def __init__(self, vector_field_module, name=None, latex_name=None,
is_identity=False):
r"""
Construct a field of tangent-space automorphisms.
TESTS::
sage: Manifold._clear_cache_() # for doctests only
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: a = M.automorphism_field(name='a') ; a
field of tangent-space automorphisms 'a' on the 2-dimensional manifold 'M'
sage: a[:] = [[1+x^2, x*y], [0, 1+y^2]]
sage: a.parent()
General linear group of the free module X(M) of vector fields on
the 2-dimensional manifold 'M'
sage: a.parent() is M.automorphism_field_group()
True
sage: TestSuite(a).run()
"""
FreeModuleAutomorphism.__init__(self, vector_field_module,
name=name, latex_name=latex_name,
is_identity=is_identity)
# TensorFieldParal attributes:
self._vmodule = vector_field_module
self._domain = vector_field_module._domain
self._ambient_domain = vector_field_module._ambient_domain
# Initialization of derived quantities:
TensorFieldParal._init_derived(self)
def __init__(self,
vector_field_module,
name=None,
latex_name=None,
is_identity=False):
r"""
Construct a field of tangent-space automorphisms.
TESTS::
sage: Manifold._clear_cache_() # for doctests only
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: a = M.automorphism_field(name='a') ; a
field of tangent-space automorphisms 'a' on the 2-dimensional manifold 'M'
sage: a[:] = [[1+x^2, x*y], [0, 1+y^2]]
sage: a.parent()
General linear group of the free module X(M) of vector fields on
the 2-dimensional manifold 'M'
sage: a.parent() is M.automorphism_field_group()
True
sage: TestSuite(a).run()
"""
FreeModuleAutomorphism.__init__(self,
vector_field_module,
name=name,
latex_name=latex_name,
is_identity=is_identity)
# TensorFieldParal attributes:
self._vmodule = vector_field_module
self._domain = vector_field_module._domain
self._ambient_domain = vector_field_module._ambient_domain
# Initialization of derived quantities:
TensorFieldParal._init_derived(self)
def _del_derived(self, del_restrictions=True):
r"""
Delete the derived quantities
INPUT:
- ``del_restrictions`` -- (default: True) determines whether the
restrictions of ``self`` to subdomains are deleted.
"""
TensorFieldParal._del_derived(self, del_restrictions=del_restrictions)
self._exterior_derivative = None
def _del_derived(self, del_restrictions=True):
r"""
Delete the derived quantities
INPUT:
- ``del_restrictions`` -- (default: True) determines whether the
restrictions of ``self`` to subdomains are deleted.
"""
TensorFieldParal._del_derived(self, del_restrictions=del_restrictions)
VectorField._del_derived(self)
self._del_dependencies()
def __init__(self, vector_field_module, name=None, latex_name=None):
FiniteRankFreeModuleElement.__init__(self, vector_field_module, name=name,
latex_name=latex_name)
# TensorFieldParal attributes:
self._domain = vector_field_module._domain
self._ambient_domain = vector_field_module._ambient_domain
# VectorField attributes:
self._vmodule = vector_field_module
# Initialization of derived quantities:
TensorFieldParal._init_derived(self)
VectorField._init_derived(self)
# Initialization of list of quantities depending on self:
self._init_dependencies()
def _del_derived(self, del_restrictions=True):
r"""
Delete the derived quantities
INPUT:
- ``del_restrictions`` -- (default: True) determines whether the
restrictions of ``self`` to subdomains are deleted.
"""
# Delete the derived quantities pertaining to the mother classes:
FreeModuleAutomorphism._del_derived(self)
TensorFieldParal._del_derived(self, del_restrictions=del_restrictions)
def _del_derived(self, del_restrictions=True):
r"""
Delete the derived quantities
INPUT:
- ``del_restrictions`` -- (default: True) determines whether the
restrictions of ``self`` to subdomains are deleted.
"""
# Delete the derived quantities pertaining to the mother classes:
FreeModuleAutomorphism._del_derived(self)
TensorFieldParal._del_derived(self, del_restrictions=del_restrictions)
def __init__(self, vector_field_module, name=None, latex_name=None):
FiniteRankFreeModuleElement.__init__(self,
vector_field_module,
name=name,
latex_name=latex_name)
# TensorFieldParal attributes:
self._domain = vector_field_module._domain
self._ambient_domain = vector_field_module._ambient_domain
# VectorField attributes:
self._vmodule = vector_field_module
# Initialization of derived quantities:
TensorFieldParal._init_derived(self)
VectorField._init_derived(self)
# Initialization of list of quantities depending on self:
self._init_dependencies()
def __call__(self, *args):
r"""
Redefinition of
:meth:`~sage.tensor.modules.free_module_tensor.FreeModuleTensor.__call__`
to allow for domain treatment
"""
return TensorFieldParal.__call__(self, *args)
def _init_derived(self):
r"""
Initialize the derived quantities
"""
TensorFieldParal._init_derived(self)
self._exterior_derivative = None
def restrict(self, subdomain, dest_map=None):
r"""
Return the restriction of ``self`` to some subset of its domain.
If such restriction has not been defined yet, it is constructed here.
This is a redefinition of
:meth:`sage.geometry.manifolds.tensorfield.TensorFieldParal.restrict`
to take into account the identity map.
