def __init__(self,
vector_field_module,
degree,
name=None,
latex_name=None):
r"""
Construct a differential form.
TESTS:
Construction via ``parent.element_class``, and not via a direct call
to ``DiffForm`, to fit with the category framework::
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: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y),
....: intersection_name='W', restrictions1= x>0,
....: restrictions2= u+v>0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: W = U.intersection(V)
sage: e_xy = c_xy.frame() ; e_uv = c_uv.frame()
sage: A = M.diff_form_module(2)
sage: XM = M.vector_field_module()
sage: a = A.element_class(XM, 2, name='a'); a
2-form a on the 2-dimensional differentiable manifold M
sage: a[e_xy,0,1] = x+y
sage: a.add_comp_by_continuation(e_uv, W, c_uv)
sage: TestSuite(a).run(skip='_test_pickling')
Construction with ``DifferentiableManifold.diff_form``::
sage: a1 = M.diff_form(2, name='a'); a1
2-form a on the 2-dimensional differentiable manifold M
sage: type(a1) == type(a)
True
sage: a1.parent() is a.parent()
True
.. TODO::
Fix ``_test_pickling`` (in the superclass :class:`TensorField`).
"""
TensorField.__init__(
self,
vector_field_module, (0, degree),
name=name,
latex_name=latex_name,
antisym=range(degree),
parent=vector_field_module.dual_exterior_power(degree))
self._init_derived() # initialization of derived quantities
def __init__(self, tensor, embedding, screen=None):
r"""
TESTS::
sage: M = Manifold(4, 'M', structure="Lorentzian")
sage: X.<t,x,y,z> = M.chart()
sage: S = Manifold(3, 'S', ambient=M, structure='degenerate_metric')
sage: X_S.<u,v,w> = S.chart()
sage: Phi = S.diff_map(M, {(X_S, X): [u, u, v, w]},
....: name='Phi', latex_name=r'\Phi');
sage: Phi_inv = M.diff_map(S, {(X, X_S): [x,y, z]}, name='Phi_inv',
....: latex_name=r'\Phi^{-1}');
sage: S.set_immersion(Phi, inverse=Phi_inv); S.declare_embedding()
sage: g = M.metric()
sage: g[0,0], g[1,1], g[2,2], g[3,3] = -1,1,1,1
sage: v = M.vector_field(); v[1] = 1
sage: S.set_transverse(rigging=v)
sage: xi = M.vector_field(); xi[0] = 1; xi[1] = 1
sage: U = M.vector_field(); U[2] = 1; V = M.vector_field(); V[3] = 1
sage: Sc = S.screen('Sc', (U,V), xi);
sage: T1 = M.tensor_field(1,1).along(Phi); T1[0,0] = 1
sage: from sage.manifolds.differentiable.degenerate_submanifold import TangentTensor
sage: T2 = TangentTensor(T1, Phi); T2
Tensor field of type (1,1) along the degenerate hypersurface S embedded in
4-dimensional differentiable manifold M with values on the 4-dimensional
Lorentzian manifold M
"""
if not isinstance(tensor, TensorField):
raise TypeError("the second argument must be a tensor field")
self._tensor = tensor
self._tensor.set_name(tensor._name, latex_name=tensor._latex_name)
self._embedding = embedding
try:
tensor = tensor.along(embedding)
except ValueError:
pass
if isinstance(tensor, TensorFieldParal):
TensorFieldParal.__init__(self, tensor._vmodule, tensor._tensor_type, name=tensor._name,
latex_name=tensor._latex_name, sym=tensor._sym, antisym=tensor._antisym)
else:
TensorField.__init__(self, tensor._vmodule, tensor._tensor_type, name=tensor._name,
latex_name=tensor._latex_name, sym=tensor._sym, antisym=tensor._antisym)
f = tensor._domain._ambient.default_frame().along(embedding)
self[f, :] = tensor[f, :]
frame = self._domain.adapted_frame(screen)
self.display(frame)
for i in self._domain._ambient.index_generator(tensor.tensor_rank()):
for j in range(len(i)):
if i[j]==self._domain._ambient._dim-self._domain._sindex-1:
self[frame, i] = 0
def _init_derived(self):
r"""
Initialize the derived quantities.
TESTS::
sage: M = Manifold(2, 'M')
sage: a = M.automorphism_field(name='a')
sage: a._init_derived()
"""
TensorField._init_derived(self)
self._inverse = None # inverse not set yet
def _init_derived(self):
r"""
Initialize the derived quantities.
