def sqrt_abs_det(self, frame=None):
r"""
Square root of the absolute value of the determinant of the metric
components in the specified frame.
INPUT:
- ``frame`` -- (default: None) vector frame with
respect to which the components `g_{ij}` of ``self`` are defined;
if None, the domain's default frame is used. If a chart is
provided, the associated coordinate frame is used
OUTPUT:
- `\sqrt{|\det (g_{ij})|}`, as an instance of :class:`ScalarField`
EXAMPLES:
Standard metric in the Euclidean space `\RR^3` with spherical
coordinates::
sage: m = Manifold(3, 'M', start_index=1)
sage: c_spher.<r,th,ph> = m.chart(r'r:[0,+oo) th:[0,pi]:\theta ph:[0,2*pi):\phi')
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, r^2, (r*sin(th))^2
sage: g.view()
g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
sage: g.sqrt_abs_det().expr()
r^2*sin(th)
Metric determinant on a 2-dimensional manifold::
sage: M = Manifold(2, 'M', start_index=1)
sage: X.<x,y> = M.chart('x y')
sage: g = Metric(M, 'g')
sage: g[1,1], g[1, 2], g[2, 2] = 1+x, x*y , 1-y
sage: g[:]
[ x + 1 x*y]
[ x*y -y + 1]
sage: s = g.sqrt_abs_det() ; s
scalar field on the 2-dimensional manifold 'M'
sage: s.expr()
sqrt(-x^2*y^2 - (x + 1)*y + x + 1)
Determinant in a frame different from the default's one::
sage: Y.<u,v> = M.chart('u v')
sage: ch_X_Y = CoordChange(X, Y, x+y, x-y)
sage: ch_X_Y.inverse()
coordinate change from chart (M, (u, v)) to chart (M, (x, y))
sage: g.comp(Y.frame)[:, Y]
[ 1/8*u^2 - 1/8*v^2 + 1/4*v + 1/2 1/4*u]
[ 1/4*u -1/8*u^2 + 1/8*v^2 + 1/4*v + 1/2]
sage: g.sqrt_abs_det(Y.frame).expr()
1/2*sqrt(-x^2*y^2 - (x + 1)*y + x + 1)
sage: g.sqrt_abs_det(Y.frame).expr(Y)
1/8*sqrt(-u^4 - v^4 + 2*(u^2 + 2)*v^2 - 4*u^2 + 16*v + 16)
A chart can be passed instead of a frame::
sage: g.sqrt_abs_det(Y) is g.sqrt_abs_det(Y.frame)
True
The metric determinant depends on the frame::
sage: g.sqrt_abs_det(X.frame) == g.sqrt_abs_det(Y.frame)
False
"""
from sage.functions.other import sqrt
from scalarfield import ScalarField
from utilities import simplify_chain
manif = self.manifold
dom = self.domain
if frame is None:
frame = dom.def_frame
if frame in dom.atlas:
# frame is actually a chart and is changed to the associated
# coordinate frame:
frame = frame.frame
if frame not in self._sqrt_abs_dets:
# a new computation is necessary
detg = self.determinant(frame)
resu = ScalarField(dom)
for chart in detg.express:
x = self._indic_signat * detg.express[chart].express # |g|
x = simplify_chain(sqrt(x))
resu.add_expr(x, chart=chart)
self._sqrt_abs_dets[frame] = resu
return self._sqrt_abs_dets[frame]
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 determinant(self, frame=None):
r"""
Determinant of the metric components in the specified frame.
