def _new_comp(self, frame):
r"""
Create some components in the given frame.
"""
from component import CompFullySym
return CompFullySym(frame, 2)
def pushforward(self, tensor):
r"""
Pushforward operator associated with the embedding.
INPUT:
- ``tensor`` -- instance of :class:`TensorField` representing a fully
contravariant tensor field `T` on the submanifold, i.e. a tensor
field of type (k,0), with k a positive integer. The case k=0
corresponds to a scalar field.
OUTPUT:
- instance of :class:`TensorField` representing a field of fully
contravariant tensors of the ambient manifold, field defined on
the domain occupied by the submanifold.
EXAMPLES:
Pushforward of a vector field defined on `S^2`, submanifold of `\RR^3`::
sage: M = Manifold(3, 'R^3', r'\RR^3', start_index=1)
sage: c_cart.<x,y,z> = M.chart('x y z') # Cartesian coordinates on R^3
sage: S = Submanifold(M, 2, 'S^2', start_index=1)
sage: U = S.open_domain('U') # U = S minus two poles
sage: c_spher.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi') # spherical coordinates on U
sage: S.def_embedding( DiffMapping(S, M, [sin(th)*cos(ph), sin(th)*sin(ph), cos(th)], name='i', latex_name=r'\iota') )
sage: v = VectorField(S, 'v')
sage: v[2] = 1 ; v.view() # azimuthal vector field on S^2
v = d/dph
sage: iv = S.pushforward(v) ; iv
vector field 'i_*(v)' on the domain 'S^2' on the 3-dimensional manifold 'R^3'
sage: iv.view(c_cart.frame, c_spher) # the pushforward expanded on the Cartesian frame, with components expressed in terms of (th,ph) coordinates
i_*(v) = -sin(ph)*sin(th) d/dx + cos(ph)*sin(th) d/dy
The components of the pushforward vector are scalar fields on the submanifold::
sage: iv.comp(c_cart.frame)[[1]]
scalar field on the open domain 'U' on the 2-dimensional submanifold 'S^2' of the 3-dimensional manifold 'R^3'
Pushforward of a tangent vector to a helix, submanifold of `\RR^3`::
sage: H = Submanifold(M, 1, 'H')
sage: c_t.<t> = H.chart('t')
sage: H.def_embedding( DiffMapping(H, M, [cos(t), sin(t), t], name='iH') )
sage: u = VectorField(H, 'u')
sage: u[0] = 1 ; u.view() # tangent vector to the helix
u = d/dt
sage: iu = H.pushforward(u) ; iu
vector field 'iH_*(u)' on the domain 'H' on the 3-dimensional manifold 'R^3'
sage: iu.view(c_cart.frame, c_t)
iH_*(u) = -sin(t) d/dx + cos(t) d/dy + d/dz
"""
from component import Components, CompWithSym, CompFullySym, \
CompFullyAntiSym
if tensor.manifold != self:
raise TypeError("The tensor field is not defined on the " +
"submanifold.")
(ncon, ncov) = tensor.tensor_type
if ncov != 0:
raise TypeError("The pushforward cannot be taken on a tensor " +
"with some covariant part.")
dom1 = tensor.domain
embed = self.embedding
resu_name = None
resu_latex_name = None
if embed.name is not None and tensor.name is not None:
resu_name = embed.name + '_*(' + tensor.name + ')'
if embed.latex_name is not None and tensor.latex_name is not None:
resu_latex_name = embed.latex_name + '_*' + tensor.latex_name
if ncon == 0:
raise NotImplementedError("The case of a scalar field is not " +
"implemented yet.")
# A pair of charts (chart1, chart2) where the computation
# is feasable is searched, privileging the default chart of the
# start domain for chart1
chart1 = None
chart2 = None
def_chart1 = dom1.def_chart
def_chart2 = self.ambient_manifold.def_chart
if def_chart1.frame in tensor.components and \
(def_chart1, def_chart2) in embed.coord_expression:
chart1 = def_chart1
chart2 = def_chart2
else:
for (chart1n, chart2n) in embed.coord_expression:
if chart2n == def_chart2 and \
chart1n.frame in tensor.components:
chart1 = chart1n
chart2 = def_chart2
break
if chart2 is None:
# It is not possible to have def_chart2 as chart for
# expressing the result; any other chart is then looked for:
for (chart1n, chart2n) in self.coord_expression:
if chart1n.frame in tensor.components:
chart1 = chart1n
chart2 = chart2n
break
if chart1 is None:
raise ValueError("No common chart could be find to compute " +
"the pushforward of the tensor field.")
