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diff_form_module.py
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diff_form_module.py
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r"""
Differential Form Modules
The set `\Omega^p(U, \Phi)` of `p`-forms along a differentiable manifold `U`
with values on a differentiable manifold `M` via a differentiable map
`\Phi:\ U \rightarrow M` (possibly `U = M` and `\Phi = \mathrm{Id}_M`)
is a module over the algebra `C^k(U)` of differentiable scalar fields on `U`.
It is a free module if and only if `M` is parallelizable. Accordingly,
two classes implement `\Omega^p(U, \Phi)`:
- :class:`DiffFormModule` for differential forms with values on a generic
(in practice, not parallelizable) differentiable manifold `M`
- :class:`DiffFormFreeModule` for differential forms with values on a
parallelizable manifold `M`
AUTHORS:
- Eric Gourgoulhon (2015): initial version
- Travis Scrimshaw (2016): review tweaks
REFERENCES:
- [KN1963]_
- [Lee2013]_
"""
# *****************************************************************************
# Copyright (C) 2015 Eric Gourgoulhon <eric.gourgoulhon@obspm.fr>
# 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.
# https://www.gnu.org/licenses/
# *****************************************************************************
from sage.misc.cachefunc import cached_method
from sage.structure.unique_representation import UniqueRepresentation
from sage.structure.parent import Parent
from sage.categories.modules import Modules
from sage.tensor.modules.ext_pow_free_module import ExtPowerDualFreeModule
from sage.manifolds.differentiable.diff_form import DiffForm, DiffFormParal
from sage.manifolds.differentiable.tensorfield import TensorField
from sage.manifolds.differentiable.tensorfield_paral import TensorFieldParal
class DiffFormModule(UniqueRepresentation, Parent):
r"""
Module of differential forms of a given degree `p` (`p`-forms) along a
differentiable manifold `U` with values on a differentiable manifold `M`.
Given a differentiable manifold `U` and a differentiable map
`\Phi: U \rightarrow M` to a differentiable manifold `M`, the set
`\Omega^p(U, \Phi)` of `p`-forms along `U` with values on `M` is
a module over `C^k(U)`, the commutative algebra of differentiable
scalar fields on `U` (see
:class:`~sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra`).
The standard case of `p`-forms *on* a differentiable manifold `M`
corresponds to `U = M` and `\Phi = \mathrm{Id}_M`. Other common cases
are `\Phi` being an immersion and `\Phi` being a curve in `M`
(`U` is then an open interval of `\RR`).
.. NOTE::
This class implements `\Omega^p(U,\Phi)` in the case where `M` is
not assumed to be parallelizable; the module `\Omega^p(U, \Phi)`
is then not necessarily free. If `M` is parallelizable, the class
:class:`DiffFormFreeModule` must be used instead.
INPUT:
- ``vector_field_module`` -- module `\mathfrak{X}(U, \Phi)` of vector
fields along `U` with values on `M` via the map `\Phi: U \rightarrow M`
- ``degree`` -- positive integer; the degree `p` of the differential forms
EXAMPLES:
Module of 2-forms on a non-parallelizable 2-dimensional manifold::
sage: M = Manifold(2, 'M')
sage: U = M.open_subset('U') ; V = M.open_subset('V')
sage: M.declare_union(U,V) # M is the union of U and V
sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart()
sage: transf = c_xy.transition_map(c_uv, (x+y, x-y),
....: intersection_name='W', restrictions1= x>0, restrictions2= u+v>0)
sage: inv = transf.inverse()
sage: W = U.intersection(V)
sage: eU = c_xy.frame() ; eV = c_uv.frame()
sage: XM = M.vector_field_module() ; XM
Module X(M) of vector fields on the 2-dimensional differentiable
manifold M
sage: A = M.diff_form_module(2) ; A
Module Omega^2(M) of 2-forms on the 2-dimensional differentiable
manifold M
sage: latex(A)
\Omega^{2}\left(M\right)
``A`` is nothing but the second exterior power of the dual of ``XM``, i.e.