TESTS::
sage: M = Manifold(2, 'M')
sage: a = M.automorphism_field(name='a')
sage: a._init_derived()
"""
TensorField._init_derived(self)
self._inverse = None # inverse not set yet
def _del_derived(self):
r"""
Delete the derived quantities.
TESTS::
sage: M = Manifold(3, 'M')
sage: a = M.diff_form(2, name='a')
sage: a._del_derived()
"""
TensorField._del_derived(self)
self.exterior_derivative.clear_cache()
def _del_derived(self):
r"""
Delete the derived quantities.
TESTS::
sage: M = Manifold(3, 'M')
sage: a = M.diff_form(2, name='a')
sage: a._del_derived()
"""
TensorField._del_derived(self)
self.exterior_derivative.clear_cache()
def _del_derived(self):
r"""
Delete the derived quantities.
TESTS::
sage: M = Manifold(2, 'M')
sage: a = M.automorphism_field(name='a')
sage: a._del_derived()
"""
# First delete the derived quantities pertaining to the mother class:
TensorField._del_derived(self)
# then deletes the inverse automorphism:
self._inverse = None
def _del_derived(self):
r"""
Delete the derived quantities.
TESTS::
sage: M = Manifold(2, 'M')
sage: a = M.automorphism_field(name='a')
sage: a._del_derived()
"""
# First delete the derived quantities pertaining to the mother class:
TensorField._del_derived(self)
# then deletes the inverse automorphism:
self._inverse = None
def __init__(self, vector_field_module, degree, name=None, latex_name=None):
r"""
Construct a differential form.
TESTS:
Construction via ``parent.element_class``, and not via a direct call
to ``DiffForm`, to fit with the category framework::
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: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y),
....: intersection_name='W', restrictions1= x>0,
....: restrictions2= u+v>0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: W = U.intersection(V)
sage: e_xy = c_xy.frame() ; e_uv = c_uv.frame()
sage: A = M.diff_form_module(2)
sage: XM = M.vector_field_module()
sage: a = A.element_class(XM, 2, name='a'); a
2-form a on the 2-dimensional differentiable manifold M
sage: a[e_xy,0,1] = x+y
sage: a.add_comp_by_continuation(e_uv, W, c_uv)
sage: TestSuite(a).run(skip='_test_pickling')
Construction with ``DifferentiableManifold.diff_form``::
sage: a1 = M.diff_form(2, name='a'); a1
2-form a on the 2-dimensional differentiable manifold M
sage: type(a1) == type(a)
True
sage: a1.parent() is a.parent()
True
.. TODO::
Fix ``_test_pickling`` (in the superclass :class:`TensorField`).
"""
TensorField.__init__(self, vector_field_module, (0,degree), name=name,
latex_name=latex_name, antisym=range(degree),
parent=vector_field_module.dual_exterior_power(degree))
self._init_derived() # initialization of derived quantities
def __call__(self, *args):
r"""
TESTS::
sage: M = Manifold(4, 'M', structure="Lorentzian")
sage: X.<t,x,y,z> = M.chart()
sage: S = Manifold(3, 'S', ambient=M, structure='degenerate_metric')
sage: X_S.<u,v,w> = S.chart()
sage: Phi = S.diff_map(M, {(X_S, X): [u, u, v, w]},
....: name='Phi', latex_name=r'\Phi');
sage: Phi_inv = M.diff_map(S, {(X, X_S): [x,y, z]}, name='Phi_inv',
....: latex_name=r'\Phi^{-1}');
sage: S.set_immersion(Phi, inverse=Phi_inv); S.declare_embedding()
sage: g = M.metric()
sage: g[0,0], g[1,1], g[2,2], g[3,3] = -1,1,1,1
sage: v = M.vector_field(); v[1] = 1
sage: S.set_transverse(rigging=v)
sage: xi = M.vector_field(); xi[0] = 1; xi[1] = 1
sage: U = M.vector_field(); U[2] = 1; V = M.vector_field(); V[3] = 1
sage: Sc = S.screen('Sc', (U,V), xi);
sage: T1 = M.tensor_field(1,1).along(Phi); T1[0,0] = 1
sage: from sage.manifolds.differentiable.degenerate_submanifold import TangentTensor
sage: T2 = TangentTensor(T1, Phi); T2(xi.along(Phi))
Vector field along the degenerate hypersurface S embedded in
4-dimensional differentiable manifold M with values on the 4-dimensional
Lorentzian manifold M
"""
for vector in args:
try:
vector = vector.along(self._embedding)
except ValueError:
pass
if not self._domain.is_tangent(vector):
raise ValueError("The provided vector field is not " +
"tangent to {}".format(self._domain._name))
try:
return TensorField.__call__(self._tensor.along(self._embedding),
*args)
except ValueError:
return TensorField.__call__(self._tensor, *args)
def __init__(self, vector_field_module, name=None, latex_name=None):
r"""
Construct a vector field with values on a non-parallelizable manifold.