frame1 = chart1.frame
frame2 = chart2.frame
# Computation at the component level:
tcomp = tensor.components[frame1]
# Construction of the pushforward components (ptcomp):
if isinstance(tcomp, CompFullySym):
ptcomp = CompFullySym(frame2, ncon)
elif isinstance(tcomp, CompFullyAntiSym):
ptcomp = CompFullyAntiSym(frame2, ncon)
elif isinstance(tcomp, CompWithSym):
ptcomp = CompWithSym(frame2,
ncon,
sym=tcomp.sym,
antisym=tcomp.antisym)
else:
ptcomp = Components(frame2, ncon)
phi = embed.coord_expression[(chart1, chart2)]
jacob = phi.jacobian()
# X2 coordinates expressed in terms of X1 ones via the mapping:
# coord2_1 = phi(*(chart1.xx))
si1 = self.sindex
si2 = self.ambient_manifold.sindex
for ind_new in ptcomp.non_redundant_index_generator():
res = 0
for ind_old in self.index_generator(ncon):
t = tcomp[[ind_old]].function_chart(chart1)
for i in range(ncon):
t *= jacob[ind_new[i] - si2][ind_old[i] - si1]
res += t
ptcomp[ind_new] = res
resu = ptcomp.tensor_field((ncon, 0),
name=resu_name,
latex_name=resu_latex_name)
resu.domain = self.domain_amb # the domain is resctrited to the submanifold
return resu
def pushforward(self, tensor):
r"""
Pushforward operator associated with the embedding.
INPUT:
- ``tensor`` -- instance of :class:`TensorField` representing a fully
contravariant tensor field `T` on the submanifold, i.e. a tensor
field of type (k,0), with k a positive integer. The case k=0
corresponds to a scalar field.
OUTPUT:
- instance of :class:`TensorField` representing a field of fully
contravariant tensors of the ambient manifold, field defined on
the domain occupied by the submanifold.
EXAMPLES:
Pushforward of a vector field defined on `S^2`, submanifold of `\RR^3`::
sage: M = Manifold(3, 'R^3', r'\RR^3', start_index=1)
sage: c_cart.<x,y,z> = M.chart('x y z') # Cartesian coordinates on R^3
sage: S = Submanifold(M, 2, 'S^2', start_index=1)
sage: U = S.open_domain('U') # U = S minus two poles
sage: c_spher.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi') # spherical coordinates on U
sage: S.def_embedding( DiffMapping(S, M, [sin(th)*cos(ph), sin(th)*sin(ph), cos(th)], name='i', latex_name=r'\iota') )
sage: v = VectorField(S, 'v')
sage: v[2] = 1 ; v.view() # azimuthal vector field on S^2
v = d/dph
sage: iv = S.pushforward(v) ; iv
vector field 'i_*(v)' on the domain 'S^2' on the 3-dimensional manifold 'R^3'
sage: iv.view(c_cart.frame, c_spher) # the pushforward expanded on the Cartesian frame, with components expressed in terms of (th,ph) coordinates
i_*(v) = -sin(ph)*sin(th) d/dx + cos(ph)*sin(th) d/dy
The components of the pushforward vector are scalar fields on the submanifold::
sage: iv.comp(c_cart.frame)[[1]]
scalar field on the open domain 'U' on the 2-dimensional submanifold 'S^2' of the 3-dimensional manifold 'R^3'
Pushforward of a tangent vector to a helix, submanifold of `\RR^3`::
sage: H = Submanifold(M, 1, 'H')
sage: c_t.<t> = H.chart('t')
sage: H.def_embedding( DiffMapping(H, M, [cos(t), sin(t), t], name='iH') )
sage: u = VectorField(H, 'u')
sage: u[0] = 1 ; u.view() # tangent vector to the helix
u = d/dt
sage: iu = H.pushforward(u) ; iu
vector field 'iH_*(u)' on the domain 'H' on the 3-dimensional manifold 'R^3'
sage: iu.view(c_cart.frame, c_t)
iH_*(u) = -sin(t) d/dx + cos(t) d/dy + d/dz
"""
from component import Components, CompWithSym, CompFullySym, \
CompFullyAntiSym
if tensor.manifold != self:
raise TypeError("The tensor field is not defined on the " +
"submanifold.")