we have `\Omega^{2}(M) = \Lambda^2(\mathfrak{X}(M)^*)`::
sage: A is XM.dual_exterior_power(2)
True
Modules of differential forms are unique::
sage: A is M.diff_form_module(2)
True
`\Omega^2(M)` is a module over the algebra `C^k(M)` of (differentiable)
scalar fields on `M`::
sage: A.category()
Category of modules over Algebra of differentiable scalar fields on
the 2-dimensional differentiable manifold M
sage: CM = M.scalar_field_algebra() ; CM
Algebra of differentiable scalar fields on the 2-dimensional
differentiable manifold M
sage: A in Modules(CM)
True
sage: A.base_ring() is CM
True
sage: A.base_module()
Module X(M) of vector fields on the 2-dimensional differentiable
manifold M
sage: A.base_module() is XM
True
Elements can be constructed from ``A()``. In particular, ``0`` yields
the zero element of ``A``::
sage: z = A(0) ; z
2-form zero on the 2-dimensional differentiable manifold M
sage: z.display(eU)
zero = 0
sage: z.display(eV)
zero = 0
sage: z is A.zero()
True
while non-zero elements are constructed by providing their components in a
given vector frame::
sage: a = A([[0,3*x],[-3*x,0]], frame=eU, name='a') ; a
2-form a on the 2-dimensional differentiable manifold M
sage: a.add_comp_by_continuation(eV, W, c_uv) # finishes initializ. of a
sage: a.display(eU)
a = 3*x dx∧dy
sage: a.display(eV)
a = (-3/4*u - 3/4*v) du∧dv
An alternative is to construct the 2-form from an empty list of
components and to set the nonzero nonredundant components afterwards::
sage: a = A([], name='a')
sage: a[eU,0,1] = 3*x
sage: a.add_comp_by_continuation(eV, W, c_uv)
sage: a.display(eU)
a = 3*x dx∧dy
sage: a.display(eV)
a = (-3/4*u - 3/4*v) du∧dv
The module `\Omega^1(M)` is nothing but the dual of `\mathfrak{X}(M)`
(the module of vector fields on `M`)::
sage: L1 = M.diff_form_module(1) ; L1
Module Omega^1(M) of 1-forms on the 2-dimensional differentiable
manifold M
sage: L1 is XM.dual()
True
Since any tensor field of type `(0,1)` is a 1-form, there is a coercion
map from the set `T^{(0,1)}(M)` of such tensors to `\Omega^1(M)`::
sage: T01 = M.tensor_field_module((0,1)) ; T01
Module T^(0,1)(M) of type-(0,1) tensors fields on the 2-dimensional
differentiable manifold M
sage: L1.has_coerce_map_from(T01)
True
There is also a coercion map in the reverse direction::
sage: T01.has_coerce_map_from(L1)
True
For a degree `p \geq 2`, the coercion holds only in the direction
`\Omega^p(M)\rightarrow T^{(0,p)}(M)`::
sage: T02 = M.tensor_field_module((0,2)) ; T02
Module T^(0,2)(M) of type-(0,2) tensors fields on the 2-dimensional
differentiable manifold M
sage: T02.has_coerce_map_from(A)
True
sage: A.has_coerce_map_from(T02)
False
The coercion map `T^{(0,1)}(M) \rightarrow \Omega^1(M)` in action::
sage: b = T01([y,x], frame=eU, name='b') ; b
Tensor field b of type (0,1) on the 2-dimensional differentiable
manifold M
sage: b.add_comp_by_continuation(eV, W, c_uv)
sage: b.display(eU)
b = y dx + x dy
sage: b.display(eV)
b = 1/2*u du - 1/2*v dv
sage: lb = L1(b) ; lb
1-form b on the 2-dimensional differentiable manifold M
sage: lb.display(eU)
b = y dx + x dy
sage: lb.display(eV)
b = 1/2*u du - 1/2*v dv
The coercion map `\Omega^1(M) \rightarrow T^{(0,1)}(M)` in action::
sage: tlb = T01(lb) ; tlb
Tensor field b of type (0,1) on the 2-dimensional differentiable
manifold M
sage: tlb.display(eU)
b = y dx + x dy
sage: tlb.display(eV)
b = 1/2*u du - 1/2*v dv
sage: tlb == b
True
The coercion map `\Omega^2(M) \rightarrow T^{(0,2)}(M)` in action::
sage: ta = T02(a) ; ta
Tensor field a of type (0,2) on the 2-dimensional differentiable
manifold M
sage: ta.display(eU)
a = 3*x dx⊗dy - 3*x dy⊗dx
sage: a.display(eU)
a = 3*x dx∧dy
sage: ta.display(eV)
a = (-3/4*u - 3/4*v) du⊗dv + (3/4*u + 3/4*v) dv⊗du
sage: a.display(eV)
a = (-3/4*u - 3/4*v) du∧dv
There is also coercion to subdomains, which is nothing but the restriction
of the differential form to some subset of its domain::
sage: L2U = U.diff_form_module(2) ; L2U
Free module Omega^2(U) of 2-forms on the Open subset U of the
2-dimensional differentiable manifold M
sage: L2U.has_coerce_map_from(A)
True
sage: a_U = L2U(a) ; a_U
2-form a on the Open subset U of the 2-dimensional differentiable
manifold M
sage: a_U.display(eU)
a = 3*x dx∧dy
"""
Element = DiffForm
def __init__(self, vector_field_module, degree):
r"""
Construction a module of differential forms.