INPUT:
- ``subdomain`` -- open subset `U` of ``self._domain`` (must be an
instance of :class:`~sage.geometry.manifolds.domain.ManifoldOpenSubset`)
- ``dest_map`` -- (default: ``None``) destination map
`\Phi:\ U \rightarrow V`, where `V` is a subset of
``self._codomain``
(type: :class:`~sage.geometry.manifolds.diffmapping.DiffMapping`)
If None, the restriction of ``self._vmodule._dest_map`` to `U` is
used.
OUTPUT:
- instance of :class:`AutomorphismFieldParal` representing the
restriction.
EXAMPLES:
Restriction of an automorphism field defined on `\RR^2` to a disk::
sage: M = Manifold(2, 'R^2')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
sage: D = M.open_subset('D') # the unit open disc
sage: c_cart_D = c_cart.restrict(D, x^2+y^2<1)
sage: a = M.automorphism_field(name='a') ; a
field of tangent-space automorphisms 'a' on the 2-dimensional manifold 'R^2'
sage: a[:] = [[1, x*y], [0, 3]]
sage: a.restrict(D)
field of tangent-space automorphisms 'a' on the open subset 'D' of the 2-dimensional manifold 'R^2'
sage: a.restrict(D)[:]
[ 1 x*y]
[ 0 3]
Restriction to the disk of the field of tangent-space identity maps::
sage: id = M.tangent_identity_field() ; id
field of tangent-space identity maps on the 2-dimensional manifold 'R^2'
sage: id.restrict(D)
field of tangent-space identity maps on the open subset 'D' of the 2-dimensional manifold 'R^2'
sage: id.restrict(D)[:]
[1 0]
[0 1]
sage: id.restrict(D) == D.tangent_identity_field()
True
"""
if subdomain == self._domain:
return self
if subdomain not in self._restrictions:
if not self._is_identity:
return TensorFieldParal.restrict(self, subdomain,
dest_map=dest_map)
# Special case of the identity map:
if not subdomain.is_subset(self._domain):
raise ValueError("The provided domain is not a subset of " +
"the field's domain.")
if dest_map is None:
dest_map = self._fmodule._dest_map.restrict(subdomain)
elif not dest_map._codomain.is_subset(self._ambient_domain):
raise ValueError("Argument dest_map not compatible with " +
"self._ambient_domain")
smodule = subdomain.vector_field_module(dest_map=dest_map)
self._restrictions[subdomain] = smodule.identity_map()
return self._restrictions[subdomain]
def restrict(self, subdomain, dest_map=None):
r"""
Return the restriction of ``self`` to some subset of its domain.
If such restriction has not been defined yet, it is constructed here.
This is a redefinition of
:meth:`sage.geometry.manifolds.tensorfield.TensorFieldParal.restrict`
to take into account the identity map.
INPUT:
- ``subdomain`` -- open subset `U` of ``self._domain`` (must be an
instance of :class:`~sage.geometry.manifolds.domain.ManifoldOpenSubset`)
- ``dest_map`` -- (default: ``None``) destination map
`\Phi:\ U \rightarrow V`, where `V` is a subset of
``self._codomain``
(type: :class:`~sage.geometry.manifolds.diffmapping.DiffMapping`)
If None, the restriction of ``self._vmodule._dest_map`` to `U` is
used.
OUTPUT:
- instance of :class:`AutomorphismFieldParal` representing the
restriction.
EXAMPLES:
Restriction of an automorphism field defined on `\RR^2` to a disk::
sage: M = Manifold(2, 'R^2')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
sage: D = M.open_subset('D') # the unit open disc
sage: c_cart_D = c_cart.restrict(D, x^2+y^2<1)
sage: a = M.automorphism_field(name='a') ; a
field of tangent-space automorphisms 'a' on the 2-dimensional manifold 'R^2'
sage: a[:] = [[1, x*y], [0, 3]]
sage: a.restrict(D)
field of tangent-space automorphisms 'a' on the open subset 'D' of the 2-dimensional manifold 'R^2'
sage: a.restrict(D)[:]
[ 1 x*y]
[ 0 3]
Restriction to the disk of the field of tangent-space identity maps::
sage: id = M.tangent_identity_field() ; id
field of tangent-space identity maps on the 2-dimensional manifold 'R^2'
sage: id.restrict(D)
field of tangent-space identity maps on the open subset 'D' of the 2-dimensional manifold 'R^2'
sage: id.restrict(D)[:]
[1 0]
[0 1]
sage: id.restrict(D) == D.tangent_identity_field()
True
"""
if subdomain == self._domain:
return self
if subdomain not in self._restrictions:
if not self._is_identity:
return TensorFieldParal.restrict(self,
subdomain,
dest_map=dest_map)
# Special case of the identity map:
if not subdomain.is_subset(self._domain):
raise ValueError("The provided domain is not a subset of " +
"the field's domain.")
if dest_map is None:
dest_map = self._fmodule._dest_map.restrict(subdomain)
elif not dest_map._codomain.is_subset(self._ambient_domain):
raise ValueError("Argument dest_map not compatible with " +
"self._ambient_domain")
smodule = subdomain.vector_field_module(dest_map=dest_map)
self._restrictions[subdomain] = smodule.identity_map()
return self._restrictions[subdomain]