(ncon, ncov) = tensor.tensor_type
if ncov != 0:
raise TypeError("The pushforward cannot be taken on a tensor " +
"with some covariant part.")
dom1 = tensor.domain
embed = self.embedding
resu_name = None ; resu_latex_name = None
if embed.name is not None and tensor.name is not None:
resu_name = embed.name + '_*(' + tensor.name + ')'
if embed.latex_name is not None and tensor.latex_name is not None:
resu_latex_name = embed.latex_name + '_*' + tensor.latex_name
if ncon == 0:
raise NotImplementedError("The case of a scalar field is not " +
"implemented yet.")
# A pair of charts (chart1, chart2) where the computation
# is feasable is searched, privileging the default chart of the
# start domain for chart1
chart1 = None; chart2 = None
def_chart1 = dom1.def_chart
def_chart2 = self.ambient_manifold.def_chart
if def_chart1.frame in tensor.components and \
(def_chart1, def_chart2) in embed.coord_expression:
chart1 = def_chart1
chart2 = def_chart2
else:
for (chart1n, chart2n) in embed.coord_expression:
if chart2n == def_chart2 and \
chart1n.frame in tensor.components:
chart1 = chart1n
chart2 = def_chart2
break
if chart2 is None:
# It is not possible to have def_chart2 as chart for
# expressing the result; any other chart is then looked for:
for (chart1n, chart2n) in self.coord_expression:
if chart1n.frame in tensor.components:
chart1 = chart1n
chart2 = chart2n
break
if chart1 is None:
raise ValueError("No common chart could be find to compute " +
"the pushforward of the tensor field.")
frame1 = chart1.frame
frame2 = chart2.frame
# Computation at the component level:
tcomp = tensor.components[frame1]
# Construction of the pushforward components (ptcomp):
if isinstance(tcomp, CompFullySym):
ptcomp = CompFullySym(frame2, ncon)
elif isinstance(tcomp, CompFullyAntiSym):
ptcomp = CompFullyAntiSym(frame2, ncon)
elif isinstance(tcomp, CompWithSym):
ptcomp = CompWithSym(frame2, ncon, sym=tcomp.sym,
antisym=tcomp.antisym)
else:
ptcomp = Components(frame2, ncon)
phi = embed.coord_expression[(chart1, chart2)]
jacob = phi.jacobian()
# X2 coordinates expressed in terms of X1 ones via the mapping:
# coord2_1 = phi(*(chart1.xx))
si1 = self.sindex
si2 = self.ambient_manifold.sindex
for ind_new in ptcomp.non_redundant_index_generator():
res = 0
for ind_old in self.index_generator(ncon):
t = tcomp[[ind_old]].function_chart(chart1)
for i in range(ncon):
t *= jacob[ind_new[i]-si2][ind_old[i]-si1]
res += t
ptcomp[ind_new] = res
resu = ptcomp.tensor_field((ncon, 0), name=resu_name,
latex_name=resu_latex_name)
resu.domain = self.domain_amb # the domain is resctrited to the submanifold
return resu
def pullback(self, tensor):
r"""
Pullback operator associated with the differentiable mapping.
INPUT:
- ``tensor`` -- instance of :class:`TensorField` representing a fully
covariant tensor field `T` on the *arrival* domain, i.e. a tensor
field of type (0,p), with p a positive or zero integer. The case p=0
corresponds to a scalar field.
OUTPUT:
- instance of :class:`TensorField` representing a fully
covariant tensor field on the *start* domain that is the
pullback of `T` given by ``self``.
