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r""" Multivector Field Modules
The set `A^p(U, \Phi)` of `p`-vector fields 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 `A^p(U,\Phi)`:
- :class:`MultivectorModule` for `p`-vector fields with values on a generic (in practice, not parallelizable) differentiable manifold `M` - :class:`MultivectorFreeModule` for `p`-vector fields with values on a parallelizable manifold `M`
AUTHORS:
- Eric Gourgoulhon (2017): initial version
REFERENCES:
- \R. L. Bishop and S. L. Goldberg (1980) [BG1980]_ - \C.-M. Marle (1997) [Mar1997]_
""" #****************************************************************************** # Copyright (C) 2017 Eric Gourgoulhon <eric.gourgoulhon@obspm.fr> # # 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/ #******************************************************************************
MultivectorField, MultivectorFieldParal)
r""" Module of multivector fields of a given degree `p` (`p`-vector fields) 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 `A^p(U, \Phi)` of `p`-vector fields (i.e. alternating tensor fields of type `(p,0)`) 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`-vector fields *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 `A^p(U,\Phi)` in the case where `M` is not assumed to be parallelizable; the module `A^p(U, \Phi)` is then not necessarily free. If `M` is parallelizable, the class :class:`MultivectorFreeModule` 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 multivector fields
EXAMPLES:
Module of 2-vector fields 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.multivector_module(2) ; A Module A^2(M) of 2-vector fields on the 2-dimensional differentiable manifold M sage: latex(A) A^{2}\left(M\right)
``A`` is nothing but the second exterior power of of ``XM``, i.e. we have `A^{2}(M) = \Lambda^2(\mathfrak{X}(M))`::
sage: A is XM.exterior_power(2) True
Modules of multivector fields are unique::
sage: A is M.multivector_module(2) True
`A^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-vector field 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-vector field 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 d/dx/\d/dy sage: a.display(eV) a = (-3*u - 3*v) d/du/\d/dv
An alternative is to construct the 2-vector field 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 d/dx/\d/dy sage: a.display(eV) a = (-3*u - 3*v) d/du/\d/dv
The module `A^1(M)` is nothing but the dual of `\mathfrak{X}(M)` (the module of vector fields on `M`)::
sage: A1 = M.multivector_module(1) ; A1 Module X(M) of vector fields on the 2-dimensional differentiable manifold M sage: A1 is XM True
There is a coercion map `A^p(M)\rightarrow T^{(p,0)}(M)`::
sage: T20 = M.tensor_field_module((2,0)) ; T20 Module T^(2,0)(M) of type-(2,0) tensors fields on the 2-dimensional differentiable manifold M sage: T20.has_coerce_map_from(A) True
but of course not in the reverse direction, since not all contravariant tensor field is alternating::
sage: A.has_coerce_map_from(T20) False
The coercion map `A^2(M) \rightarrow T^{(2,0)}(M)` in action::
sage: ta = T20(a) ; ta Tensor field a of type (2,0) on the 2-dimensional differentiable manifold M sage: ta.display(eU) a = 3*x d/dx*d/dy - 3*x d/dy*d/dx sage: a.display(eU) a = 3*x d/dx/\d/dy sage: ta.display(eV) a = (-3*u - 3*v) d/du*d/dv + (3*u + 3*v) d/dv*d/du sage: a.display(eV) a = (-3*u - 3*v) d/du/\d/dv
There is also coercion to subdomains, which is nothing but the restriction of the multivector field to some subset of its domain::
sage: A2U = U.multivector_module(2) ; A2U Free module A^2(U) of 2-vector fields on the Open subset U of the 2-dimensional differentiable manifold M sage: A2U.has_coerce_map_from(A) True sage: a_U = A2U(a) ; a_U 2-vector field a on the Open subset U of the 2-dimensional differentiable manifold M sage: a_U.display(eU) a = 3*x d/dx/\d/dy
"""
r""" Construction a module of multivector fields.
TESTS:
Module of 2-vector fields 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.multivector_module import \ ....: MultivectorModule sage: A = MultivectorModule(M.vector_field_module(), 2) ; A Module A^2(M) of 2-vector fields 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`)
""" domain._latex_name) 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()): category=Modules(self._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 multivector field.
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.multivector_module(2) sage: a = A([[0, x*y], [-x*y, 0]], name='a'); a 2-vector field a on the 2-dimensional differentiable manifold M sage: a.display(c_xy.frame()) a = x*y d/dx/\d/dy sage: A(0) is A.zero() True
""" # coercion by domain restriction if (self._degree == comp._tensor_type[0] 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)) # standard construction name=name, latex_name=latex_name)
r""" Construct some (unnamed) multivector field.
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.multivector_module(2) sage: A._an_element_() 2-vector field on the 2-dimensional differentiable manifold M
""" # Non-trivial open covers of the domain: # is trivial # selected dest_map=self._dest_map.restrict(dom))
r""" Determine whether coercion to ``self`` exists from other parent.
TESTS::
sage: M = Manifold(3, 'M') sage: A2 = M.multivector_module(2) sage: A2._coerce_map_from_(M.tensor_field_module((2,0))) False sage: U = M.open_subset('U') sage: A2U = U.multivector_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))
def zero(self): """ Return the zero of ``self``.
