def __init__(self, domain, coordinates='', names=None, calc_method=None):
r"""
Construct a chart on a real differentiable manifold.
TESTS::
sage: forget() # for doctests only
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: X
Chart (M, (x, y))
sage: type(X)
<class 'sage.manifolds.differentiable.chart.RealDiffChart'>
sage: assumptions() # assumptions set in X._init_coordinates
[x is real, y is real]
sage: TestSuite(X).run()
"""
RealChart.__init__(self,
domain,
coordinates=coordinates,
names=names,
calc_method=calc_method)
# Construction of the coordinate frame associated to the chart:
self._frame = CoordFrame(self)
self._coframe = self._frame._coframe
def __init__(self, domain, coordinates='', names=None):
r"""
Construct a chart on a real differentiable manifold.
TESTS::
sage: forget() # for doctests only
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: X
Chart (M, (x, y))
sage: type(X)
<class 'sage.manifolds.differentiable.chart.RealDiffChart'>
sage: assumptions() # assumptions set in X._init_coordinates
[x is real, y is real]
sage: TestSuite(X).run()
"""
RealChart.__init__(self, domain, coordinates=coordinates, names=names)
def restrict(self, subset, restrictions=None):
r"""
Return the restriction of the chart to some subset.
If the current chart is `(U, \varphi)`, a *restriction* (or
*subchart*) is a chart `(V, \psi)` such that `V \subset U`
and `\psi = \varphi |_V`.
If such subchart has not been defined yet, it is constructed here.
The coordinates of the subchart bare the same names as the
coordinates of the original chart.
INPUT:
- ``subset`` -- open subset `V` of the chart domain `U`
- ``restrictions`` -- (default: ``None``) list of coordinate
restrictions defining the subset `V`
A restriction can be any symbolic equality or inequality involving
the coordinates, such as ``x > y`` or ``x^2 + y^2 != 0``. The items
of the list ``restrictions`` are combined with the ``and`` operator;
if some restrictions are to be combined with the ``or`` operator
instead, they have to be passed as a tuple in some single item
of the list ``restrictions``. For example::
restrictions = [x > y, (x != 0, y != 0), z^2 < x]
means ``(x > y) and ((x != 0) or (y != 0)) and (z^2 < x)``.
If the list ``restrictions`` contains only one item, this
item can be passed as such, i.e. writing ``x > y`` instead
of the single element list ``[x > y]``.
OUTPUT:
- a :class:`RealDiffChart` `(V, \psi)`
EXAMPLES:
Cartesian coordinates on the unit open disc in `\RR^2` as a subchart
of the global Cartesian coordinates::
sage: M = Manifold(2, 'R^2')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
sage: D = M.open_subset('D') # the unit open disc
sage: c_cart_D = c_cart.restrict(D, x^2+y^2<1)
sage: p = M.point((1/2, 0))
sage: p in D
True
sage: q = M.point((1, 2))
sage: q in D
False
Cartesian coordinates on the annulus `1 < \sqrt{x^2+y^2} < 2`::
sage: A = M.open_subset('A')
sage: c_cart_A = c_cart.restrict(A, [x^2+y^2>1, x^2+y^2<4])
sage: p in A, q in A
(False, False)
sage: a = M.point((3/2,0))
sage: a in A
True
"""
if subset == self._domain:
return self
if subset not in self._dom_restrict:
resu = RealChart.restrict(self, subset, restrictions=restrictions)
# Update of superframes and subframes:
resu._frame._superframes.update(self._frame._superframes)
for sframe in self._frame._superframes:
sframe._subframes.add(resu._frame)
sframe._restrictions[subset] = resu._frame
# The subchart frame is not a "top frame" in the supersets
# (including self._domain):
for dom in self._domain._supersets:
if resu._frame in dom._top_frames:
# it was added by the Chart constructor invoked in
# Chart.restrict above
dom._top_frames.remove(resu._frame)
return self._dom_restrict[subset]
def restrict(self, subset, restrictions=None):
r"""
Return the restriction of the chart to some subset.
If the current chart is `(U, \varphi)`, a *restriction* (or
*subchart*) is a chart `(V, \psi)` such that `V \subset U`
and `\psi = \varphi |_V`.
If such subchart has not been defined yet, it is constructed here.
The coordinates of the subchart bare the same names as the
coordinates of the original chart.
INPUT:
- ``subset`` -- open subset `V` of the chart domain `U`
- ``restrictions`` -- (default: ``None``) list of coordinate
restrictions defining the subset `V`
A restriction can be any symbolic equality or inequality involving
the coordinates, such as ``x > y`` or ``x^2 + y^2 != 0``. The items
of the list ``restrictions`` are combined with the ``and`` operator;
if some restrictions are to be combined with the ``or`` operator
instead, they have to be passed as a tuple in some single item
of the list ``restrictions``. For example::
restrictions = [x > y, (x != 0, y != 0), z^2 < x]
means ``(x > y) and ((x != 0) or (y != 0)) and (z^2 < x)``.
If the list ``restrictions`` contains only one item, this
item can be passed as such, i.e. writing ``x > y`` instead
of the single element list ``[x > y]``.
OUTPUT:
- a :class:`RealDiffChart` `(V, \psi)`
EXAMPLES:
Cartesian coordinates on the unit open disc in `\RR^2` as a subchart
of the global Cartesian coordinates::
sage: M = Manifold(2, 'R^2')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
sage: D = M.open_subset('D') # the unit open disc
sage: c_cart_D = c_cart.restrict(D, x^2+y^2<1)
sage: p = M.point((1/2, 0))
sage: p in D
True
sage: q = M.point((1, 2))
sage: q in D
False
Cartesian coordinates on the annulus `1 < \sqrt{x^2+y^2} < 2`::
sage: A = M.open_subset('A')
sage: c_cart_A = c_cart.restrict(A, [x^2+y^2>1, x^2+y^2<4])
sage: p in A, q in A
(False, False)
sage: a = M.point((3/2,0))
sage: a in A
True
"""
if subset == self._domain:
return self
if subset not in self._dom_restrict:
resu = RealChart.restrict(self, subset, restrictions=restrictions)
# Update of superframes and subframes:
resu._frame._superframes.update(self._frame._superframes)
for sframe in self._frame._superframes:
sframe._subframes.add(resu._frame)
sframe._restrictions[subset] = resu._frame
# The subchart frame is not a "top frame" in the supersets
# (including self._domain):
for dom in self._domain._supersets:
if resu._frame in dom._top_frames:
# it was added by the Chart constructor invoked in
# Chart.restrict above
dom._top_frames.remove(resu._frame)
return self._dom_restrict[subset]