EXAMPLES:
Pullback on `S^2` of a scalar field defined on `R^3`::
sage: m = Manifold(2, 'S^2', start_index=1)
sage: c_spher.<th,ph> = m.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi') # spherical coord. on S^2
sage: n = Manifold(3, 'R^3', r'\RR^3', start_index=1)
sage: c_cart.<x,y,z> = n.chart('x y z') # Cartesian coord. on R^3
sage: Phi = DiffMapping(m, n, (sin(th)*cos(ph), sin(th)*sin(ph), cos(th)), name='Phi', latex_name=r'\Phi')
sage: f = ScalarField(n, x*y*z, name='f') ; f
scalar field 'f' on the 3-dimensional manifold 'R^3'
sage: f.view()
f: (x, y, z) |--> x*y*z
sage: pf = Phi.pullback(f) ; pf
scalar field 'Phi_*(f)' on the 2-dimensional manifold 'S^2'
sage: pf.view()
Phi_*(f): (th, ph) |--> cos(ph)*cos(th)*sin(ph)*sin(th)^2
Pullback on `S^2` of the standard Euclidean metric on `R^3`::
sage: g = SymBilinFormField(n, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, 1, 1
sage: g.view()
g = dx*dx + dy*dy + dz*dz
sage: pg = Phi.pullback(g) ; pg
field of symmetric bilinear forms 'Phi_*(g)' on the 2-dimensional manifold 'S^2'
sage: pg.view()
Phi_*(g) = dth*dth + sin(th)^2 dph*dph
Pullback on `S^2` of a 3-form on `R^3`::
sage: a = DiffForm(n, 3, 'A')
sage: a[1,2,3] = f
sage: a.view()
A = x*y*z dx/\dy/\dz
sage: pa = Phi.pullback(a) ; pa
3-form 'Phi_*(A)' on the 2-dimensional manifold 'S^2'
sage: pa.view() # should be zero (as any 3-form on a 2-dimensional manifold)
Phi_*(A) = 0
"""
from scalarfield import ScalarField
from vectorframe import CoordFrame
from rank2field import SymBilinFormField
from diffform import DiffForm, OneForm
if not isinstance(tensor, TensorField):
raise TypeError("The argument 'tensor' must be a tensor field.")
dom1 = self.domain1
dom2 = self.domain2
if not tensor.domain.is_subdomain(dom2):
raise TypeError("The tensor field is not defined on the mapping " +
"arrival domain.")
(ncon, ncov) = tensor.tensor_type
if ncon != 0:
raise TypeError("The pullback cannot be taken on a tensor " +
"with some contravariant part.")
resu_name = None ; resu_latex_name = None
if self.name is not None and tensor.name is not None:
resu_name = self.name + '_*(' + tensor.name + ')'
if self.latex_name is not None and tensor.latex_name is not None:
resu_latex_name = self.latex_name + '_*' + tensor.latex_name
if ncov == 0:
# Case of a scalar field
# ----------------------
resu = ScalarField(dom1, name=resu_name,
latex_name=resu_latex_name)
for chart2 in tensor.express:
for chart1 in dom1.atlas:
if (chart1, chart2) in self.coord_expression:
phi = self.coord_expression[(chart1, chart2)]
coord1 = chart1.xx
ff = tensor.express[chart2]
resu.add_expr( ff(*(phi(*coord1))), chart1)
return resu
else:
# Case of tensor field of rank >= 1
# ---------------------------------
if isinstance(tensor, OneForm):
resu = OneForm(dom1, name=resu_name,
latex_name=resu_latex_name)
elif isinstance(tensor, DiffForm):
resu = DiffForm(dom1, ncov, name=resu_name,
latex_name=resu_latex_name)
elif isinstance(tensor, SymBilinFormField):
resu = SymBilinFormField(dom1, name=resu_name,
latex_name=resu_latex_name)
else:
resu = TensorField(dom1, 0, ncov, name=resu_name,
latex_name=resu_latex_name, sym=tensor.sym,
antisym=tensor.antisym)
for frame2 in tensor.components:
if isinstance(frame2, CoordFrame):
chart2 = frame2.chart
for chart1 in dom1.atlas:
if (chart1, chart2) in self.coord_expression:
# Computation at the component level:
frame1 = chart1.frame
tcomp = tensor.components[frame2]
if isinstance(tcomp, CompFullySym):
ptcomp = CompFullySym(frame1, ncov)
elif isinstance(tcomp, CompFullyAntiSym):
ptcomp = CompFullyAntiSym(frame1, ncov)
elif isinstance(tcomp, CompWithSym):
ptcomp = CompWithSym(frame1, ncov, sym=tcomp.sym,
antisym=tcomp.antisym)
else:
ptcomp = Components(frame1, ncov)
phi = self.coord_expression[(chart1, chart2)]
jacob = phi.jacobian()
# X2 coordinates expressed in terms of X1 ones via the mapping:
coord2_1 = phi(*(chart1.xx))
si1 = dom1.manifold.sindex
si2 = dom2.manifold.sindex
for ind_new in ptcomp.non_redundant_index_generator():
res = 0
for ind_old in dom2.manifold.index_generator(ncov):
ff = tcomp[[ind_old]].function_chart(chart2)
t = FunctionChart(chart1, ff(*coord2_1))
for i in range(ncov):
t *= jacob[ind_old[i]-si2][ind_new[i]-si1]
res += t
ptcomp[ind_new] = res
resu.components[frame1] = ptcomp
return resu
def inverse(self):
r"""
Return the inverse metric.