TESTS:
Construction via ``parent.element_class``, and not via a direct call
to ``VectorField``, to fit with the category framework::
sage: M = Manifold(2, 'M') # 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: XM = M.vector_field_module()
sage: a = XM.element_class(XM, name='a'); a
Vector field a on the 2-dimensional differentiable manifold M
sage: a[c_xy.frame(),:] = [x, y]
sage: a[c_uv.frame(),:] = [-u, -v]
sage: TestSuite(a).run(skip='_test_pickling')
Construction with ``DifferentiableManifold.vector_field``::
sage: a1 = M.vector_field(name='a'); a1
Vector field a on the 2-dimensional differentiable manifold M
sage: type(a1) == type(a)
True
.. TODO::
Fix ``_test_pickling`` (in the superclass :class:`TensorField`).
"""
TensorField.__init__(self,
vector_field_module, (1, 0),
name=name,
latex_name=latex_name)
# Initialization of derived quantities:
TensorField._init_derived(self)
# Initialization of list of quantities depending on self:
self._init_dependencies()
def __init__(self, vector_field_module, name=None, latex_name=None):
r"""
Construct a vector field with values on a non-parallelizable manifold.
TESTS:
Construction via ``parent.element_class``, and not via a direct call
to ``VectorField``, to fit with the category framework::
sage: M = Manifold(2, 'M') # 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: XM = M.vector_field_module()
sage: a = XM.element_class(XM, name='a'); a
Vector field a on the 2-dimensional differentiable manifold M
sage: a[c_xy.frame(),:] = [x, y]
sage: a[c_uv.frame(),:] = [-u, -v]
sage: TestSuite(a).run(skip='_test_pickling')
Construction with ``DifferentiableManifold.vector_field``::
sage: a1 = M.vector_field(name='a'); a1
Vector field a on the 2-dimensional differentiable manifold M
sage: type(a1) == type(a)
True
.. TODO::
Fix ``_test_pickling`` (in the superclass :class:`TensorField`).
"""
TensorField.__init__(self, vector_field_module, (1,0), name=name,
latex_name=latex_name)
# Initialization of derived quantities:
TensorField._init_derived(self)
# Initialization of list of quantities depending on self:
self._init_dependencies()
def __mul__(self, other):
r"""
Redefinition of
:meth:`~sage.manifolds.differentiable.tensorfield.TensorField.__mul__`
so that ``*`` dispatches either to automorphism composition or
to the tensor product.
TESTS::
sage: M = Manifold(2, 'M') # 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: e_xy = c_xy.frame(); e_uv = c_uv.frame()
sage: a = M.automorphism_field(name='a')
sage: a[e_xy, :] = [[-1, 0], [0, 1]]
sage: a.add_comp_by_continuation(e_uv, U.intersection(V), c_uv)
sage: b = M.automorphism_field(name='b')
sage: b[e_uv, :] = [[1, 0], [0, -2]]
sage: b.add_comp_by_continuation(e_xy, U.intersection(V), c_xy)
sage: w = M.vector_field(name='w')
sage: w[e_xy, :] = [3, 1]
sage: w.add_comp_by_continuation(e_uv, U.intersection(V), c_uv)
sage: s = a.__mul__(b); s # automorphism composition
Field of tangent-space automorphisms on the 2-dimensional differentiable manifold M
sage: s(w) == a(b(w)) # long time
True
sage: s = a.__mul__(w); s # tensor product
Tensor field of type (2,1) on the 2-dimensional differentiable manifold M
"""
if isinstance(other, AutomorphismField):
return self._mul_(other) # general linear group law
else:
return TensorField.__mul__(self, other) # tensor product
def __mul__(self, other):
r"""
Redefinition of
:meth:`~sage.manifolds.differentiable.tensorfield.TensorField.__mul__`
so that ``*`` dispatches either to automorphism composition or
to the tensor product.
