def __invert__(self):
r"""
Return the inverse automorphism.
"""
from sage.matrix.constructor import matrix
from sage.tensor.modules.comp import Components
from vectorframe import CoordFrame
from utilities import simplify_chain
if self._is_identity:
return self
if self._inverse is None:
if self._name is None:
inv_name = None
else:
inv_name = self._name + '^(-1)'
if self._latex_name is None:
inv_latex_name = None
else:
inv_latex_name = self._latex_name + r'^{-1}'
fmodule = self._fmodule
si = fmodule._sindex
nsi = fmodule._rank + si
self._inverse = fmodule.automorphism(name=inv_name,
latex_name=inv_latex_name)
for frame in self._components:
if isinstance(frame, CoordFrame):
chart = frame._chart
else:
chart = self._domain._def_chart #!# to be improved
try:
mat_self = matrix([[
self.comp(frame)[i, j, chart]._express
for j in range(si, nsi)
] for i in range(si, nsi)])
except (KeyError, ValueError):
continue
mat_inv = mat_self.inverse()
cinv = Components(fmodule._ring,
frame,
2,
start_index=si,
output_formatter=fmodule._output_formatter)
for i in range(si, nsi):
for j in range(si, nsi):
cinv[i, j] = {
chart: simplify_chain(mat_inv[i - si, j - si])
}
self._inverse._components[frame] = cinv
return self._inverse
def inverse(self):
r"""
Return the inverse automorphism.
"""
from sage.matrix.constructor import matrix
from component import Components
from utilities import simplify_chain
if self._inverse is None:
if self.name is None:
inv_name = None
else:
inv_name = 'inv-' + self.name
if self.latex_name is None:
inv_latex_name = None
else:
inv_latex_name = self.latex_name + r'^{-1}'
manif = self.manifold
dom = self.domain
si = manif.sindex
nsi = manif.dim + si
self._inverse = AutomorphismField(dom, inv_name, inv_latex_name)
for frame_name in self.components:
if frame_name[-2:] == '_b':
# coordinate basis
chart_name = frame_name[:-2]
else:
chart_name = dom.def_chart.name #!# to be improved
try:
mat_self = matrix([[
self.comp(frame_name)[i, j, chart_name].express
for j in range(si, nsi)
] for i in range(si, nsi)])
except (KeyError, ValueError):
continue
mat_inv = mat_self.inverse()
cinv = Components(dom.frames[frame_name], 2)
for i in range(si, nsi):
for j in range(si, nsi):
cinv[i, j,
chart_name] = simplify_chain(mat_inv[i - si,
j - si])
self._inverse.components[frame_name] = cinv
return self._inverse
def __invert__(self):
r"""
Return the inverse automorphism.
"""
from sage.matrix.constructor import matrix
from sage.tensor.modules.comp import Components
from vectorframe import CoordFrame
from utilities import simplify_chain
if self._is_identity:
return self
if self._inverse is None:
if self._name is None:
inv_name = None
else:
inv_name = self._name + '^(-1)'
if self._latex_name is None:
inv_latex_name = None
else:
inv_latex_name = self._latex_name + r'^{-1}'
fmodule = self._fmodule
si = fmodule._sindex ; nsi = fmodule._rank + si
self._inverse = fmodule.automorphism(name=inv_name,
latex_name=inv_latex_name)
for frame in self._components:
if isinstance(frame, CoordFrame):
chart = frame._chart
else:
chart = self._domain._def_chart #!# to be improved
try:
mat_self = matrix(
[[self.comp(frame)[i, j, chart]._express
for j in range(si, nsi)] for i in range(si, nsi)])
except (KeyError, ValueError):
continue
mat_inv = mat_self.inverse()
cinv = Components(fmodule._ring, frame, 2, start_index=si,
output_formatter=fmodule._output_formatter)
for i in range(si, nsi):
for j in range(si, nsi):
cinv[i, j] = {chart: simplify_chain(mat_inv[i-si,j-si])}
self._inverse._components[frame] = cinv
return self._inverse
def inverse(self):
r"""
Return the inverse automorphism.
"""
from sage.matrix.constructor import matrix
from component import Components
from utilities import simplify_chain
if self._inverse is None:
if self.name is None:
inv_name = None
else:
inv_name = 'inv-' + self.name
if self.latex_name is None:
inv_latex_name = None
else:
inv_latex_name = self.latex_name + r'^{-1}'
manif = self.manifold
dom = self.domain
si = manif.sindex ; nsi = manif.dim + si
self._inverse = AutomorphismField(dom, inv_name, inv_latex_name)
for frame_name in self.components:
if frame_name[-2:] == '_b':
# coordinate basis
chart_name = frame_name[:-2]
else:
chart_name = dom.def_chart.name #!# to be improved
try:
mat_self = matrix(
[[self.comp(frame_name)[i, j, chart_name].express
for j in range(si, nsi)] for i in range(si, nsi)])
except (KeyError, ValueError):
continue
mat_inv = mat_self.inverse()
cinv = Components(dom.frames[frame_name], 2)
for i in range(si, nsi):
for j in range(si, nsi):
cinv[i, j, chart_name] = simplify_chain(
mat_inv[i-si,j-si])
self._inverse.components[frame_name] = cinv
return self._inverse
def inverse(self, chartname1=None, chartname2=None):
r"""
Returns the inverse diffeomorphism.