EXAMPLES:
Inverse metric on a 2-dimensional manifold::
sage: m = Manifold(2, 'M', start_index=1)
sage: c_xy = Chart(m, 'x y', 'xy')
sage: g = Metric(m, 'g')
sage: g[1,1], g[1,2], g[2,2] = 1+x, x*y, 1-x
sage: g[:] # components in the manifold's default frame
[ x + 1 x*y]
[ x*y -x + 1]
sage: ig = g.inverse() ; ig
tensor field 'inv_g' of type (2,0) on the 2-dimensional manifold 'M'
sage: ig[:]
[ (x - 1)/(x^2*y^2 + x^2 - 1) x*y/(x^2*y^2 + x^2 - 1)]
[ x*y/(x^2*y^2 + x^2 - 1) -(x + 1)/(x^2*y^2 + x^2 - 1)]
If the metric is modified, the inverse metric is automatically updated::
sage: g[1,2] = 0 ; g[:]
[ x + 1 0]
[ 0 -x + 1]
sage: g.inverse()[:]
[ 1/(x + 1) 0]
[ 0 -1/(x - 1)]
"""
from sage.matrix.constructor import matrix
from component import CompFullySym
from utilities import simplify_chain
# Is the inverse metric up to date ?
for frame_name in self.components:
if frame_name not in self._inverse.components:
# the computation is necessary
manif = self.manifold
if frame_name[-2:] == '_b':
# coordinate basis
chart_name = frame_name[:-2]
else:
chart_name = manif.def_chart.name
si = manif.sindex
nsi = manif.dim + si
try:
gmat = matrix([[
self.comp(frame_name)[i, j, chart_name].express
for j in range(si, nsi)
] for i in range(si, nsi)])
except KeyError:
continue
gmat_inv = gmat.inverse()
cinv = CompFullySym(manif, 2, frame_name)
for i in range(si, nsi):
for j in range(i, nsi): # symmetry taken into account
cinv[i, j,
chart_name] = simplify_chain(gmat_inv[i - si,
j - si])
self._inverse.components[frame_name] = cinv
return self._inverse
def pullback(self, tensor):
r"""
Pullback operator associated with the differentiable mapping.
INPUT:
- ``tensor`` -- instance of :class:`TensorField` representing a fully
covariant tensor field `T` on the *arrival* manifold, i.e. a tensor
field of type (0,p), with p a positive or zero integer. The case p=0
corresponds to a scalar field.
OUTPUT:
- instance of :class:`TensorField` representing a fully
covariant tensor field on the *start* manifold that is the
pullback of `T` given by ``self``.
EXAMPLES:
Pullback on `S^2` of a scalar field defined on `R^3`::
sage: m = Manifold(2, 'S^2', start_index=1)
sage: c_spher = Chart(m, r'th:[0,pi]:\theta ph:[0,2*pi):\phi', 'spher') # spherical coord. on S^2
sage: n = Manifold(3, 'R^3', r'\RR^3', start_index=1)
sage: c_cart = Chart(n, 'x y z', 'cart') # Cartesian coord. on R^3
sage: Phi = DiffMapping(m, n, (sin(th)*cos(ph), sin(th)*sin(ph), cos(th)), name='Phi', latex_name=r'\Phi')
sage: f = ScalarField(n, x*y*z, name='f') ; f
scalar field 'f' on the 3-dimensional manifold 'R^3'
sage: f.show()
f: (x, y, z) |--> x*y*z
sage: pf = Phi.pullback(f) ; pf
scalar field 'Phi_*(f)' on the 2-dimensional manifold 'S^2'
sage: pf.show()
Phi_*(f): (th, ph) |--> cos(ph)*cos(th)*sin(ph)*sin(th)^2
Pullback on `S^2` of the standard Euclidean metric on `R^3`::
sage: g = SymBilinFormField(n, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, 1, 1
sage: g.show()
g = dx*dx + dy*dy + dz*dz
sage: pg = Phi.pullback(g) ; pg
field of symmetric bilinear forms 'Phi_*(g)' on the 2-dimensional manifold 'S^2'
sage: pg.show()
Phi_*(g) = dth*dth + sin(th)^2 dph*dph
Pullback on `S^2` of a 3-form on `R^3`::
sage: a = DiffForm(n, 3, 'A')
sage: a[1,2,3] = f
sage: a.show()
A = x*y*z dx/\dy/\dz
sage: pa = Phi.pullback(a) ; pa
3-form 'Phi_*(A)' on the 2-dimensional manifold 'S^2'
sage: pa.show() # should be zero (as any 3-form on a 2-dimensional manifold)
Phi_*(A) = 0
"""
from scalarfield import ScalarField
if not isinstance(tensor, TensorField):
raise TypeError("The argument 'tensor' must be a tensor field.")