TESTS::
sage: M = Manifold(2, 'M') # 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: e_xy = c_xy.frame(); e_uv = c_uv.frame()
sage: a = M.automorphism_field(name='a')
sage: a[e_xy, :] = [[-1, 0], [0, 1]]
sage: a.add_comp_by_continuation(e_uv, U.intersection(V), c_uv)
sage: b = M.automorphism_field(name='b')
sage: b[e_uv, :] = [[1, 0], [0, -2]]
sage: b.add_comp_by_continuation(e_xy, U.intersection(V), c_xy)
sage: w = M.vector_field(name='w')
sage: w[e_xy, :] = [3, 1]
sage: w.add_comp_by_continuation(e_uv, U.intersection(V), c_uv)
sage: s = a.__mul__(b); s # automorphism composition
Field of tangent-space automorphisms on the 2-dimensional differentiable manifold M
sage: s(w) == a(b(w)) # long time
True
sage: s = a.__mul__(w); s # tensor product
Tensor field of type (2,1) on the 2-dimensional differentiable manifold M
"""
if isinstance(other, AutomorphismField):
return self._mul_(other) # general linear group law
else:
return TensorField.__mul__(self, other) # tensor product
def __call__(self, *arg):
r"""
Redefinition of
:meth:`~sage.manifolds.differentiable.tensorfield.TensorField.__call__`
to allow for a proper treatment of the identity map and of the call
with a single argument
TESTS:
Field of identity maps on the 2-sphere::
sage: M = Manifold(2, 'M') # 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: e_xy = c_xy.frame(); e_uv = c_uv.frame()
sage: w = M.vector_field(name='w')
sage: w[e_xy, :] = [3, 1]
sage: w.add_comp_by_continuation(e_uv, U.intersection(V), c_uv)
sage: z = M.one_form(name='z')
sage: z[e_xy, :] = [-y, x]
sage: z.add_comp_by_continuation(e_uv, U.intersection(V), c_uv)
sage: Id = M.tangent_identity_field()
sage: s = Id(w); s
Vector field w on the 2-dimensional differentiable manifold M
sage: s == w
True
sage: s = Id(z, w); s
Scalar field z(w) on the 2-dimensional differentiable manifold M
sage: s == z(w)
True
Field of automorphisms on the 2-sphere::
sage: a = M.automorphism_field(name='a')
sage: a[e_xy, :] = [[-1, 0], [0, 1]]
sage: a.add_comp_by_continuation(e_uv, U.intersection(V), c_uv)
Call with a single argument::
sage: s = a(w); s
Vector field a(w) on the 2-dimensional differentiable manifold M
sage: s.display(e_xy)
a(w) = -3 d/dx + d/dy
sage: s.display(e_uv)
a(w) = (3*u^2 - 2*u*v - 3*v^2) d/du + (u^2 + 6*u*v - v^2) d/dv
sage: s.restrict(U) == a.restrict(U)(w.restrict(U))
True
sage: s.restrict(V) == a.restrict(V)(w.restrict(V))
True
sage: s.restrict(U) == a(w.restrict(U))
True
sage: s.restrict(U) == a.restrict(U)(w)
True
Call with two arguments::
sage: s = a(z, w); s
Scalar field a(z,w) on the 2-dimensional differentiable manifold M
sage: s.display()
a(z,w): M --> R
on U: (x, y) |--> x + 3*y
on V: (u, v) |--> (u + 3*v)/(u^2 + v^2)
sage: s.restrict(U) == a.restrict(U)(z.restrict(U), w.restrict(U))
True
sage: s.restrict(V) == a.restrict(V)(z.restrict(V), w.restrict(V))
True
sage: s.restrict(U) == a(z.restrict(U), w.restrict(U))
True
sage: s.restrict(U) == a(z, w.restrict(U))
True
"""
if self._is_identity:
if len(arg) == 1:
# The identity map acting as such, on a vector field:
vector = arg[0]
if vector._tensor_type != (1,0):
raise TypeError("the argument must be a vector field")
dom = self._domain.intersection(vector._domain)
return vector.restrict(dom)
elif len(arg) == 2:
# self acting as a type-(1,1) tensor on a pair
# (1-form, vector field), returning a scalar field:
oneform = arg[0]
vector = arg[1]
dom = self._domain.intersection(
oneform._domain).intersection(vector._domain)
return oneform.restrict(dom)(vector.restrict(dom))
else:
raise TypeError("wrong number of arguments")
# Generic case
if len(arg) == 1:
# The field of automorphisms acting on a vector field:
vector = arg[0]
if vector._tensor_type != (1,0):
raise TypeError("the argument must be a vector field")
dom = self._domain.intersection(vector._domain)
vector_dom = vector.restrict(dom)
if dom != self._domain:
return self.restrict(dom)(vector_dom)
resu = dom.vector_field()
if self._name is not None and vector._name is not None:
resu._name = self._name + "(" + vector._name + ")"
if self._latex_name is not None and vector._latex_name is not None:
resu._latex_name = self._latex_name + r"\left(" + \
vector._latex_name + r"\right)"
for sdom, automorph in self._restrictions.items():
resu._restrictions[sdom] = automorph(vector_dom.restrict(sdom))
return resu
# Case of 2 arguments:
return TensorField.__call__(self, *arg)
def __call__(self, *arg):
r"""
Redefinition of
:meth:`~sage.manifolds.differentiable.tensorfield.TensorField.__call__`
to allow for a proper treatment of the identity map and of the call
with a single argument
TESTS:
Field of identity maps on the 2-sphere::
sage: M = Manifold(2, 'M') # 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: e_xy = c_xy.frame(); e_uv = c_uv.frame()
sage: w = M.vector_field(name='w')
sage: w[e_xy, :] = [3, 1]
sage: w.add_comp_by_continuation(e_uv, U.intersection(V), c_uv)
sage: z = M.one_form(name='z')
sage: z[e_xy, :] = [-y, x]
sage: z.add_comp_by_continuation(e_uv, U.intersection(V), c_uv)
sage: Id = M.tangent_identity_field()
sage: s = Id(w); s
Vector field w on the 2-dimensional differentiable manifold M
sage: s == w
True
sage: s = Id(z, w); s
Scalar field z(w) on the 2-dimensional differentiable manifold M
sage: s == z(w)
True
Field of automorphisms on the 2-sphere::
sage: a = M.automorphism_field(name='a')
sage: a[e_xy, :] = [[-1, 0], [0, 1]]
sage: a.add_comp_by_continuation(e_uv, U.intersection(V), c_uv)
Call with a single argument::
sage: s = a(w); s
Vector field a(w) on the 2-dimensional differentiable manifold M
sage: s.display(e_xy)
a(w) = -3 d/dx + d/dy
sage: s.display(e_uv)
a(w) = (3*u^2 - 2*u*v - 3*v^2) d/du + (u^2 + 6*u*v - v^2) d/dv
sage: s.restrict(U) == a.restrict(U)(w.restrict(U))
True
sage: s.restrict(V) == a.restrict(V)(w.restrict(V))
True
sage: s.restrict(U) == a(w.restrict(U))
True
sage: s.restrict(U) == a.restrict(U)(w)
True
Call with two arguments::
sage: s = a(z, w); s
Scalar field a(z,w) on the 2-dimensional differentiable manifold M
sage: s.display()
a(z,w): M --> R
on U: (x, y) |--> x + 3*y
on V: (u, v) |--> (u + 3*v)/(u^2 + v^2)
sage: s.restrict(U) == a.restrict(U)(z.restrict(U), w.restrict(U))
True
sage: s.restrict(V) == a.restrict(V)(z.restrict(V), w.restrict(V))
True
sage: s.restrict(U) == a(z.restrict(U), w.restrict(U))
True
sage: s.restrict(U) == a(z, w.restrict(U))
True
"""
if self._is_identity:
if len(arg) == 1:
# The identity map acting as such, on a vector field:
vector = arg[0]
if vector._tensor_type != (1,0):
raise TypeError("the argument must be a vector field")
dom = self._domain.intersection(vector._domain)
return vector.restrict(dom)
elif len(arg) == 2:
# self acting as a type-(1,1) tensor on a pair
# (1-form, vector field), returning a scalar field:
oneform = arg[0]
vector = arg[1]
dom = self._domain.intersection(
oneform._domain).intersection(vector._domain)
return oneform.restrict(dom)(vector.restrict(dom))
else:
raise TypeError("wrong number of arguments")
# Generic case
if len(arg) == 1:
# The field of automorphisms acting on a vector field:
vector = arg[0]
if vector._tensor_type != (1,0):
raise TypeError("the argument must be a vector field")
dom = self._domain.intersection(vector._domain)
vector_dom = vector.restrict(dom)
if dom != self._domain:
return self.restrict(dom)(vector_dom)
resu = dom.vector_field()
if self._name is not None and vector._name is not None:
resu._name = self._name + "(" + vector._name + ")"
if self._latex_name is not None and vector._latex_name is not None:
resu._latex_name = self._latex_name + r"\left(" + \
vector._latex_name + r"\right)"
for sdom, automorph in self._restrictions.items():
resu._restrictions[sdom] = automorph(vector_dom.restrict(sdom))
return resu
# Case of 2 arguments:
return TensorField.__call__(self, *arg)
def __init__(self, vector_field_module, name=None, latex_name=None,
is_identity=False):
r"""
Construct a field of tangent-space automorphisms on a
non-parallelizable manifold.