INPUT:
- ``chartname1`` -- (default: None) string defining the chart in which
the computation of the inverse is performed; if none is provided, the
default chart of self.manifold1 will be used
- ``chartname2`` -- (default: None) string defining the chart in which
the computation of the inverse is performed; if none is provided, the
default chart of self.manifold2 will be used
OUTPUT:
- the inverse diffeomorphism
EXAMPLES:
The inverse of a rotation in the plane::
sage: m = Manifold(2, "plane")
sage: c_cart = Chart(m, 'x y', 'cart')
sage: # A pi/3 rotation around the origin:
sage: rot = Diffeomorphism(m, m, ((x - sqrt(3)*y)/2, (sqrt(3)*x + y)/2))
sage: p = Point(m,(1,2))
sage: q = rot(p)
sage: irot = rot.inverse()
sage: p1 = irot(q)
sage: p1 == p
True
"""
from sage.symbolic.ring import SR
from sage.symbolic.relation import solve
from utilities import simplify_chain
if self._inverse is not None:
return self._inverse
if chartname1 is None: chartname1 = self.manifold1.def_chart.name
if chartname2 is None: chartname2 = self.manifold2.def_chart.name
coord_map = self.coord_expression[(chartname1, chartname2)]
chart1 = self.manifold1.atlas[chartname1]
chart2 = self.manifold2.atlas[chartname2]
n1 = len(chart1.xx)
n2 = len(chart2.xx)
# New symbolic variables (different from chart2.xx to allow for a
# correct solution even when chart2 = chart1):
x2 = [ SR.var('xxxx' + str(i)) for i in range(n2) ]
equations = [x2[i] == coord_map.functions[i] for i in range(n2) ]
solutions = solve(equations, chart1.xx, solution_dict=True)
if len(solutions) == 0:
raise ValueError("No solution found")
if len(solutions) > 1:
raise ValueError("Non-unique solution found")
#!# This should be the Python 2.7 form:
# substitutions = {x2[i]: chart2.xx[i] for i in range(n2)}
#
# Here we use a form compatible with Python 2.6:
substitutions = dict([(x2[i], chart2.xx[i]) for i in range(n2)])
inv_functions = [solutions[0][chart1.xx[i]].subs(substitutions)
for i in range(n1)]
for i in range(n1):
x = inv_functions[i]
try:
inv_functions[i] = simplify_chain(x)
except AttributeError:
pass
self._inverse = Diffeomorphism(self.manifold2, self.manifold1,
inv_functions, chartname2, chartname1)
return self._inverse
def inverse(self, chart1=None, chart2=None):
r"""
Return the inverse diffeomorphism.
INPUT:
- ``chart1`` -- (default: None) chart in which the computation of the
inverse is performed if necessary; if none is provided, the default
chart of the start domain will be used
- ``chart2`` -- (default: None) chart in which the computation of the
inverse is performed if necessary; if none is provided, the default
chart of the arrival domain will be used
OUTPUT:
- the inverse diffeomorphism
EXAMPLES:
The inverse of a rotation in the Euclidean plane::
sage: m = Manifold(2, 'R^2', r'\RR^2')
sage: c_cart.<x,y> = m.chart('x y')
sage: # A pi/3 rotation around the origin:
sage: rot = Diffeomorphism(m, m, ((x - sqrt(3)*y)/2, (sqrt(3)*x + y)/2), name='R')
sage: rot.inverse()
diffeomorphism 'R^(-1)' on the 2-dimensional manifold 'R^2'
sage: rot.inverse().view()
R^(-1): R^2 --> R^2, (x, y) |--> (1/2*sqrt(3)*y + 1/2*x, -1/2*sqrt(3)*x + 1/2*y)
Checking that applying successively the diffeomorphism and its
inverse results in the identity::
sage: (a, b) = var('a b')
sage: p = Point(m, (a,b)) # a generic point on M
sage: q = rot(p)
sage: p1 = rot.inverse()(q)
sage: p1 == p
True
"""
from sage.symbolic.ring import SR
from sage.symbolic.relation import solve
from utilities import simplify_chain
if self._inverse is not None:
return self._inverse
if chart1 is None: chart1 = self.domain1.def_chart
if chart2 is None: chart2 = self.domain2.def_chart
coord_map = self.coord_expression[(chart1, chart2)]
n1 = len(chart1.xx)
n2 = len(chart2.xx)
# New symbolic variables (different from chart2.xx to allow for a
# correct solution even when chart2 = chart1):
x2 = [ SR.var('xxxx' + str(i)) for i in range(n2) ]
equations = [ x2[i] == coord_map.functions[i].express
for i in range(n2) ]
solutions = solve(equations, chart1.xx, solution_dict=True)
if len(solutions) == 0:
raise ValueError("No solution found")
if len(solutions) > 1:
raise ValueError("Non-unique solution found")
#!# This should be the Python 2.7 form:
# substitutions = {x2[i]: chart2.xx[i] for i in range(n2)}
#
# Here we use a form compatible with Python 2.6:
substitutions = dict([(x2[i], chart2.xx[i]) for i in range(n2)])
inv_functions = [solutions[0][chart1.xx[i]].subs(substitutions)
for i in range(n1)]
for i in range(n1):
x = inv_functions[i]
try:
inv_functions[i] = simplify_chain(x)
except AttributeError:
pass
if self.name is None:
name = None
else:
name = self.name + '^(-1)'
if self.latex_name is None:
latex_name = None
else:
latex_name = self.latex_name + r'^{-1}'
self._inverse = Diffeomorphism(self.domain2, self.domain1,
inv_functions, chart2, chart1,
name=name, latex_name=latex_name)
return self._inverse
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 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 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.<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-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 vectorframe import CoordFrame