TESTS:
Module of 2-forms on a non-parallelizable 2-dimensional manifold::
sage: M = Manifold(2, 'M')
sage: U = M.open_subset('U') ; V = M.open_subset('V')
sage: M.declare_union(U,V) # M is the union of U and V
sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart()
sage: transf = c_xy.transition_map(c_uv, (x+y, x-y),
....: intersection_name='W', restrictions1= x>0,
....: restrictions2= u+v>0)
sage: inv = transf.inverse()
sage: from sage.manifolds.differentiable.diff_form_module import \
....: DiffFormModule
sage: A = DiffFormModule(M.vector_field_module(), 2) ; A
Module Omega^2(M) of 2-forms on the 2-dimensional differentiable
manifold M
sage: TestSuite(A).run(skip='_test_elements')
In the above test suite, ``_test_elements`` is skipped because of the
``_test_pickling`` error of the elements (to be fixed in
:class:`sage.manifolds.differentiable.tensorfield.TensorField`)
"""
domain = vector_field_module._domain
dest_map = vector_field_module._dest_map
name = "Omega^{}(".format(degree) + domain._name
latex_name = r"\Omega^{{{}}}\left({}".format(degree, domain._latex_name)
if dest_map is not domain.identity_map():
dm_name = dest_map._name
dm_latex_name = dest_map._latex_name
if dm_name is None:
dm_name = "unnamed map"
if dm_latex_name is None:
dm_latex_name = r"\mathrm{unnamed\; map}"
name += "," + dm_name
latex_name += "," + dm_latex_name
self._name = name + ")"
self._latex_name = latex_name + r"\right)"
self._vmodule = vector_field_module
self._degree = degree
# the member self._ring is created for efficiency (to avoid calls to
# self.base_ring()):
self._ring = domain.scalar_field_algebra()
Parent.__init__(self, base=self._ring, category=Modules(self._ring))
self._domain = domain
self._dest_map = dest_map
self._ambient_domain = vector_field_module._ambient_domain
# NB: self._zero_element is not constructed here, since no element
# can be constructed here, to avoid some infinite recursion.
#### Parent methods
def _element_constructor_(self, comp=[], frame=None, name=None,
latex_name=None):
r"""
Construct a differential form.
TESTS::
sage: M = Manifold(2, 'M')
sage: U = M.open_subset('U'); V = M.open_subset('V')
sage: c_xy.<x,y> = U.chart(); c_uv.<u,v> = V.chart()
sage: M.declare_union(U,V)
sage: A = M.diff_form_module(2)
sage: a = A([[0, x*y], [-x*y, 0]], name='a'); a
2-form a on the 2-dimensional differentiable manifold M
sage: a.display(c_xy.frame())
a = x*y dx∧dy
sage: A(0) is A.zero()
True
"""
try:
if comp.is_trivial_zero():
return self.zero()
except AttributeError:
if comp == 0:
return self.zero()
if isinstance(comp, (DiffForm, DiffFormParal)):
# coercion by domain restriction
if (self._degree == comp._tensor_type[1]
and self._domain.is_subset(comp._domain)
and self._ambient_domain.is_subset(comp._ambient_domain)):
return comp.restrict(self._domain)
else:
raise TypeError("cannot convert the {} ".format(comp) +
"to an element of {}".format(self))
if isinstance(comp, TensorField):
# coercion of a tensor of type (0,1) to a linear form
tensor = comp # for readability
if (tensor.tensor_type() == (0,1) and self._degree == 1
and tensor._vmodule is self._vmodule):
resu = self.element_class(self._vmodule, 1, name=tensor._name,
latex_name=tensor._latex_name)
for dom, rst in tensor._restrictions.items():
resu._restrictions[dom] = dom.diff_form_module(1)(rst)
return resu
else:
raise TypeError("cannot convert the {} ".format(tensor) +
"to an element of {}".format(self))
if not isinstance(comp, (list, tuple)):
raise TypeError("cannot convert the {} ".format(comp) +
"to an element of {}".format(self))
# standard construction
resu = self.element_class(self._vmodule, self._degree, name=name,
latex_name=latex_name)
if comp:
resu.set_comp(frame)[:] = comp
return resu
def _an_element_(self):
r"""
Construct some (unnamed) differential form.