TESTS:
Construction via ``parent.element_class``, and not via a direct call
to ``AutomorphismField``, to fit with the category framework::
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: XM = M.vector_field_module()
sage: GL = XM.general_linear_group()
sage: a = GL.element_class(XM, name='a'); a
Field of tangent-space automorphisms a on the 2-dimensional
differentiable manifold M
sage: a[c_xy.frame(), :] = [[1+x^2, 0], [0, 1+y^2]]
sage: a.add_comp_by_continuation(c_uv.frame(), U.intersection(V), c_uv)
sage: TestSuite(a).run(skip='_test_pickling')
Construction of the identity field::
sage: b = GL.element_class(XM, is_identity=True); b
Field of tangent-space identity maps on the 2-dimensional
differentiable manifold M
sage: TestSuite(b).run(skip='_test_pickling')
Construction with ``DifferentiableManifold.automorphism_field``::
sage: a1 = M.automorphism_field(name='a'); a1
Field of tangent-space automorphisms a on the 2-dimensional
differentiable manifold M
sage: type(a1) == type(a)
True
.. TODO::
Fix ``_test_pickling`` (in the superclass :class:`TensorField`).
"""
if is_identity:
if name is None:
name = 'Id'
if latex_name is None and name == 'Id':
latex_name = r'\mathrm{Id}'
TensorField.__init__(self, vector_field_module, (1,1), name=name,
latex_name=latex_name,
parent=vector_field_module.general_linear_group())
self._is_identity = is_identity
self._init_derived() # initialization of derived quantities
# Specific initializations for the field of identity maps:
if self._is_identity:
self._inverse = self
for dom in self._domain._subsets:
if dom.is_manifestly_parallelizable():
fmodule = dom.vector_field_module()
self._restrictions[dom] = fmodule.identity_map(name=name,
latex_name=latex_name)
def __init__(self, vector_field_module, name=None, latex_name=None,
is_identity=False):
r"""
Construct a field of tangent-space automorphisms on a
non-parallelizable manifold.
TESTS:
Construction via ``parent.element_class``, and not via a direct call
to ``AutomorphismField``, to fit with the category framework::
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: XM = M.vector_field_module()
sage: GL = XM.general_linear_group()
sage: a = GL.element_class(XM, name='a'); a
Field of tangent-space automorphisms a on the 2-dimensional
differentiable manifold M
sage: a[c_xy.frame(), :] = [[1+x^2, 0], [0, 1+y^2]]
sage: a.add_comp_by_continuation(c_uv.frame(), U.intersection(V), c_uv)
sage: TestSuite(a).run(skip='_test_pickling')
Construction of the identity field::
sage: b = GL.element_class(XM, is_identity=True); b
Field of tangent-space identity maps on the 2-dimensional
differentiable manifold M
sage: TestSuite(b).run(skip='_test_pickling')
Construction with ``DifferentiableManifold.automorphism_field``::
sage: a1 = M.automorphism_field(name='a'); a1
Field of tangent-space automorphisms a on the 2-dimensional
differentiable manifold M
sage: type(a1) == type(a)
True
.. TODO::
Fix ``_test_pickling`` (in the superclass :class:`TensorField`).
"""
if is_identity:
if name is None:
name = 'Id'
if latex_name is None and name == 'Id':
latex_name = r'\mathrm{Id}'
TensorField.__init__(self, vector_field_module, (1,1), name=name,
latex_name=latex_name,
parent=vector_field_module.general_linear_group())
self._is_identity = is_identity
self._init_derived() # initialization of derived quantities
# Specific initializations for the field of identity maps:
if self._is_identity:
self._inverse = self
for dom in self._domain._subsets:
if dom.is_manifestly_parallelizable():
fmodule = dom.vector_field_module()
self._restrictions[dom] = fmodule.identity_map(name=name,
latex_name=latex_name)
def restrict(self, subdomain, dest_map=None):
r"""
Return the restriction of ``self`` to some subdomain.