from utilities import simplify_chain
# Is the inverse metric up to date ?
for frame in self.components:
if frame not in self._inverse.components:
# the computation is necessary
manif = self.manifold
dom = self.domain
if isinstance(frame, CoordFrame):
chart = frame.chart
else:
chart = dom.def_chart
si = manif.sindex
nsi = manif.dim + si
try:
gmat = matrix(
[[self.comp(frame)[i, j, chart].express
for j in range(si, nsi)] for i in range(si, nsi)])
except (KeyError, ValueError):
continue
gmat_inv = gmat.inverse()
cinv = CompFullySym(frame, 2)
for i in range(si, nsi):
for j in range(i, nsi): # symmetry taken into account
cinv[i, j, chart] = simplify_chain(
gmat_inv[i-si,j-si])
self._inverse.components[frame] = cinv
return self._inverse
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 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 inverse(self, chartname1=None, chartname2=None):
r"""
Returns the inverse diffeomorphism.
INPUT:
- ``chartname1`` -- (default: None) string defining the chart in which
the computation of the inverse is performed; if none is provided, the
default chart of self.manifold1 will be used
- ``chartname2`` -- (default: None) string defining the chart in which
the computation of the inverse is performed; if none is provided, the
default chart of self.manifold2 will be used
OUTPUT:
- the inverse diffeomorphism
EXAMPLES:
The inverse of a rotation in the plane::
sage: m = Manifold(2, "plane")
sage: c_cart = Chart(m, 'x y', 'cart')
sage: # A pi/3 rotation around the origin:
sage: rot = Diffeomorphism(m, m, ((x - sqrt(3)*y)/2, (sqrt(3)*x + y)/2))
sage: p = Point(m,(1,2))
sage: q = rot(p)
sage: irot = rot.inverse()
sage: p1 = irot(q)
sage: p1 == p
True
"""
from sage.symbolic.ring import SR
from sage.symbolic.relation import solve
from utilities import simplify_chain
if self._inverse is not None:
return self._inverse
if chartname1 is None: chartname1 = self.manifold1.def_chart.name
if chartname2 is None: chartname2 = self.manifold2.def_chart.name
coord_map = self.coord_expression[(chartname1, chartname2)]
chart1 = self.manifold1.atlas[chartname1]
chart2 = self.manifold2.atlas[chartname2]
n1 = len(chart1.xx)
n2 = len(chart2.xx)
# New symbolic variables (different from chart2.xx to allow for a
# correct solution even when chart2 = chart1):
x2 = [SR.var('xxxx' + str(i)) for i in range(n2)]
equations = [x2[i] == coord_map.functions[i] for i in range(n2)]
solutions = solve(equations, chart1.xx, solution_dict=True)
if len(solutions) == 0:
raise ValueError("No solution found")
if len(solutions) > 1:
raise ValueError("Non-unique solution found")
#!# This should be the Python 2.7 form:
# substitutions = {x2[i]: chart2.xx[i] for i in range(n2)}
#
# Here we use a form compatible with Python 2.6:
substitutions = dict([(x2[i], chart2.xx[i]) for i in range(n2)])
inv_functions = [
solutions[0][chart1.xx[i]].subs(substitutions) for i in range(n1)
]
for i in range(n1):
x = inv_functions[i]
try:
inv_functions[i] = simplify_chain(x)
except AttributeError:
pass
self._inverse = Diffeomorphism(self.manifold2, self.manifold1,
inv_functions, chartname2, chartname1)
return self._inverse
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]