manif1 = self.manifold1
manif2 = self.manifold2
if tensor.manifold != manif2:
raise TypeError("The tensor field is not defined on the mapping " +
"arrival manifold.")
(ncon, ncov) = tensor.tensor_type
if ncon != 0:
raise TypeError("The pullback cannot be taken on a tensor " +
"with some contravariant part.")
resu_name = None
resu_latex_name = None
if self.name is not None and tensor.name is not None:
resu_name = self.name + '_*(' + tensor.name + ')'
if self.latex_name is not None and tensor.latex_name is not None:
resu_latex_name = self.latex_name + '_*' + tensor.latex_name
chart1_name = None
chart2_name = None
def_chart1 = manif1.def_chart
def_chart1_name = def_chart1.name
def_chart2 = manif2.def_chart
def_chart2_name = def_chart2.name
if ncov == 0:
# Case of a scalar field
# ----------------------
# A pair of chart (chart1_name, chart2_name) where the computation
# is feasable is searched, privileging the default chart of the
# start manifold for chart1_name
if def_chart2_name in tensor.express and \
(def_chart1_name, def_chart2_name) in self.coord_expression:
chart1_name = def_chart1_name
chart2_name = def_chart2_name
else:
for (chart1n, chart2n) in self.coord_expression:
if chart1n == def_chart1_name and chart2n in tensor.express:
chart1_name = def_chart1_name
chart2_name = chart2n
break
if chart1_name is None:
# It is not possible to have def_chart1 as chart for
# expressing the result; any other chart is then looked for:
for (chart1n, chart2n) in self.coord_expression:
if chart2n in tensor.express:
chart1_name = chart1n
chart2_name = chart2n
break
if chart1_name is None:
raise ValueError("No common chart could be find to compute " +
"the pullback of the scalar field.")
phi = self.coord_expression[(chart1_name, chart2_name)]
coord1 = manif1.atlas[chart1_name].xx
ff = tensor.express[chart2_name]
return ScalarField(manif1, ff(*(phi(*coord1))), chart1_name,
resu_name, resu_latex_name)
else:
# Case of tensor field of rank >= 1
# ---------------------------------
# A pair of chart (chart1_name, chart2_name) where the computation
# is feasable is searched, privileging the default chart of the
# start manifold for chart1_name
if def_chart2.frame.name in tensor.components and \
(def_chart1_name, def_chart2_name) in self.coord_expression:
chart1_name = def_chart1_name
chart2_name = def_chart2_name
else:
for (chart1n, chart2n) in self.coord_expression:
if chart1n == def_chart1_name and \
manif2.atlas[chart2n].frame.name in tensor.components:
chart1_name = def_chart1_name
chart2_name = chart2n
break
if chart1_name is None:
# It is not possible to have def_chart1 as chart for
# expressing the result; any other chart is then looked for:
for (chart1n, chart2n) in self.coord_expression:
if manif2.atlas[chart2n].frame.name in tensor.components:
chart1_name = chart1n
chart2_name = chart2n
break
if chart1_name is None:
raise ValueError("No common chart could be find to compute " +
"the pullback of the tensor field.")