TESTS::
sage: M = Manifold(2, 'M')
sage: U = M.open_subset('U'); V = M.open_subset('V')
sage: c_xy.<x,y> = U.chart(); c_uv.<u,v> = V.chart()
sage: M.declare_union(U,V)
sage: A = M.diff_form_module(2)
sage: A._an_element_()
2-form on the 2-dimensional differentiable manifold M
"""
resu = self.element_class(self._vmodule, self._degree)
for oc in self._domain.open_covers(trivial=False):
# the first non-trivial open cover is selected
for dom in oc:
vmodule_dom = dom.vector_field_module(
dest_map=self._dest_map.restrict(dom))
dmodule_dom = vmodule_dom.dual_exterior_power(self._degree)
resu.set_restriction(dmodule_dom._an_element_())
return resu
return resu
def _coerce_map_from_(self, other):
r"""
Determine whether coercion to ``self`` exists from other parent.
TESTS::
sage: M = Manifold(3, 'M')
sage: A1 = M.diff_form_module(1)
sage: A1._coerce_map_from_(M.tensor_field_module((0,1)))
True
sage: A2 = M.diff_form_module(2)
sage: A2._coerce_map_from_(M.tensor_field_module((0,2)))
False
sage: U = M.open_subset('U')
sage: A2U = U.diff_form_module(2)
sage: A2U._coerce_map_from_(A2)
True
sage: A2._coerce_map_from_(A2U)
False
"""
if isinstance(other, (DiffFormModule, DiffFormFreeModule)):
# coercion by domain restriction
return (self._degree == other._degree
and self._domain.is_subset(other._domain)
and self._ambient_domain.is_subset(other._ambient_domain))
from sage.manifolds.differentiable.tensorfield_module import TensorFieldModule
if isinstance(other, TensorFieldModule):
# coercion of a type-(0,1) tensor to a linear form
return (self._vmodule is other._vmodule and self._degree == 1
and other.tensor_type() == (0,1))
return False
@cached_method
def zero(self):
"""
Return the zero of ``self``.
EXAMPLES::
sage: M = Manifold(3, 'M')
sage: A2 = M.diff_form_module(2)
sage: A2.zero()
2-form zero on the 3-dimensional differentiable manifold M
"""
zero = self._element_constructor_(name='zero', latex_name='0')
for frame in self._domain._frames:
if self._dest_map.restrict(frame._domain) == frame._dest_map:
zero.add_comp(frame)
# (since new components are initialized to zero)
zero._is_zero = True # This element is certainly zero
zero.set_immutable()
return zero
#### End of Parent methods
def _repr_(self):
r"""
Return a string representation of the object.
TESTS::
sage: M = Manifold(3, 'M')
sage: A2 = M.diff_form_module(2)
sage: A2
Module Omega^2(M) of 2-forms on
the 3-dimensional differentiable manifold M
"""
description = "Module "
if self._name is not None:
description += self._name + " "
description += "of {}-forms ".format(self._degree)
if self._dest_map is self._domain.identity_map():
description += "on the {}".format(self._domain)
else:
description += "along the {} mapped into the {}".format(
self._domain, self._ambient_domain)
return description
def _latex_(self):
r"""
Return a LaTeX representation of the object.