INPUT:
- ``frame`` -- (default: None) vector frame with
respect to which the components `g_{ij}` of ``self`` are defined;
if None, the domain's default frame is used. If a chart is
provided, the associated coordinate frame is used
OUTPUT:
- the determinant `\det (g_{ij})`, as an instance of :class:`ScalarField`
EXAMPLES:
Metric determinant on a 2-dimensional manifold::
sage: M = Manifold(2, 'M', start_index=1)
sage: X.<x,y> = M.chart('x y')
sage: g = Metric(M, 'g')
sage: g[1,1], g[1, 2], g[2, 2] = 1+x, x*y , 1-y
sage: g[:]
[ x + 1 x*y]
[ x*y -y + 1]
sage: s = g.determinant() # determinant in M's default frame
sage: s.expr()
-x^2*y^2 - (x + 1)*y + x + 1
Determinant in a frame different from the default's one::
sage: Y.<u,v> = M.chart('u v')
sage: ch_X_Y = CoordChange(X, Y, x+y, x-y)
sage: ch_X_Y.inverse()
coordinate change from chart (M, (u, v)) to chart (M, (x, y))
sage: g.comp(Y.frame)[:, Y]
[ 1/8*u^2 - 1/8*v^2 + 1/4*v + 1/2 1/4*u]
[ 1/4*u -1/8*u^2 + 1/8*v^2 + 1/4*v + 1/2]
sage: g.determinant(Y.frame).expr()
-1/4*x^2*y^2 - 1/4*(x + 1)*y + 1/4*x + 1/4
sage: g.determinant(Y.frame).expr(Y)
-1/64*u^4 - 1/64*v^4 + 1/32*(u^2 + 2)*v^2 - 1/16*u^2 + 1/4*v + 1/4
A chart can be passed instead of a frame::
sage: g.determinant(X) is g.determinant(X.frame)
True
sage: g.determinant(Y) is g.determinant(Y.frame)
True
The metric determinant depends on the frame::
sage: g.determinant(X.frame) == g.determinant(Y.frame)
False
"""
from sage.matrix.constructor import matrix
from scalarfield import ScalarField
from utilities import simple_determinant, simplify_chain
manif = self.manifold
dom = self.domain
if frame is None:
frame = dom.def_frame
if frame in dom.atlas:
# frame is actually a chart and is changed to the associated
# coordinate frame:
frame = frame.frame
if frame not in self._determinants:
# a new computation is necessary
resu = ScalarField(dom)
gg = self.comp(frame)
i1 = manif.sindex
for chart in gg[[i1, i1]].express:
gm = matrix( [[ gg[i, j, chart].express
for j in manif.irange()] for i in manif.irange()] )
detgm = simplify_chain(simple_determinant(gm))
resu.add_expr(detgm, chart=chart)
self._determinants[frame] = resu
return self._determinants[frame]
def sqrt_abs_det(self, frame_name=None):
r"""
Square root of the absolute value of the determinant of the metric
components in the specified frame.
INPUT:
- ``frame_name`` -- (default: None) name of the vector frame with
respect to which the components `g_{ij}` of ``self`` are defined;
if None, the manifold's default frame is used. If a chart name is
provided, the associated coordinate basis is used
OUTPUT:
- `\sqrt{|\det (g_{ij})|}`, as an instance of :class:`ScalarField`
EXAMPLES:
Standard metric in the Euclidean space `\RR^3` with spherical
coordinates::
sage: m = Manifold(3, 'M', start_index=1)
sage: c_spher = Chart(m, r'r:positive th:positive:\theta ph:\phi', 'spher')
sage: assume(th>=0); assume(th<=pi)
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, r^2, (r*sin(th))^2
sage: g.show()
g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
sage: g.sqrt_abs_det().expr()
r^2*sin(th)
Metric determinant on a 2-dimensional manifold::
sage: M = Manifold(2, 'M', start_index=1)
sage: X = 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-y
sage: g[:]
[ x + 1 x*y]
[ x*y -y + 1]
sage: s = g.sqrt_abs_det() ; s
scalar field on the 2-dimensional manifold 'M'
sage: s.expr()
sqrt(-x^2*y^2 - (x + 1)*y + x + 1)
Determinant in a frame different from the default's one::
sage: Y = Chart(M, 'u v', 'uv')
sage: ch_X_Y = CoordChange(X, Y, x+y, x-y)
sage: ch_X_Y.inverse()
coordinate change from chart 'uv' (u, v) to chart 'xy' (x, y)
sage: g.comp('uv_b')[:, 'uv']
[ 1/8*u^2 - 1/8*v^2 + 1/4*v + 1/2 1/4*u]
[ 1/4*u -1/8*u^2 + 1/8*v^2 + 1/4*v + 1/2]
sage: g.sqrt_abs_det('uv_b').expr()
