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 __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)