TESTS::
sage: M = Manifold(3, 'M', latex_name=r'\mathcal{M}')
sage: A2 = M.diff_form_module(2)
sage: A2._latex_()
'\\Omega^{2}\\left(\\mathcal{M}\\right)'
sage: latex(A2) # indirect doctest
\Omega^{2}\left(\mathcal{M}\right)
"""
if self._latex_name is None:
return r'\mbox{' + str(self) + r'}'
else:
return self._latex_name
def base_module(self):
r"""
Return the vector field module on which the differential form module
``self`` is constructed.
OUTPUT:
- a
:class:`~sage.manifolds.differentiable.vectorfield_module.VectorFieldModule`
representing the module on which ``self`` is defined
EXAMPLES::
sage: M = Manifold(3, 'M')
sage: A2 = M.diff_form_module(2) ; A2
Module Omega^2(M) of 2-forms on the 3-dimensional differentiable
manifold M
sage: A2.base_module()
Module X(M) of vector fields on the 3-dimensional differentiable
manifold M
sage: A2.base_module() is M.vector_field_module()
True
sage: U = M.open_subset('U')
sage: A2U = U.diff_form_module(2) ; A2U
Module Omega^2(U) of 2-forms on the Open subset U of the
3-dimensional differentiable manifold M
sage: A2U.base_module()
Module X(U) of vector fields on the Open subset U of the
3-dimensional differentiable manifold M
"""
return self._vmodule
def degree(self):
r"""
Return the degree of the differential forms in ``self``.
OUTPUT:
- integer `p` such that ``self`` is a set of `p`-forms
EXAMPLES::
sage: M = Manifold(3, 'M')
sage: M.diff_form_module(1).degree()
1
sage: M.diff_form_module(2).degree()
2
sage: M.diff_form_module(3).degree()
3
"""
return self._degree
# *****************************************************************************
class DiffFormFreeModule(ExtPowerDualFreeModule):
r"""
Free module of differential forms of a given degree `p` (`p`-forms) along
a differentiable manifold `U` with values on a parallelizable manifold `M`.
Given a differentiable manifold `U` and a differentiable map
`\Phi:\; U \rightarrow M` to a parallelizable manifold `M` of dimension
`n`, the set `\Omega^p(U, \Phi)` of `p`-forms along `U` with values on `M`
is a free module of rank `\binom{n}{p}` over `C^k(U)`, the commutative
algebra of differentiable scalar fields on `U` (see
:class:`~sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra`).
The standard case of `p`-forms *on* a differentiable manifold `M`
corresponds to `U = M` and `\Phi = \mathrm{Id}_M`. Other common cases are
`\Phi` being an immersion and `\Phi` being a curve in `M` (`U` is then an
open interval of `\RR`).
.. NOTE::
This class implements `\Omega^p(U, \Phi)` in the case where `M` is
parallelizable; `\Omega^p(U, \Phi)` is then a *free* module. If `M`
is not parallelizable, the class :class:`DiffFormModule` must be used
instead.
INPUT:
- ``vector_field_module`` -- free module `\mathfrak{X}(U,\Phi)` of vector
fields along `U` associated with the map `\Phi: U \rightarrow V`
- ``degree`` -- positive integer; the degree `p` of the differential forms
EXAMPLES:
Free module of 2-forms on a parallelizable 3-dimensional manifold::
sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: XM = M.vector_field_module() ; XM
Free module X(M) of vector fields on the 3-dimensional differentiable
manifold M
sage: A = M.diff_form_module(2) ; A
Free module Omega^2(M) of 2-forms on the 3-dimensional differentiable
manifold M
sage: latex(A)
\Omega^{2}\left(M\right)
``A`` is nothing but the second exterior power of the dual of ``XM``, i.e.