This is a redefinition of
:meth:`sage.manifolds.differentiable.tensorfield.TensorField.restrict`
to take into account the identity map.
INPUT:
- ``subdomain`` --
:class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`
open subset `V` of ``self._domain``
- ``dest_map`` -- (default: ``None``)
:class:`~sage.manifolds.differentiable.diff_map.DiffMap`;
destination map `\Phi:\ V \rightarrow N`, where `N` is a
subdomain of ``self._codomain``; if ``None``, the restriction
of ``self.base_module().destination_map()`` to `V` is used
OUTPUT:
- a :class:`AutomorphismField` representing the restriction
EXAMPLES:
Restrictions of an automorphism field on the 2-sphere::
sage: M = Manifold(2, 'S^2', start_index=1)
sage: U = M.open_subset('U') # the complement of the North pole
sage: stereoN.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: eN = stereoN.frame() # the associated vector frame
sage: V = M.open_subset('V') # the complement of the South pole
sage: stereoS.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: eS = stereoS.frame() # the associated vector frame
sage: transf = stereoN.transition_map(stereoS, (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: inv = transf.inverse() # transformation from stereoS to stereoN
sage: W = U.intersection(V) # the complement of the North and South poles
sage: stereoN_W = W.atlas()[0] # restriction of stereographic coord. from North pole to W
sage: stereoS_W = W.atlas()[1] # restriction of stereographic coord. from South pole to W
sage: eN_W = stereoN_W.frame() ; eS_W = stereoS_W.frame()
sage: a = M.automorphism_field(name='a') ; a
Field of tangent-space automorphisms a on the 2-dimensional
differentiable manifold S^2
sage: a[eN,:] = [[1, atan(x^2+y^2)], [0,3]]
sage: a.add_comp_by_continuation(eS, W, chart=stereoS)
sage: a.restrict(U)
Field of tangent-space automorphisms a on the Open subset U of the
2-dimensional differentiable manifold S^2
sage: a.restrict(U)[eN,:]
[ 1 arctan(x^2 + y^2)]
[ 0 3]
sage: a.restrict(V)
Field of tangent-space automorphisms a on the Open subset V of the
2-dimensional differentiable manifold S^2
sage: a.restrict(V)[eS,:]
[ (u^4 + 10*u^2*v^2 + v^4 + 2*(u^3*v - u*v^3)*arctan(1/(u^2 + v^2)))/(u^4 + 2*u^2*v^2 + v^4) -(4*u^3*v - 4*u*v^3 + (u^4 - 2*u^2*v^2 + v^4)*arctan(1/(u^2 + v^2)))/(u^4 + 2*u^2*v^2 + v^4)]
[ 4*(u^2*v^2*arctan(1/(u^2 + v^2)) - u^3*v + u*v^3)/(u^4 + 2*u^2*v^2 + v^4) (3*u^4 - 2*u^2*v^2 + 3*v^4 - 2*(u^3*v - u*v^3)*arctan(1/(u^2 + v^2)))/(u^4 + 2*u^2*v^2 + v^4)]
sage: a.restrict(W)
Field of tangent-space automorphisms a on the Open subset W of the
2-dimensional differentiable manifold S^2
sage: a.restrict(W)[eN_W,:]
[ 1 arctan(x^2 + y^2)]
[ 0 3]
Restrictions 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
differentiable manifold S^2
sage: id.restrict(U)
Field of tangent-space identity maps on the Open subset U of the
2-dimensional differentiable manifold S^2
sage: id.restrict(U)[eN,:]
[1 0]
[0 1]
sage: id.restrict(V)
Field of tangent-space identity maps on the Open subset V of the
2-dimensional differentiable manifold S^2
sage: id.restrict(V)[eS,:]
[1 0]
[0 1]
sage: id.restrict(W)[eN_W,:]
[1 0]
[0 1]
sage: id.restrict(W)[eS_W,:]
[1 0]
[0 1]
"""
if subdomain == self._domain:
return self
if subdomain not in self._restrictions:
if not self._is_identity:
return TensorField.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._vmodule._dest_map.restrict(subdomain)
elif not dest_map._codomain.is_subset(self._ambient_domain):
raise ValueError("the argument 'dest_map' is not compatible " +
"with the ambient domain of " +
"the {}".format(self))
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 subdomain.