1/2*sqrt(-x^2*y^2 - (x + 1)*y + x + 1)
sage: g.sqrt_abs_det('uv_b').expr('uv')
1/8*sqrt(-u^4 - 4*u^2 - v^4 + 2*(u^2 + 2)*v^2 + 16*v + 16)
The name of a chart can be passed instead of the name of a frame::
sage: g.sqrt_abs_det('uv') is g.sqrt_abs_det('uv_b')
True
The metric determinant depends on the frame::
sage: g.sqrt_abs_det('xy_b') == g.sqrt_abs_det('uv_b')
False
"""
from sage.functions.other import sqrt
from scalarfield import ScalarField
from utilities import simplify_chain
manif = self.manifold
if frame_name is None:
frame_name = manif.def_frame.name
if frame_name in manif.atlas:
# frame_name is actually the name of a chart and is changed to the
# name of the associated coordinate basis:
frame_name = manif.atlas[frame_name].frame.name
if frame_name not in self._sqrt_abs_dets:
# a new computation is necessary
detg = self.determinant(frame_name)
resu = ScalarField(manif)
for chart_name in detg.express:
x = self._indic_signat * detg.express[chart_name].express # |g|
x = simplify_chain(sqrt(x))
resu.set_expr(x, chart_name=chart_name, delete_others=False)
self._sqrt_abs_dets[frame_name] = resu
return self._sqrt_abs_dets[frame_name]
def determinant(self, frame_name=None):
r"""
Determinant of the metric components in the specified frame.
INPUT:
- ``frame_name`` -- (default: None) name of the vector frame with
respect to which the components `g_{ij}` of ``self`` are defined;
if None, the manifold's default frame is used. If a chart name is
provided, the associated coordinate basis is used
OUTPUT:
- the determinant `\det (g_{ij})`, as an instance of :class:`ScalarField`
EXAMPLES:
Metric determinant on a 2-dimensional manifold::
sage: M = Manifold(2, 'M', start_index=1)
sage: X = 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-y
sage: g[:]
[ x + 1 x*y]
[ x*y -y + 1]
sage: s = g.determinant() # determinant in M's default frame
sage: s.expr()
-x^2*y^2 - (x + 1)*y + x + 1
Determinant in a frame different from the default's one::
sage: Y = Chart(M, 'u v', 'uv')
sage: ch_X_Y = CoordChange(X, Y, x+y, x-y)
sage: ch_X_Y.inverse()
coordinate change from chart 'uv' (u, v) to chart 'xy' (x, y)
sage: g.comp('uv_b')[:, 'uv']
[ 1/8*u^2 - 1/8*v^2 + 1/4*v + 1/2 1/4*u]
[ 1/4*u -1/8*u^2 + 1/8*v^2 + 1/4*v + 1/2]
sage: g.determinant('uv_b').expr()
-1/4*x^2*y^2 - 1/4*(x + 1)*y + 1/4*x + 1/4
sage: g.determinant('uv_b').expr('uv')
-1/64*u^4 - 1/16*u^2 - 1/64*v^4 + 1/32*(u^2 + 2)*v^2 + 1/4*v + 1/4
The name of a chart can be passed instead of the name of a frame::
sage: g.determinant('xy') is g.determinant('xy_b')
True
sage: g.determinant('uv') is g.determinant('uv_b')
True
The metric determinant depends on the frame::
sage: g.determinant('xy_b') == g.determinant('uv_b')
False
"""
from sage.matrix.constructor import matrix
from scalarfield import ScalarField
from utilities import simple_determinant, simplify_chain
manif = self.manifold
if frame_name is None:
frame_name = manif.def_frame.name
if frame_name in manif.atlas:
# frame_name is actually the name of a chart and is changed to the
# name of the associated coordinate basis:
frame_name = manif.atlas[frame_name].frame.name
if frame_name not in self._determinants:
# a new computation is necessary
resu = ScalarField(manif)
gg = self.comp(frame_name)
i1 = manif.sindex
for chart_name in gg[[i1, i1]].express:
gm = matrix(
[[gg[i, j, chart_name].express for j in manif.irange()]
for i in manif.irange()])
detgm = simplify_chain(simple_determinant(gm))
resu.set_expr(detgm,
chart_name=chart_name,
delete_others=False)
self._determinants[frame_name] = resu
return self._determinants[frame_name]
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 sqrt_abs_det(self, frame=None):
r"""
Square root of the absolute value of the determinant of the metric
components in the specified frame.