we have `\Omega^{2}(M) = \Lambda^2(\mathfrak{X}(M)^*)` (see
:class:`~sage.tensor.modules.ext_pow_free_module.ExtPowerDualFreeModule`)::
sage: A is XM.dual_exterior_power(2)
True
`\Omega^{2}(M)` is a module over the algebra `C^k(M)` of (differentiable)
scalar fields on `M`::
sage: A.category()
Category of finite dimensional modules over Algebra of differentiable
scalar fields on the 3-dimensional differentiable manifold M
sage: CM = M.scalar_field_algebra() ; CM
Algebra of differentiable scalar fields on the 3-dimensional
differentiable manifold M
sage: A in Modules(CM)
True
sage: A.base_ring()
Algebra of differentiable scalar fields on
the 3-dimensional differentiable manifold M
sage: A.base_module()
Free module X(M) of vector fields on
the 3-dimensional differentiable manifold M
sage: A.base_module() is XM
True
sage: A.rank()
3
Elements can be constructed from `A`. In particular, ``0`` yields
the zero element of `A`::
sage: A(0)
2-form zero on the 3-dimensional differentiable manifold M
sage: A(0) is A.zero()
True
while non-zero elements are constructed by providing their components
in a given vector frame::
sage: comp = [[0,3*x,-z],[-3*x,0,4],[z,-4,0]]
sage: a = A(comp, frame=X.frame(), name='a') ; a
2-form a on the 3-dimensional differentiable manifold M
sage: a.display()
a = 3*x dx∧dy - z dx∧dz + 4 dy∧dz
An alternative is to construct the 2-form from an empty list of
components and to set the nonzero nonredundant components afterwards::
sage: a = A([], name='a')
sage: a[0,1] = 3*x # component in the manifold's default frame
sage: a[0,2] = -z
sage: a[1,2] = 4
sage: a.display()
a = 3*x dx∧dy - z dx∧dz + 4 dy∧dz
The module `\Omega^1(M)` is nothing but the dual of `\mathfrak{X}(M)`
(the free module of vector fields on `M`)::
sage: L1 = M.diff_form_module(1) ; L1
Free module Omega^1(M) of 1-forms on the 3-dimensional differentiable
manifold M
sage: L1 is XM.dual()
True
Since any tensor field of type `(0,1)` is a 1-form, there is a coercion
map from the set `T^{(0,1)}(M)` of such tensors to `\Omega^1(M)`::
sage: T01 = M.tensor_field_module((0,1)) ; T01
Free module Omega^1(M) of 1-forms on the 3-dimensional differentiable manifold M
sage: L1.has_coerce_map_from(T01)
True
There is also a coercion map in the reverse direction::
sage: T01.has_coerce_map_from(L1)
True
For a degree `p \geq 2`, the coercion holds only in the direction
`\Omega^p(M) \rightarrow T^{(0,p)}(M)`::
sage: T02 = M.tensor_field_module((0,2)); T02
Free module T^(0,2)(M) of type-(0,2) tensors fields on the
3-dimensional differentiable manifold M
sage: T02.has_coerce_map_from(A)
True
sage: A.has_coerce_map_from(T02)
False
The coercion map `T^{(0,1)}(M) \rightarrow \Omega^1(M)` in action::
sage: b = T01([-x,2,3*y], name='b'); b
1-form b on the 3-dimensional differentiable manifold M
sage: b.display()
b = -x dx + 2 dy + 3*y dz
sage: lb = L1(b) ; lb
1-form b on the 3-dimensional differentiable manifold M
sage: lb.display()
b = -x dx + 2 dy + 3*y dz
The coercion map `\Omega^1(M) \rightarrow T^{(0,1)}(M)` in action::
sage: tlb = T01(lb); tlb
1-form b on the 3-dimensional differentiable manifold M
sage: tlb == b
True
The coercion map `\Omega^2(M) \rightarrow T^{(0,2)}(M)` in action::
sage: T02 = M.tensor_field_module((0,2)) ; T02
Free module T^(0,2)(M) of type-(0,2) tensors fields on the
3-dimensional differentiable manifold M
sage: ta = T02(a) ; ta
Tensor field a of type (0,2) on the 3-dimensional differentiable
manifold M
sage: ta.display()
a = 3*x dx⊗dy - z dx⊗dz - 3*x dy⊗dx + 4 dy⊗dz + z dz⊗dx - 4 dz⊗dy
sage: a.display()
a = 3*x dx∧dy - z dx∧dz + 4 dy∧dz
sage: ta.symmetries() # the antisymmetry is preserved
no symmetry; antisymmetry: (0, 1)
There is also coercion to subdomains, which is nothing but the
restriction of the differential form to some subset of its domain::
sage: U = M.open_subset('U', coord_def={X: x^2+y^2<1})
sage: B = U.diff_form_module(2) ; B
Free module Omega^2(U) of 2-forms on the Open subset U of the
3-dimensional differentiable manifold M
sage: B.has_coerce_map_from(A)
True
sage: a_U = B(a) ; a_U
2-form a on the Open subset U of the 3-dimensional differentiable
manifold M
sage: a_U.display()
a = 3*x dx∧dy - z dx∧dz + 4 dy∧dz
"""
Element = DiffFormParal
def __init__(self, vector_field_module, degree):
r"""
Construct a free module of differential forms.