chart1 = manif1.atlas[chart1_name]
chart2 = manif2.atlas[chart2_name]
frame1_name = chart1.frame.name
frame2_name = chart2.frame.name
# Computation at the component level:
tcomp = tensor.components[frame2_name]
if isinstance(tcomp, CompFullySym):
ptcomp = CompFullySym(manif1, ncov, frame1_name)
elif isinstance(tcomp, CompFullyAntiSym):
ptcomp = CompFullyAntiSym(manif1, ncov, frame1_name)
elif isinstance(tcomp, CompWithSym):
ptcomp = CompWithSym(manif1,
ncov,
frame1_name,
sym=tcomp.sym,
antisym=tcomp.antisym)
else:
ptcomp = Components(manif1, ncov, frame1_name)
phi = self.coord_expression[(chart1_name, chart2_name)]
jacob = phi.jacobian()
# X2 coordinates expressed in terms of X1 ones via the mapping:
coord2_1 = phi(*(chart1.xx))
si1 = manif1.sindex
si2 = manif2.sindex
for ind_new in ptcomp.non_redundant_index_generator():
res = 0
for ind_old in manif2.index_generator(ncov):
ff = tcomp[[ind_old]].function_chart(chart2_name)
t = FunctionChart(chart1, ff(*coord2_1))
for i in range(ncov):
t *= jacob[ind_old[i] - si2][ind_new[i] - si1]
res += t
ptcomp[ind_new] = res
resu = tensor_field_from_comp(ptcomp, (0, ncov))
resu.set_name(resu_name, resu_latex_name)
return resu
def pullback(self, tensor):
r"""
Pullback operator associated with the differentiable mapping.
INPUT:
- ``tensor`` -- instance of :class:`TensorField` representing a fully
covariant tensor field `T` on the *arrival* manifold, i.e. a tensor
field of type (0,p), with p a positive or zero integer. The case p=0
corresponds to a scalar field.
OUTPUT:
- instance of :class:`TensorField` representing a fully
covariant tensor field on the *start* manifold that is the
pullback of `T` given by ``self``.
EXAMPLES:
Pullback on `S^2` of a scalar field defined on `R^3`::
sage: m = Manifold(2, 'S^2', start_index=1)
sage: c_spher = Chart(m, r'th:[0,pi]:\theta ph:[0,2*pi):\phi', 'spher') # spherical coord. on S^2
sage: n = Manifold(3, 'R^3', r'\RR^3', start_index=1)
sage: c_cart = Chart(n, 'x y z', 'cart') # Cartesian coord. on R^3
sage: Phi = DiffMapping(m, n, (sin(th)*cos(ph), sin(th)*sin(ph), cos(th)), name='Phi', latex_name=r'\Phi')
sage: f = ScalarField(n, x*y*z, name='f') ; f
scalar field 'f' on the 3-dimensional manifold 'R^3'
sage: f.show()
f: (x, y, z) |--> x*y*z
sage: pf = Phi.pullback(f) ; pf
scalar field 'Phi_*(f)' on the 2-dimensional manifold 'S^2'
sage: pf.show()
Phi_*(f): (th, ph) |--> cos(ph)*cos(th)*sin(ph)*sin(th)^2
Pullback on `S^2` of the standard Euclidean metric on `R^3`::
sage: g = SymBilinFormField(n, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, 1, 1
sage: g.show()
g = dx*dx + dy*dy + dz*dz
sage: pg = Phi.pullback(g) ; pg
field of symmetric bilinear forms 'Phi_*(g)' on the 2-dimensional manifold 'S^2'
sage: pg.show()
Phi_*(g) = dth*dth + sin(th)^2 dph*dph
Pullback on `S^2` of a 3-form on `R^3`::
sage: a = DiffForm(n, 3, 'A')
sage: a[1,2,3] = f
sage: a.show()
A = x*y*z dx/\dy/\dz
sage: pa = Phi.pullback(a) ; pa
3-form 'Phi_*(A)' on the 2-dimensional manifold 'S^2'
sage: pa.show() # should be zero (as any 3-form on a 2-dimensional manifold)
Phi_*(A) = 0
"""
from scalarfield import ScalarField
if not isinstance(tensor, TensorField):
raise TypeError("The argument 'tensor' must be a tensor field.")
manif1 = self.manifold1
manif2 = self.manifold2
if tensor.manifold != manif2:
raise TypeError("The tensor field is not defined on the mapping " +
"arrival manifold.")
