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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. # http://www.gnu.org/licenses/ #******************************************************************************
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
"""
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.differentialbe.tensorfield.TensorField`)
""" 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 # the member self._ring is created for efficiency (to avoid calls to # self.base_ring()): # NB: self._zero_element is not constructed here, since no element # can be constructed here, to avoid some infinite recursion.
#### Parent methods
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
""" # 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)) # coercion of a tensor of type (0,1) to a linear form and tensor._vmodule is self._vmodule): latex_name=tensor._latex_name) else: raise TypeError("cannot convert the {} ".format(tensor) + "to an element of {}".format(self)) # standard construction latex_name=latex_name)
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
""" # Non-trivial open covers of the domain: # is trivial dest_map=self._dest_map.restrict(dom))
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
""" # coercion by domain restriction and self._domain.is_subset(other._domain) and self._ambient_domain.is_subset(other._ambient_domain))
# coercion of a type-(0,1) tensor to a linear form and other.tensor_type() == (0,1))
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
""" latex_name='0') # (since new components are initialized to zero)
#### End of Parent methods
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
""" else: description += "along the {} mapped into the {}".format( self._domain, self._ambient_domain)
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)
""" return r'\mbox{' + str(self) + r'}' else:
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
"""
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
"""
#******************************************************************************
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 T^(0,1)(M) of type-(0,1) tensors fields 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 Tensor field b of type (0,1) 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 Tensor field b of type (0,1) 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
"""
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()
""" name=name, latex_name=latex_name)
#### Parent methods
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
""" # coercion by domain restriction and self._domain.is_subset(comp._domain) and self._ambient_domain.is_subset(comp._ambient_domain)): else: raise TypeError("cannot convert the {} ".format(comp) + "to a differential form in {}".format(self)) # coercion of a tensor of type (0,1) to a linear form and tensor._fmodule is self._fmodule): latex_name=tensor._latex_name) else: raise TypeError("cannot convert the {} ".format(tensor) + "to an element of {}".format(self)) latex_name=latex_name)
# Rem: _an_element_ is declared in the superclass ExtPowerDualFreeModule
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
""" # coercion by domain restriction and self._domain.is_subset(other._domain) and self._ambient_domain.is_subset(other._ambient_domain))
# coercion of a type-(0,1) tensor to a linear form and other.tensor_type() == (0,1))
#### End of Parent methods
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
""" else: description += "along the {} mapped into the {}".format( self._domain, self._ambient_domain)
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