EXAMPLES::
sage: M = Manifold(3, 'M') sage: A2 = M.multivector_module(2) sage: A2.zero() 2-vector field zero on the 3-dimensional differentiable manifold M
""" name='zero', 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.multivector_module(2) sage: A2 Module A^2(M) of 2-vector fields on the 3-dimensional differentiable manifold M
""" else: description += "along the {} mapped into the {}".format( elf._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.multivector_module(2) sage: A2._latex_() 'A^{2}\\left(\\mathcal{M}\\right)' sage: latex(A2) # indirect doctest A^{2}\left(\mathcal{M}\right)
""" return r'\mbox{' + str(self) + r'}' else:
r""" Return the vector field module on which the multivector field 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.multivector_module(2) ; A2 Module A^2(M) of 2-vector fields 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.multivector_module(2) ; A2U Module A^2(U) of 2-vector fields 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 multivector fields in ``self``.
OUTPUT:
- integer `p` such that ``self`` is a set of `p`-vector fields
EXAMPLES::
sage: M = Manifold(3, 'M') sage: M.multivector_module(2).degree() 2 sage: M.multivector_module(3).degree() 3
"""
#***********************************************************************
r""" Free module of multivector fields of a given degree `p` (`p`-vector fields) 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 `A^p(U, \Phi)` of `p`-vector fields (i.e. alternating tensor fields of type `(p,0)`) 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`-vector fields *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 `A^p(U, \Phi)` in the case where `M` is parallelizable; `A^p(U, \Phi)` is then a *free* module. If `M` is not parallelizable, the class :class:`MultivectorModule` 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 multivector fields
EXAMPLES:
Free module of 2-vector fields 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.multivector_module(2) ; A Free module A^2(M) of 2-vector fields on the 3-dimensional differentiable manifold M sage: latex(A) A^{2}\left(M\right)
``A`` is nothing but the second exterior power of ``XM``, i.e. we have `A^{2}(M) = \Lambda^2(\mathfrak{X}(M))` (see :class:`~sage.tensor.modules.ext_pow_free_module.ExtPowerFreeModule`)::
sage: A is XM.exterior_power(2) True
`A^{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-vector field 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-vector field a on the 3-dimensional differentiable manifold M sage: a.display() a = 3*x d/dx/\d/dy - z d/dx/\d/dz + 4 d/dy/\d/dz
An alternative is to construct the 2-vector field 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 d/dx/\d/dy - z d/dx/\d/dz + 4 d/dy/\d/dz
The module `A^1(M)` is nothing but `\mathfrak{X}(M)` (the free module of vector fields on `M`)::
sage: A1 = M.multivector_module(1) ; A1 Free module X(M) of vector fields on the 3-dimensional differentiable manifold M sage: A1 is XM True
There is a coercion map `A^p(M) \rightarrow T^{(p,0)}(M)`::
sage: T20 = M.tensor_field_module((2,0)); T20 Free module T^(2,0)(M) of type-(2,0) tensors fields on the 3-dimensional differentiable manifold M sage: T20.has_coerce_map_from(A) True
but of course not in the reverse direction, since not all contravariant tensor field is alternating::
sage: A.has_coerce_map_from(T20) False
The coercion map `A^2(M) \rightarrow T^{(2,0)}(M)` in action::
sage: T20 = M.tensor_field_module((2,0)) ; T20 Free module T^(2,0)(M) of type-(2,0) tensors fields on the 3-dimensional differentiable manifold M sage: ta = T20(a) ; ta Tensor field a of type (2,0) on the 3-dimensional differentiable manifold M sage: ta.display() a = 3*x d/dx*d/dy - z d/dx*d/dz - 3*x d/dy*d/dx + 4 d/dy*d/dz + z d/dz*d/dx - 4 d/dz*d/dy sage: a.display() a = 3*x d/dx/\d/dy - z d/dx/\d/dz + 4 d/dy/\d/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 multivector field to some subset of its domain::
sage: U = M.open_subset('U', coord_def={X: x^2+y^2<1}) sage: B = U.multivector_module(2) ; B Free module A^2(U) of 2-vector fields 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-vector field a on the Open subset U of the 3-dimensional differentiable manifold M sage: a_U.display() a = 3*x d/dx/\d/dy - z d/dx/\d/dz + 4 d/dy/\d/dz
"""
r""" Construct a free module of multivector fields.
TESTS::
sage: M = Manifold(3, 'M') sage: X.<x,y,z> = M.chart() sage: from sage.manifolds.differentiable.multivector_module \ ....: import MultivectorFreeModule sage: A = MultivectorFreeModule(M.vector_field_module(), 2) sage: A Free module A^2(M) of 2-vector fields on the 3-dimensional differentiable manifold M sage: TestSuite(A).run()
""" domain._latex_name) 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=name, latex_name=latex_name)
#### Parent methods
latex_name=None): r""" Construct a multivector field.
TESTS::
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() # makes M parallelizable sage: A = M.multivector_module(2) sage: a = A([[0, x], [-x, 0]], name='a'); a 2-vector field a on the 2-dimensional differentiable manifold M sage: a.display() a = x d/dx/\d/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 multivector field in {}".format(self)) # standard construction latex_name=latex_name)
# Rem: _an_element_ is declared in the superclass ExtPowerFreeModule
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.multivector_module(2) sage: U = M.open_subset('U', coord_def = {X: z<0}) sage: A2U = U.multivector_module(2) sage: A2U._coerce_map_from_(A2) True sage: A2._coerce_map_from_(A2U) False sage: A1 = M.multivector_module(1) sage: A2U._coerce_map_from_(A1) False sage: A2._coerce_map_from_(M.tensor_field_module((2,0))) False
""" # coercion by domain restriction and self._domain.is_subset(other._domain) and self._ambient_domain.is_subset( other._ambient_domain))
#### 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.multivector_module(2) sage: A Free module A^2(M) of 2-vector fields on the 3-dimensional differentiable manifold M
""" else: description += "along the {} mapped into the {}".format( self._domain, self._ambient_domain) |