TESTS::
sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: from sage.manifolds.differentiable.diff_form_module import DiffFormFreeModule
sage: A = DiffFormFreeModule(M.vector_field_module(), 2) ; A
Free module Omega^2(M) of 2-forms on
the 3-dimensional differentiable manifold M
sage: TestSuite(A).run()
"""
domain = vector_field_module._domain
dest_map = vector_field_module._dest_map
name = "Omega^{}(".format(degree) + domain._name
latex_name = r"\Omega^{{{}}}\left({}".format(degree, domain._latex_name)
if dest_map is not domain.identity_map():
dm_name = dest_map._name
dm_latex_name = dest_map._latex_name
if dm_name is None:
dm_name = "unnamed map"
if dm_latex_name is None:
dm_latex_name = r"\mathrm{unnamed\; map}"
name += "," + dm_name
latex_name += "," + dm_latex_name
name += ")"
latex_name += r"\right)"
ExtPowerDualFreeModule.__init__(self, vector_field_module, degree,
name=name, latex_name=latex_name)
self._domain = domain
self._dest_map = dest_map
self._ambient_domain = vector_field_module._ambient_domain
#### Parent methods
def _element_constructor_(self, comp=[], frame=None, name=None,
latex_name=None):
r"""
Construct a differential form.
TESTS::
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart() # makes M parallelizable
sage: A = M.diff_form_module(2)
sage: a = A([[0, x], [-x, 0]], name='a'); a
2-form a on the 2-dimensional differentiable manifold M
sage: a.display()
a = x dx∧dy
sage: A(0) is A.zero()
True
Check that :trac:`27658` is fixed::
sage: f = M.scalar_field(x)
sage: f in A
False
"""
try:
if comp.is_trivial_zero():
return self.zero()
except AttributeError:
if comp == 0:
return self.zero()
if isinstance(comp, (DiffForm, DiffFormParal)):
# coercion by domain restriction
if (self._degree == comp._tensor_type[1]
and self._domain.is_subset(comp._domain)
and self._ambient_domain.is_subset(comp._ambient_domain)):
return comp.restrict(self._domain)
else:
raise TypeError("cannot convert the {} ".format(comp) +
"to a differential form in {}".format(self))
if isinstance(comp, TensorFieldParal):
# coercion of a tensor of type (0,1) to a linear form
tensor = comp # for readability
if (tensor.tensor_type() == (0,1) and self._degree == 1
and tensor._fmodule is self._fmodule):
resu = self.element_class(self._fmodule, 1, name=tensor._name,
latex_name=tensor._latex_name)
for frame, comp in tensor._components.items():
resu._components[frame] = comp.copy()
return resu
else:
raise TypeError("cannot convert the {} ".format(tensor) +
"to an element of {}".format(self))
if not isinstance(comp, (list, tuple)):
raise TypeError("cannot convert the {} ".format(comp) +
"to an element of {}".format(self))
# standard construction
resu = self.element_class(self._fmodule, self._degree, name=name,
latex_name=latex_name)
if comp:
resu.set_comp(frame)[:] = comp
return resu
# Rem: _an_element_ is declared in the superclass ExtPowerDualFreeModule
def _coerce_map_from_(self, other):
r"""
Determine whether coercion to ``self`` exists from other parent.
TESTS::
sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: A2 = M.diff_form_module(2)
sage: U = M.open_subset('U', coord_def = {X: z<0})
sage: A2U = U.diff_form_module(2)
sage: A2U._coerce_map_from_(A2)
True
sage: A2._coerce_map_from_(A2U)
False
sage: A1 = M.diff_form_module(1)
sage: A2U._coerce_map_from_(A1)
False
sage: A1._coerce_map_from_(M.tensor_field_module((0,1)))
True
sage: A1._coerce_map_from_(M.tensor_field_module((1,0)))
False
"""
if isinstance(other, (DiffFormModule, DiffFormFreeModule)):
# coercion by domain restriction
return (self._degree == other._degree
and self._domain.is_subset(other._domain)
and self._ambient_domain.is_subset(other._ambient_domain))
from sage.manifolds.differentiable.tensorfield_module import TensorFieldFreeModule
if isinstance(other, TensorFieldFreeModule):
# coercion of a type-(0,1) tensor to a linear form
return (self._fmodule is other._fmodule and self._degree == 1
and other.tensor_type() == (0,1))
return False
#### End of Parent methods
def _repr_(self):
r"""
Return a string representation of ``self``.