This is a redefinition of
:meth:`sage.manifolds.differentiable.tensorfield.TensorField.restrict`
to take into account the identity map.
INPUT:
- ``subdomain`` --
:class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`
open subset `V` of ``self._domain``
- ``dest_map`` -- (default: ``None``)
:class:`~sage.manifolds.differentiable.diff_map.DiffMap`;
destination map `\Phi:\ V \rightarrow N`, where `N` is a
subdomain of ``self._codomain``; if ``None``, the restriction
of ``self.base_module().destination_map()`` to `V` is used
OUTPUT:
- a :class:`AutomorphismField` representing the restriction
EXAMPLES:
Restrictions of an automorphism field on the 2-sphere::
sage: M = Manifold(2, 'S^2', start_index=1)
sage: U = M.open_subset('U') # the complement of the North pole
sage: stereoN.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: eN = stereoN.frame() # the associated vector frame
sage: V = M.open_subset('V') # the complement of the South pole
sage: stereoS.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: eS = stereoS.frame() # the associated vector frame
sage: transf = stereoN.transition_map(stereoS, (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: inv = transf.inverse() # transformation from stereoS to stereoN
sage: W = U.intersection(V) # the complement of the North and South poles
sage: stereoN_W = W.atlas()[0] # restriction of stereographic coord. from North pole to W
sage: stereoS_W = W.atlas()[1] # restriction of stereographic coord. from South pole to W
sage: eN_W = stereoN_W.frame() ; eS_W = stereoS_W.frame()
sage: a = M.automorphism_field(name='a') ; a
Field of tangent-space automorphisms a on the 2-dimensional
differentiable manifold S^2
sage: a[eN,:] = [[1, atan(x^2+y^2)], [0,3]]
sage: a.add_comp_by_continuation(eS, W, chart=stereoS)
sage: a.restrict(U)
Field of tangent-space automorphisms a on the Open subset U of the
2-dimensional differentiable manifold S^2
sage: a.restrict(U)[eN,:]
[ 1 arctan(x^2 + y^2)]
[ 0 3]
sage: a.restrict(V)
Field of tangent-space automorphisms a on the Open subset V of the
2-dimensional differentiable manifold S^2
sage: a.restrict(V)[eS,:]
[ (u^4 + 10*u^2*v^2 + v^4 + 2*(u^3*v - u*v^3)*arctan(1/(u^2 + v^2)))/(u^4 + 2*u^2*v^2 + v^4) -(4*u^3*v - 4*u*v^3 + (u^4 - 2*u^2*v^2 + v^4)*arctan(1/(u^2 + v^2)))/(u^4 + 2*u^2*v^2 + v^4)]
[ 4*(u^2*v^2*arctan(1/(u^2 + v^2)) - u^3*v + u*v^3)/(u^4 + 2*u^2*v^2 + v^4) (3*u^4 - 2*u^2*v^2 + 3*v^4 - 2*(u^3*v - u*v^3)*arctan(1/(u^2 + v^2)))/(u^4 + 2*u^2*v^2 + v^4)]
sage: a.restrict(W)
Field of tangent-space automorphisms a on the Open subset W of the
2-dimensional differentiable manifold S^2
sage: a.restrict(W)[eN_W,:]
[ 1 arctan(x^2 + y^2)]
[ 0 3]
Restrictions 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
differentiable manifold S^2
sage: id.restrict(U)
Field of tangent-space identity maps on the Open subset U of the
2-dimensional differentiable manifold S^2
sage: id.restrict(U)[eN,:]
[1 0]
[0 1]
sage: id.restrict(V)
Field of tangent-space identity maps on the Open subset V of the
2-dimensional differentiable manifold S^2
sage: id.restrict(V)[eS,:]
[1 0]
[0 1]
sage: id.restrict(W)[eN_W,:]
[1 0]
[0 1]
sage: id.restrict(W)[eS_W,:]
[1 0]
[0 1]
"""
if subdomain == self._domain:
return self
if subdomain not in self._restrictions:
if not self._is_identity:
return TensorField.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._vmodule._dest_map.restrict(subdomain)
elif not dest_map._codomain.is_subset(self._ambient_domain):
raise ValueError("the argument 'dest_map' is not compatible " +
"with the ambient domain of " +
"the {}".format(self))
smodule = subdomain.vector_field_module(dest_map=dest_map)
self._restrictions[subdomain] = smodule.identity_map()
return self._restrictions[subdomain]
