Coverage for local/lib/python2.7/site-packages/sage/tensor/modules/free_module_element.py : 56%
Hot-keys on this page
r m x p toggle line displays
j k next/prev highlighted chunk
0 (zero) top of page
1 (one) first highlighted chunk
|
r""" Elements of free modules of finite rank
The class :class:`FiniteRankFreeModuleElement` implements elements of free modules of finite rank over a commutative ring.
AUTHORS:
- Eric Gourgoulhon, Michal Bejger (2014-2015): initial version - Eric Gourgoulhon (2017): class :class:`FiniteRankFreeModuleElement` inherits from :class:`~sage.tensor.modules.alternating_contr_tensor.AlternatingContrTensor`
REFERENCES:
- Chap. 21 of R. Godement : *Algebra* [God1968]_ - Chap. 12 of J. M. Lee: *Introduction to Smooth Manifolds* [Lee2013]_ (only when the free module is a vector space) - Chap. 2 of B. O'Neill: *Semi-Riemannian Geometry* [ONe1983]_
"""
#****************************************************************************** # Copyright (C) 2015 Eric Gourgoulhon <eric.gourgoulhon@obspm.fr> # Copyright (C) 2015 Michal Bejger <bejger@camk.edu.pl> # # Distributed under the terms of the GNU General Public License (GPL) # as published by the Free Software Foundation; either version 2 of # the License, or (at your option) any later version. # http://www.gnu.org/licenses/ #******************************************************************************
from sage.tensor.modules.alternating_contr_tensor import AlternatingContrTensor from sage.tensor.modules.comp import Components
class FiniteRankFreeModuleElement(AlternatingContrTensor): r""" Element of a free module of finite rank over a commutative ring.
This is a Sage *element* class, the corresponding *parent* class being :class:`~sage.tensor.modules.finite_rank_free_module.FiniteRankFreeModule`.
The class :class:`FiniteRankFreeModuleElement` inherits from :class:`~sage.tensor.modules.alternating_contr_tensor.AlternatingContrTensor` because the elements of a free module `M` of finite rank over a commutative ring `R` are identified with tensors of type `(1,0)` on `M` via the canonical map
.. MATH::
\begin{array}{lllllll} \Phi: & M & \longrightarrow & M^{**} & & & \\ & v & \longmapsto & \bar v : & M^* & \longrightarrow & R \\ & & & & a & \longmapsto & a(v) \end{array}
Note that for free modules of finite rank, this map is actually an isomorphism, enabling the canonical identification: `M^{**}= M`.
INPUT:
- ``fmodule`` -- free module `M` of finite rank over a commutative ring `R`, as an instance of :class:`~sage.tensor.modules.finite_rank_free_module.FiniteRankFreeModule` - ``name`` -- (default: ``None``) name given to the element - ``latex_name`` -- (default: ``None``) LaTeX symbol to denote the element; if none is provided, the LaTeX symbol is set to ``name``
EXAMPLES:
Let us consider a rank-3 free module `M` over `\ZZ`::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M') sage: e = M.basis('e') ; e Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring
There are three ways to construct an element of the free module `M`: the first one (recommended) is using the free module::
sage: v = M([2,0,-1], basis=e, name='v') ; v Element v of the Rank-3 free module M over the Integer Ring sage: v.display() # expansion on the default basis (e) v = 2 e_0 - e_2 sage: v.parent() is M True
The second way is to construct a tensor of type `(1,0)` on `M` (cf. the canonical identification `M^{**} = M` recalled above)::
sage: v2 = M.tensor((1,0), name='v') sage: v2[0], v2[2] = 2, -1 ; v2 Element v of the Rank-3 free module M over the Integer Ring sage: v2.display() v = 2 e_0 - e_2 sage: v2 == v True
Finally, the third way is via some linear combination of the basis elements::
sage: v3 = 2*e[0] - e[2] sage: v3.set_name('v') ; v3 # in this case, the name has to be set separately Element v of the Rank-3 free module M over the Integer Ring sage: v3.display() v = 2 e_0 - e_2 sage: v3 == v True
