological manifold M
sage: h.display()
M --> R
on U: (x, y) |--> sqrt(2)*pi
on V: (u, v) |--> sqrt(2)*pi
sage: a = var('a')
sage: h = CM(a); h.display()
M --> R
on U: (x, y) |--> a
on V: (u, v) |--> a
If the symbolic expression involves some coordinate of one of the
manifold's charts, the outcome is initialized only on the chart domain::
sage: h = CM(a+x); h.display()
M --> R
on U: (x, y) |--> a + x
on W: (u, v) |--> (a*u^2 + a*v^2 + u)/(u^2 + v^2)
sage: h = CM(a+u); h.display()
M --> R
on W: (x, y) |--> (a*x^2 + a*y^2 + x)/(x^2 + y^2)
on V: (u, v) |--> a + u
If the symbolic expression involves coordinates of different charts,
the scalar field is created as a Python object, but is not initialized,
in order to avoid any ambiguity::
sage: h = CM(x+u); h.display()
M --> R
TESTS:
Ring laws::
sage: h = CM(pi*sqrt(2))
sage: s = f + h ; s
Scalar field on the 2-dimensional topological manifold M
sage: s.display()
M --> R
on U: (x, y) |--> sqrt(2)*pi + arctan(x^2 + y^2)
on V: (u, v) |--> 1/2*pi*(2*sqrt(2) + 1) - arctan(u^2 + v^2)
::
sage: s = f - h ; s
Scalar field on the 2-dimensional topological manifold M
sage: s.display()
M --> R
on U: (x, y) |--> -sqrt(2)*pi + arctan(x^2 + y^2)
on V: (u, v) |--> -1/2*pi*(2*sqrt(2) - 1) - arctan(u^2 + v^2)
::
sage: s = f*h ; s
Scalar field on the 2-dimensional topological manifold M
sage: s.display()
M --> R
on U: (x, y) |--> sqrt(2)*pi*arctan(x^2 + y^2)
on V: (u, v) |--> 1/2*sqrt(2)*(pi^2 - 2*pi*arctan(u^2 + v^2))
::
sage: s = f/h ; s
Scalar field on the 2-dimensional topological manifold M
sage: s.display()
M --> R
on U: (x, y) |--> 1/2*sqrt(2)*arctan(x^2 + y^2)/pi
on V: (u, v) |--> 1/4*sqrt(2)*(pi - 2*arctan(u^2 + v^2))/pi
::
sage: f*(h+f) == f*h + f*f
True
Ring laws with coercion::
sage: f - fW == CW.zero()
True
sage: f/fW == CW.one()
True
sage: s = f*fW ; s
Scalar field on the Open subset W of the 2-dimensional topological
manifold M
sage: s.display()
W --> R
(x, y) |--> arctan(x^2 + y^2)^2
(u, v) |--> 1/4*pi^2 - pi*arctan(u^2 + v^2) + arctan(u^2 + v^2)^2
sage: s/f == fW
True
Multiplication by a real number::
sage: s = 2*f ; s
Scalar field on the 2-dimensional topological manifold M
sage: s.display()
M --> R
on U: (x, y) |--> 2*arctan(x^2 + y^2)
on V: (u, v) |--> pi - 2*arctan(u^2 + v^2)
::
sage: 0*f == CM.zero()
True
sage: 1*f == f
True
sage: 2*(f/2) == f
True
sage: (f+2*f)/3 == f
True
sage: 1/3*(f+2*f) == f
True
The Sage test suite for algebras is passed::
sage: TestSuite(CM).run()
It is passed also for `C^0(W)`::
sage: TestSuite(CW).run()
"""
Element = ScalarField
def __init__(self, domain):
r"""
Construct an algebra of scalar fields.
TESTS::
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: CM = M.scalar_field_algebra(); CM
Algebra of scalar fields on the 2-dimensional topological
manifold M
sage: type(CM)
<class 'sage.manifolds.scalarfield_algebra.ScalarFieldAlgebra_with_category'>
sage: type(CM).__base__
<class 'sage.manifolds.scalarfield_algebra.ScalarFieldAlgebra'>
sage: TestSuite(CM).run()
"""
base_field = domain.base_field()
if domain.base_field_type() in ['real', 'complex']:
base_field = SR
Parent.__init__(self, base=base_field,
category=CommutativeAlgebras(base_field))
self._domain = domain
self._populate_coercion_lists_()
#### Methods required for any Parent
def _element_constructor_(self, coord_expression=None, chart=None,
name=None, latex_name=None):
r"""
Construct a scalar field.
INPUT:
- ``coord_expression`` -- (default: ``None``) element(s) to construct
the scalar field; this can be either
- a scalar field defined on a domain that encompass ``self._domain``;
then ``_element_constructor_`` return the restriction of
the scalar field to ``self._domain``
- a dictionary of coordinate expressions in various charts on the
domain, with the charts as keys
- a single coordinate expression; if the argument ``chart`` is
``'all'``, this expression is set to all the charts defined
on the open set; otherwise, the expression is set in the
specific chart provided by the argument ``chart``
- ``chart`` -- (default: ``None``) chart defining the coordinates used
in ``coord_expression`` when the latter is a single coordinate
expression; if none is provided (default), the default chart of the
open set is assumed. If ``chart=='all'``, ``coord_expression`` is
assumed to be independent of the chart (constant scalar field).
