r"""
Open subset of Euclidian space with coordinates
An open subset of Euclidian space with a specific set of coordinates. This
is the background on which differential forms can be defined.
AUTHORS:
- Joris Vankerschaver (2010-07-25)
EXAMPLES::
sage: x, y, z = var('x, y, z')
sage: S = CoordinatePatch((x, y, z)); S
Open subset of R^3 with coordinates x, y, z
::
sage: u, v = var('u, v')
sage: S = CoordinatePatch((u, v)); S
Open subset of R^2 with coordinates u, v
TODO:
- Add functionality for metric tensors
"""
from sage.structure.parent import Parent
class CoordinatePatch(Parent):
"""
Construct a coordinate patch, i.e. an open subset of
Euclidian space with a given set of coordinates.
EXAMPLES::
sage: x, y, z = var('x, y, z')
sage: S = CoordinatePatch((x, y, z)); S
Open subset of R^3 with coordinates x, y, z
sage: u, v = var('u, v')
sage: T = CoordinatePatch((u, v)); T
Open subset of R^2 with coordinates u, v
sage: loads(T.dumps()) == T
True
In a future release, it will be possible to specify a
metric tensor on a coordinate patch. For now, providing
any kind of metric raises an exception::
sage: x, y, z = var('x, y, z')
sage: m = matrix(SR, 3)
sage: S = CoordinatePatch((x, y, z), metric=m)
Traceback (most recent call last):
...
NotImplementedError: Metric geometry not supported yet.
"""
def __init__(self, coordinates, metric = None):
"""
An open subset of Euclidian space with a specific set of
coordinates. See ``CoordinatePatch`` for details.
INPUT::
- ``coordinates`` -- a set of symbolic variables that serve
as coordinates on this space.
- ``metric`` (default: None) -- a metric tensor on this
coordinate patch. Providing anything other than ``None``
is currently not defined.
EXAMPLES::
sage: x, y, z = var('x, y, z')
sage: S = CoordinatePatch((x, y, z)); S
Open subset of R^3 with coordinates x, y, z
"""
from sage.symbolic.ring import is_SymbolicVariable
if not all(is_SymbolicVariable(c) for c in coordinates):
raise TypeError, "%s is not a valid vector of coordinates." % \
coordinates
self._coordinates = tuple(coordinates)
dim = len(self._coordinates)
if metric is not None:
raise NotImplementedError, "Metric geometry not supported yet."
def __eq__(self, other):
"""
Return equality if and only if other has the same coordinates
as self, in the same order.
EXAMPLES::
sage: x, y, z = var('x, y, z')
sage: S = CoordinatePatch((x, y, z)); S
Open subset of R^3 with coordinates x, y, z
sage: u, v = var('u, v')
sage: T = CoordinatePatch((u, v)); T
Open subset of R^2 with coordinates u, v
sage: U = CoordinatePatch((x, y, z)); U
Open subset of R^3 with coordinates x, y, z
sage: U == S
True
sage: U == U
True
sage: U == T
False
Note that the order of the coordinates matters::
sage: x, y, z = var('x, y, z')
sage: S = CoordinatePatch((x, y, z)); S
Open subset of R^3 with coordinates x, y, z
sage: T = CoordinatePatch((x, z, y)); T
Open subset of R^3 with coordinates x, z, y
sage: S == T
False
"""
return str(self._coordinates) == str(other._coordinates)
def __ne__(self, other):
"""
Test whether two coordinate patches are not equal.
EXAMPLES::
sage: x, y, z = var('x, y, z')
sage: S = CoordinatePatch((x, y, z)); S
Open subset of R^3 with coordinates x, y, z
sage: u, v = var('u, v')
sage: T = CoordinatePatch((u, v)); T
Open subset of R^2 with coordinates u, v
sage: S != T
True
"""
return not self.__eq__(other)
def coordinates(self):
"""
Return coordinates on this coordinate patch.
OUTPUT:
- list - a list of coordinates on this space.
EXAMPLES::
sage: x, y, z = var('x, y, z')
sage: S = CoordinatePatch((x, y, z)); S
Open subset of R^3 with coordinates x, y, z
sage: S.coordinates()
(x, y, z)
"""
return self._coordinates
def coordinate(self, i=0):
"""
Return the `i^{th}` coordinate on ``self``
INPUT:
- ``i`` - integer (optional, default 0)
EXAMPLES::
sage: x, y, z = var('x, y, z')
sage: S = CoordinatePatch((x, y, z)); S
Open subset of R^3 with coordinates x, y, z
sage: S.coordinate(0)
x
sage: S.coordinate(1)
y
sage: S.coordinate(2)
z
"""
return self._coordinates[i]
def dim(self):
"""
Return the dimension of this coordinate patch, i.e. the dimension
of the Euclidian space of which this coordinate patch is an open
subset.
EXAMPLES::
sage: a, b, c, d, e = var('a, b, c, d, e')
sage: U = CoordinatePatch((a, b, c, d, e)); U
Open subset of R^5 with coordinates a, b, c, d, e
sage: U.dim()
5
"""
return len(self._coordinates)
def _repr_(self):
r"""
Return string representation of this coordinate patch.
EXAMPLES::
sage: x, y, z = var('x, y, z')
sage: S = CoordinatePatch((x, y, z)); S
Open subset of R^3 with coordinates x, y, z
sage: S._repr_()
'Open subset of R^3 with coordinates x, y, z'
sage: S.rename('coordinate patch'); S
coordinate patch
sage: S.rename(); S
Open subset of R^3 with coordinates x, y, z
"""
return r"Open subset of R^%s with coordinates %s" % \
(self.dim(), ', '.join([x._latex_() for x in self._coordinates]))
def _latex_(self):
r"""
Return latex representation of this coordinate patch.
EXAMPLES::
sage: x, y, z = var('x, y, z')
sage: S = CoordinatePatch((x, y, z)); S
Open subset of R^3 with coordinates x, y, z
sage: latex(S)
\mathbb{\RR}^3
sage: latex(S) == S._latex_()
True
"""
return "\\mathbb{\RR}^%s" % self.dim()