INPUT:
- ``frame`` -- (default: None) vector frame with
respect to which the components `g_{ij}` of ``self`` are defined;
if None, the domain's default frame is used. If a chart is
provided, the associated coordinate frame is used
OUTPUT:
- `\sqrt{|\det (g_{ij})|}`, as an instance of :class:`ScalarField`
EXAMPLES:
Standard metric in the Euclidean space `\RR^3` with spherical
coordinates::
sage: m = Manifold(3, 'M', start_index=1)
sage: c_spher.<r,th,ph> = m.chart(r'r:[0,+oo) th:[0,pi]:\theta ph:[0,2*pi):\phi')
sage: g = Metric(m, 'g')
sage: g[1,1], g[2,2], g[3,3] = 1, r^2, (r*sin(th))^2
sage: g.view()
g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
sage: g.sqrt_abs_det().expr()
r^2*sin(th)
Metric determinant on a 2-dimensional manifold::
sage: M = Manifold(2, 'M', start_index=1)
sage: X.<x,y> = M.chart('x y')
sage: g = Metric(M, 'g')
sage: g[1,1], g[1, 2], g[2, 2] = 1+x, x*y , 1-y
sage: g[:]
[ x + 1 x*y]
[ x*y -y + 1]
sage: s = g.sqrt_abs_det() ; s
scalar field on the 2-dimensional manifold 'M'
sage: s.expr()
sqrt(-x^2*y^2 - (x + 1)*y + x + 1)
Determinant in a frame different from the default's one::
sage: Y.<u,v> = M.chart('u v')
sage: ch_X_Y = CoordChange(X, Y, x+y, x-y)
sage: ch_X_Y.inverse()
coordinate change from chart (M, (u, v)) to chart (M, (x, y))
sage: g.comp(Y.frame)[:, Y]
[ 1/8*u^2 - 1/8*v^2 + 1/4*v + 1/2 1/4*u]
[ 1/4*u -1/8*u^2 + 1/8*v^2 + 1/4*v + 1/2]
sage: g.sqrt_abs_det(Y.frame).expr()
1/2*sqrt(-x^2*y^2 - (x + 1)*y + x + 1)
sage: g.sqrt_abs_det(Y.frame).expr(Y)
1/8*sqrt(-u^4 - v^4 + 2*(u^2 + 2)*v^2 - 4*u^2 + 16*v + 16)
A chart can be passed instead of a frame::
sage: g.sqrt_abs_det(Y) is g.sqrt_abs_det(Y.frame)
True
The metric determinant depends on the frame::
sage: g.sqrt_abs_det(X.frame) == g.sqrt_abs_det(Y.frame)
False
"""
from sage.functions.other import sqrt
from scalarfield import ScalarField
from utilities import simplify_chain
manif = self.manifold
dom = self.domain
if frame is None:
frame = dom.def_frame
if frame in dom.atlas:
# frame is actually a chart and is changed to the associated
# coordinate frame:
frame = frame.frame
if frame not in self._sqrt_abs_dets:
# a new computation is necessary
detg = self.determinant(frame)
resu = ScalarField(dom)
for chart in detg.express:
x = self._indic_signat * detg.express[chart].express # |g|
x = simplify_chain(sqrt(x))
resu.add_expr(x, chart=chart)
self._sqrt_abs_dets[frame] = resu
return self._sqrt_abs_dets[frame]