TESTS::
sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: A = M.diff_form_module(2)
sage: A
Free module Omega^2(M) of 2-forms on
the 3-dimensional differentiable manifold M
"""
description = "Free module "
if self._name is not None:
description += self._name + " "
description += "of {}-forms ".format(self._degree)
if self._dest_map is self._domain.identity_map():
description += "on the {}".format(self._domain)
else:
description += "along the {} mapped into the {}".format(
self._domain, self._ambient_domain)
return description
class VectorFieldDualFreeModule(DiffFormFreeModule):
r"""
Free module of differential 1-forms along a differentiable manifold `U`
with values on a parallelizable manifold `M`.
Given a differentiable manifold `U` and a differentiable map
`\Phi:\; U \rightarrow M` to a parallelizable manifold `M` of dimension
`n`, the set `\Omega^1(U, \Phi)` of 1-forms along `U` with values on `M`
is a free module of rank `n` over `C^k(U)`, the commutative
algebra of differentiable scalar fields on `U` (see
:class:`~sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra`).
The standard case of 1-forms *on* a differentiable manifold `M`
corresponds to `U = M` and `\Phi = \mathrm{Id}_M`. Other common cases are
`\Phi` being an immersion and `\Phi` being a curve in `M` (`U` is then an
open interval of `\RR`).
.. NOTE::
This class implements `\Omega^1(U, \Phi)` in the case where `M` is
parallelizable; `\Omega^1(U, \Phi)` is then a *free* module. If `M`
is not parallelizable, the class :class:`DiffFormModule` must be used
instead.
INPUT:
- ``vector_field_module`` -- free module `\mathfrak{X}(U,\Phi)` of vector
fields along `U` associated with the map `\Phi: U \rightarrow V`
EXAMPLES:
Free module of 1-forms on a parallelizable 3-dimensional manifold::
sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: XM = M.vector_field_module() ; XM
Free module X(M) of vector fields on the 3-dimensional differentiable
manifold M
sage: A = M.diff_form_module(1) ; A
Free module Omega^1(M) of 1-forms on the 3-dimensional differentiable manifold M
sage: latex(A)
\Omega^{1}\left(M\right)
``A`` is nothing but the dual of ``XM`` (the free module of vector fields on `M`)
and thus also equal to the 1st exterior
power of the dual, i.e. we have `\Omega^{1}(M) = \Lambda^1(\mathfrak{X}(M)^*)
= \mathfrak{X}(M)^*` (See
:class:`~sage.tensor.modules.ext_pow_free_module.ExtPowerDualFreeModule`)::
sage: A is XM.dual_exterior_power(1)
True
`\Omega^{1}(M)` is a module over the algebra `C^k(M)` of (differentiable)
scalar fields on `M`::
sage: A.category()
Category of finite dimensional modules over Algebra of differentiable
scalar fields on the 3-dimensional differentiable manifold M
sage: CM = M.scalar_field_algebra() ; CM
Algebra of differentiable scalar fields on the 3-dimensional
differentiable manifold M
sage: A in Modules(CM)
True
sage: A.base_ring()
Algebra of differentiable scalar fields on
the 3-dimensional differentiable manifold M
sage: A.base_module()
Free module X(M) of vector fields on
the 3-dimensional differentiable manifold M
sage: A.base_module() is XM
True
sage: A.rank()
3
Elements can be constructed from `A`. In particular, ``0`` yields
the zero element of `A`::
sage: A(0)
1-form zero on the 3-dimensional differentiable manifold M
sage: A(0) is A.zero()
True
while non-zero elements are constructed by providing their components
in a given vector frame::
sage: comp = [3*x,-z,4]
sage: a = A(comp, frame=X.frame(), name='a') ; a
1-form a on the 3-dimensional differentiable manifold M
sage: a.display()
a = 3*x dx - z dy + 4 dz
An alternative is to construct the 1-form from an empty list of
components and to set the nonzero nonredundant components afterwards::
sage: a = A([], name='a')