The canonical identification `M^{**} = M` is implemented by letting the module elements act on linear forms, providing the same result as the reverse operation (cf. the map `\Phi` defined above)::
sage: a = M.linear_form(name='a') sage: a[:] = (2, 1, -3) ; a Linear form a on the Rank-3 free module M over the Integer Ring sage: v(a) 7 sage: a(v) 7 sage: a(v) == v(a) True
.. RUBRIC:: ARITHMETIC EXAMPLES
Addition::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M') sage: e = M.basis('e') ; e Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring sage: a = M([0,1,3], name='a') ; a Element a of the Rank-3 free module M over the Integer Ring sage: a.display() a = e_1 + 3 e_2 sage: b = M([2,-2,1], name='b') ; b Element b of the Rank-3 free module M over the Integer Ring sage: b.display() b = 2 e_0 - 2 e_1 + e_2 sage: s = a + b ; s Element a+b of the Rank-3 free module M over the Integer Ring sage: s.display() a+b = 2 e_0 - e_1 + 4 e_2 sage: all(s[i] == a[i] + b[i] for i in M.irange()) True
Subtraction::
sage: s = a - b ; s Element a-b of the Rank-3 free module M over the Integer Ring sage: s.display() a-b = -2 e_0 + 3 e_1 + 2 e_2 sage: all(s[i] == a[i] - b[i] for i in M.irange()) True
Multiplication by a scalar::
sage: s = 2*a ; s Element of the Rank-3 free module M over the Integer Ring sage: s.display() 2 e_1 + 6 e_2 sage: a.display() a = e_1 + 3 e_2
Tensor product::
sage: s = a*b ; s Type-(2,0) tensor a*b on the Rank-3 free module M over the Integer Ring sage: s.symmetries() no symmetry; no antisymmetry sage: s[:] [ 0 0 0] [ 2 -2 1] [ 6 -6 3] sage: s = a*s ; s Type-(3,0) tensor a*a*b on the Rank-3 free module M over the Integer Ring sage: s[:] [[[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [2, -2, 1], [6, -6, 3]], [[0, 0, 0], [6, -6, 3], [18, -18, 9]]]
Exterior product::
sage: s = a.wedge(b) ; s Alternating contravariant tensor a/\b of degree 2 on the Rank-3 free module M over the Integer Ring sage: s.display() a/\b = -2 e_0/\e_1 - 6 e_0/\e_2 + 7 e_1/\e_2
""" def __init__(self, fmodule, name=None, latex_name=None): r""" TESTS::
sage: from sage.tensor.modules.free_module_element import FiniteRankFreeModuleElement sage: M = FiniteRankFreeModule(ZZ, 3, name='M') sage: e = M.basis('e') sage: v = FiniteRankFreeModuleElement(M, name='v') sage: v[e,:] = (-2, 1, 3) sage: TestSuite(v).run(skip="_test_category") # see below
In the above test suite, _test_category fails because v is not an instance of v.parent().category().element_class. Actually module elements must be constructed via FiniteRankFreeModule.element_class and not by a direct call to FiniteRankFreeModuleElement::
sage: v1 = M.element_class(M, name='v') sage: v1[e,:] = (-2, 1, 3) sage: TestSuite(v1).run()
""" latex_name=latex_name)
def _repr_(self): r""" Return a string representation of ``self``.
EXAMPLES::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M') sage: e = M.basis('e') sage: M([1,-2,3], name='v') Element v of the Rank-3 free module M over the Integer Ring
"""
def _new_comp(self, basis): r""" Create some (uninitialized) components of ``self`` in a given basis.
This method, which is already implemented in :meth:`FreeModuleTensor._new_comp`, is redefined here for efficiency.
INPUT:
- ``basis`` -- basis of the free module on which ``self`` is defined
OUTPUT:
- an instance of :class:`~sage.tensor.modules.comp.Components`
EXAMPLES::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M') sage: e = M.basis('e') sage: v = M([1,-2,3], name='v') sage: v._new_comp(e) 1-index components w.r.t. Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring sage: type(v._new_comp(e)) <class 'sage.tensor.modules.comp.Components'>
""" output_formatter=fmodule._output_formatter)
def _new_instance(self): r""" Create an instance of the same class as ``self``.
EXAMPLES::
sage: M = FiniteRankFreeModule(ZZ, 3, name='M') sage: e = M.basis('e') sage: v = M([1,-2,3], name='v') sage: v._new_instance() Element of the Rank-3 free module M over the Integer Ring sage: v._new_instance().parent() is v.parent() True
""" |