(ncon, ncov) = tensor.tensor_type
if ncon != 0:
raise TypeError("The pullback cannot be taken on a tensor " +
"with some contravariant part.")
resu_name = None ; resu_latex_name = None
if self.name is not None and tensor.name is not None:
resu_name = self.name + '_*(' + tensor.name + ')'
if self.latex_name is not None and tensor.latex_name is not None:
resu_latex_name = self.latex_name + '_*' + tensor.latex_name
chart1_name = None; chart2_name = None
def_chart1 = manif1.def_chart ; def_chart1_name = def_chart1.name
def_chart2 = manif2.def_chart ; def_chart2_name = def_chart2.name
if ncov == 0:
# Case of a scalar field
# ----------------------
# A pair of chart (chart1_name, chart2_name) where the computation
# is feasable is searched, privileging the default chart of the
# start manifold for chart1_name
if def_chart2_name in tensor.express and \
(def_chart1_name, def_chart2_name) in self.coord_expression:
chart1_name = def_chart1_name
chart2_name = def_chart2_name
else:
for (chart1n, chart2n) in self.coord_expression:
if chart1n == def_chart1_name and chart2n in tensor.express:
chart1_name = def_chart1_name
chart2_name = chart2n
break
if chart1_name is None:
# It is not possible to have def_chart1 as chart for
# expressing the result; any other chart is then looked for:
for (chart1n, chart2n) in self.coord_expression:
if chart2n in tensor.express:
chart1_name = chart1n
chart2_name = chart2n
break
if chart1_name is None:
raise ValueError("No common chart could be find to compute " +
"the pullback of the scalar field.")
phi = self.coord_expression[(chart1_name, chart2_name)]
coord1 = manif1.atlas[chart1_name].xx
ff = tensor.express[chart2_name]
return ScalarField(manif1, ff(*(phi(*coord1))), chart1_name,
resu_name, resu_latex_name)
else:
# Case of tensor field of rank >= 1
# ---------------------------------
# A pair of chart (chart1_name, chart2_name) where the computation
# is feasable is searched, privileging the default chart of the
# start manifold for chart1_name
if def_chart2.frame.name in tensor.components and \
(def_chart1_name, def_chart2_name) in self.coord_expression:
chart1_name = def_chart1_name
chart2_name = def_chart2_name
else:
for (chart1n, chart2n) in self.coord_expression:
if chart1n == def_chart1_name and \
manif2.atlas[chart2n].frame.name in tensor.components:
chart1_name = def_chart1_name
chart2_name = chart2n
break
if chart1_name is None:
# It is not possible to have def_chart1 as chart for
# expressing the result; any other chart is then looked for:
for (chart1n, chart2n) in self.coord_expression:
if manif2.atlas[chart2n].frame.name in tensor.components:
chart1_name = chart1n
chart2_name = chart2n
break
if chart1_name is None:
raise ValueError("No common chart could be find to compute " +
"the pullback of the tensor field.")
chart1 = manif1.atlas[chart1_name]
chart2 = manif2.atlas[chart2_name]
frame1_name = chart1.frame.name
frame2_name = chart2.frame.name
# Computation at the component level:
tcomp = tensor.components[frame2_name]
if isinstance(tcomp, CompFullySym):
ptcomp = CompFullySym(manif1, ncov, frame1_name)
elif isinstance(tcomp, CompFullyAntiSym):
ptcomp = CompFullyAntiSym(manif1, ncov, frame1_name)
elif isinstance(tcomp, CompWithSym):
ptcomp = CompWithSym(manif1, ncov, frame1_name, sym=tcomp.sym,
antisym=tcomp.antisym)
else:
ptcomp = Components(manif1, ncov, frame1_name)
phi = self.coord_expression[(chart1_name, chart2_name)]
jacob = phi.jacobian()
# X2 coordinates expressed in terms of X1 ones via the mapping:
coord2_1 = phi(*(chart1.xx))
si1 = manif1.sindex
si2 = manif2.sindex
for ind_new in ptcomp.non_redundant_index_generator():
res = 0
for ind_old in manif2.index_generator(ncov):
ff = tcomp[[ind_old]].function_chart(chart2_name)
t = FunctionChart(chart1, ff(*coord2_1))
for i in range(ncov):
t *= jacob[ind_old[i]-si2][ind_new[i]-si1]
res += t
ptcomp[ind_new] = res
resu = tensor_field_from_comp(ptcomp, (0, ncov))
resu.set_name(resu_name, resu_latex_name)
return resu