Coverage for local/lib/python2.7/site-packages/sage/tensor/coordinate_patch.py : 58%
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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 doctest:...: DeprecationWarning: Use Manifold instead. See http://trac.sagemath.org/24444 for details. 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
"""
#***************************************************************************** # Copyright (C) 2010 Joris Vankerschaver <joris.vankerschaver@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # # This code is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU # General Public License for more details. # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ #*****************************************************************************
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 doctest:...: DeprecationWarning: Use Manifold instead. See http://trac.sagemath.org/24444 for details. 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 doctest:...: DeprecationWarning: Use Manifold instead. See http://trac.sagemath.org/24444 for details. Open subset of R^3 with coordinates x, y, z """
raise TypeError("%s is not a valid vector of coordinates." % \ coordinates)
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 doctest:...: DeprecationWarning: Use Manifold instead. See http://trac.sagemath.org/24444 for details. 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 """
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 doctest:...: DeprecationWarning: Use Manifold instead. See http://trac.sagemath.org/24444 for details. 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 """
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 doctest:...: DeprecationWarning: Use Manifold instead. See http://trac.sagemath.org/24444 for details. Open subset of R^3 with coordinates x, y, z sage: S.coordinates() (x, y, z)
"""
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 doctest:...: DeprecationWarning: Use Manifold instead. See http://trac.sagemath.org/24444 for details. 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
"""
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 doctest:...: DeprecationWarning: Use Manifold instead. See http://trac.sagemath.org/24444 for details. Open subset of R^5 with coordinates a, b, c, d, e sage: U.dim() 5 """
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 doctest:...: DeprecationWarning: Use Manifold instead. See http://trac.sagemath.org/24444 for details. 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 """ (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 doctest:...: DeprecationWarning: Use Manifold instead. See http://trac.sagemath.org/24444 for details. Open subset of R^3 with coordinates x, y, z sage: latex(S) \mathbb{\RR}^3 sage: latex(S) == S._latex_() True """
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