- ``name`` -- (default: ``None``) string; name (symbol) given to the
scalar field
- ``latex_name`` -- (default: ``None``) string; LaTeX symbol to denote
the scalar field; if none is provided, the LaTeX symbol is set to
``name``
If ``coord_expression`` is ``None`` or incomplete, coordinate
expressions can be added after the creation of the object, by means
of the methods :meth:`add_expr`, :meth:`add_expr_by_continuation` and
:meth:`set_expr`
TESTS::
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: CM = M.scalar_field_algebra()
sage: f = CM({X: x+y^2}); f
Scalar field on the 2-dimensional topological manifold M
sage: f.display()
M --> R
(x, y) |--> y^2 + x
sage: f = CM({X: x+y^2}, name='f'); f
Scalar field f on the 2-dimensional topological manifold M
sage: f.display()
f: M --> R
(x, y) |--> y^2 + x
sage: U = M.open_subset('U', coord_def={X: x>0})
sage: CU = U.scalar_field_algebra()
sage: fU = CU(f); fU
Scalar field f on the Open subset U of the 2-dimensional topological
manifold M
sage: fU.display()
f: U --> R
(x, y) |--> y^2 + x
"""
try:
if coord_expression.is_trivial_zero():
return self.zero()
elif (coord_expression - 1).is_trivial_zero():
return self.one()
except AttributeError:
if coord_expression == 0:
return self.zero()
if coord_expression == 1:
return self.one()
if isinstance(coord_expression, ScalarField):
if self._domain.is_subset(coord_expression._domain):
# restriction of the scalar field to self._domain:
return coord_expression.restrict(self._domain)
else:
# Anything going wrong here should produce a readable error:
try:
# generic constructor:
resu = self.element_class(self,
coord_expression=coord_expression,
name=name, latex_name=latex_name,
chart=chart)
except TypeError:
raise TypeError("cannot convert " +
"{} to a scalar ".format(coord_expression) +
"field on {}".format(self._domain))
return resu
def _an_element_(self):
r"""
Construct some element of the algebra
TESTS::
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: CM = M.scalar_field_algebra()
sage: f = CM._an_element_(); f
Scalar field on the 2-dimensional topological manifold M
sage: f.display()
M --> R
(x, y) |--> 2
"""
return self.element_class(self, coord_expression=2, chart='all')
def _coerce_map_from_(self, other):
r"""
Determine whether coercion to ``self`` exists from ``other``.
TESTS::
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: CM = M.scalar_field_algebra()
sage: CM._coerce_map_from_(SR)
True
sage: CM._coerce_map_from_(X.function_ring())
True
sage: U = M.open_subset('U', coord_def={X: x>0})
sage: CU = U.scalar_field_algebra()
sage: CM._coerce_map_from_(CU)
False
sage: CU._coerce_map_from_(CM)
True
"""
from .chart_func import ChartFunctionRing
if other is SR:
return True # coercion from the base ring (multiplication by the
# algebra unit, i.e. self.one())
# cf. ScalarField._lmul_() for the implementation of
# the coercion map
elif isinstance(other, ScalarFieldAlgebra):
return self._domain.is_subset(other._domain)
elif isinstance(other, ChartFunctionRing):
return self._domain.is_subset(other._chart._domain)
else:
return False
#### End of methods required for any Parent
def _repr_(self):
r"""
String representation of ``self``.
TESTS::
sage: M = Manifold(2, 'M', structure='topological')
sage: CM = M.scalar_field_algebra()
sage: CM._repr_()
'Algebra of scalar fields on the 2-dimensional topological manifold M'
sage: CM
Algebra of scalar fields on the 2-dimensional topological manifold M
"""
return "Algebra of scalar fields on the {}".format(self._domain)
def _latex_(self):
r"""
LaTeX representation of the object.
TESTS::
sage: M = Manifold(2, 'M', structure='topological')
sage: CM = M.scalar_field_algebra()
sage: CM._latex_()
'C^0 \\left(M\\right)'
sage: latex(CM)
C^0 \left(M\right)
"""
return r"C^0 \left(" + self._domain._latex_() + r"\right)"
@cached_method
def zero(self):
r"""
Return the zero element of the algebra.
This is nothing but the constant scalar field `0` on the manifold,
where `0` is the zero element of the base field.
EXAMPLES::
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: CM = M.scalar_field_algebra()
sage: z = CM.zero(); z
Scalar field zero on the 2-dimensional topological manifold M
sage: z.display()
zero: M --> R
(x, y) |--> 0
The result is cached::
sage: CM.zero() is z
True
"""
coord_express = {chart: chart.zero_function()
for chart in self._domain.atlas()}
zero = self.element_class(self,
coord_expression=coord_express,
name='zero', latex_name='0')
zero._is_zero = True
zero.set_immutable()
return zero
@cached_method
def one(self):
r"""
Return the unit element of the algebra.
This is nothing but the constant scalar field `1` on the manifold,
where `1` is the unit element of the base field.
EXAMPLES::
sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: CM = M.scalar_field_algebra()
sage: h = CM.one(); h
Scalar field 1 on the 2-dimensional topological manifold M
sage: h.display()
1: M --> R
(x, y) |--> 1
The result is cached::
sage: CM.one() is h
True
"""
coord_express = {chart: chart.one_function()
for chart in self._domain.atlas()}
one = self.element_class(self, coord_expression=coord_express,
name='1', latex_name='1')
one.set_immutable()
return one