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r"""

Algebra of Differentiable Scalar Fields

The class :class:`DiffScalarFieldAlgebra` implements the commutative algebra

`C^k(M)` of differentiable scalar fields on a differentiable manifold `M` of

class `C^k` over a topological field `K` (in

most applications, `K = \RR` or `K = \CC`). By *differentiable scalar field*,

it is meant a function `M\rightarrow K` that is `k`-times continuously

differentiable. `C^k(M)` is an algebra over `K`, whose ring product is the

pointwise multiplication of `K`-valued functions, which is clearly commutative.

AUTHORS:

- Eric Gourgoulhon, Michal Bejger (2014-2015): initial version

REFERENCES:

- [KN1963]_

- [Lee2013]_

- [ONe1983]_

"""

#******************************************************************************

# 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.rings.infinity import infinity

from sage.symbolic.ring import SR

from sage.manifolds.scalarfield_algebra import ScalarFieldAlgebra

from sage.manifolds.differentiable.scalarfield import DiffScalarField

class DiffScalarFieldAlgebra(ScalarFieldAlgebra):

r"""

Commutative algebra of differentiable scalar fields on a differentiable

manifold.

If `M` is a differentiable manifold of class `C^k` over a topological

field `K`, the *commutative algebra of scalar fields on* `M` is the set

`C^k(M)` of all `k`-times continuously differentiable maps `M\rightarrow K`.

The set `C^k(M)` is an algebra over `K`, whose ring product is the

pointwise multiplication of `K`-valued functions, which is clearly

commutative.

If `K = \RR` or `K = \CC`, the field `K` over which the

algebra `C^k(M)` is constructed is represented by Sage's Symbolic Ring

``SR``, since there is no exact representation of `\RR` nor `\CC` in Sage.

Via its base class

:class:`~sage.manifolds.scalarfield_algebra.ScalarFieldAlgebra`,

the class :class:`DiffScalarFieldAlgebra` inherits from

:class:`~sage.structure.parent.Parent`, with the category set to

:class:`~sage.categories.commutative_algebras.CommutativeAlgebras`.

The corresponding *element* class is

:class:`~sage.manifolds.differentiable.scalarfield.DiffScalarField`.

INPUT:

- ``domain`` -- the differentiable manifold `M` on which the scalar fields

are defined (must be an instance of class

:class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`)

EXAMPLES:

Algebras of scalar fields on the sphere `S^2` and on some open subset of

it::

sage: M = Manifold(2, 'M') # the 2-dimensional sphere S^2

sage: U = M.open_subset('U') # complement of the North pole

sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole

sage: V = M.open_subset('V') # complement of the South pole

sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole

sage: M.declare_union(U,V) # S^2 is the union of U and V

sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)),

....: intersection_name='W', restrictions1= x^2+y^2!=0,

....: restrictions2= u^2+v^2!=0)

sage: uv_to_xy = xy_to_uv.inverse()

sage: CM = M.scalar_field_algebra() ; CM

Algebra of differentiable scalar fields on the 2-dimensional

differentiable manifold M

sage: W = U.intersection(V) # S^2 minus the two poles

sage: CW = W.scalar_field_algebra() ; CW

Algebra of differentiable scalar fields on the Open subset W of the

2-dimensional differentiable manifold M

`C^k(M)` and `C^k(W)` belong to the category of commutative

algebras over `\RR` (represented here by Sage's Symbolic Ring)::

sage: CM.category()

Category of commutative algebras over Symbolic Ring

sage: CM.base_ring()

Symbolic Ring

sage: CW.category()

Category of commutative algebras over Symbolic Ring

sage: CW.base_ring()

Symbolic Ring

The elements of `C^k(M)` are scalar fields on `M`::

sage: CM.an_element()

Scalar field on the 2-dimensional differentiable manifold M

sage: CM.an_element().display() # this sample element is a constant field

M --> R

on U: (x, y) |--> 2

on V: (u, v) |--> 2

Those of `C^k(W)` are scalar fields on `W`::

sage: CW.an_element()

Scalar field on the Open subset W of the 2-dimensional differentiable

manifold M

sage: CW.an_element().display() # this sample element is a constant field

W --> R

(x, y) |--> 2

(u, v) |--> 2

The zero element::

sage: CM.zero()

Scalar field zero on the 2-dimensional differentiable manifold M

sage: CM.zero().display()

zero: M --> R

on U: (x, y) |--> 0

on V: (u, v) |--> 0

sage: CW.zero()

Scalar field zero on the Open subset W of the 2-dimensional

differentiable manifold M

sage: CW.zero().display()

zero: W --> R

(x, y) |--> 0

(u, v) |--> 0

The unit element::

sage: CM.one()

Scalar field 1 on the 2-dimensional differentiable manifold M

sage: CM.one().display()

1: M --> R

on U: (x, y) |--> 1

on V: (u, v) |--> 1

sage: CW.one()

Scalar field 1 on the Open subset W of the 2-dimensional differentiable

manifold M

sage: CW.one().display()

1: W --> R

(x, y) |--> 1

(u, v) |--> 1

A generic element can be constructed as for any parent in Sage, namely

by means of the ``__call__`` operator on the parent (here with the

dictionary of the coordinate expressions defining the scalar field)::

sage: f = CM({c_xy: atan(x^2+y^2), c_uv: pi/2 - atan(u^2+v^2)}); f

Scalar field on the 2-dimensional differentiable manifold M

sage: f.display()

M --> R

on U: (x, y) |--> arctan(x^2 + y^2)

on V: (u, v) |--> 1/2*pi - arctan(u^2 + v^2)

sage: f.parent()

Algebra of differentiable scalar fields on the 2-dimensional

differentiable manifold M

Specific elements can also be constructed in this way::

sage: CM(0) == CM.zero()

True

sage: CM(1) == CM.one()

True

Note that the zero scalar field is cached::

sage: CM(0) is CM.zero()

True

Elements can also be constructed by means of the method

:meth:`~sage.manifolds.manifold.TopologicalManifold.scalar_field` acting on

the domain (this allows one to set the name of the scalar field at the

construction)::

sage: f1 = M.scalar_field({c_xy: atan(x^2+y^2), c_uv: pi/2 - atan(u^2+v^2)},

....: name='f')

sage: f1.parent()

Algebra of differentiable scalar fields on the 2-dimensional

differentiable manifold M

sage: f1 == f

True

sage: M.scalar_field(0, chart='all') == CM.zero()

True

The algebra `C^k(M)` coerces to `C^k(W)` since `W` is an open

subset of `M`::

sage: CW.has_coerce_map_from(CM)

True

The reverse is of course false::

sage: CM.has_coerce_map_from(CW)

False

The coercion map is nothing but the restriction to `W` of scalar fields

on `M`::

sage: fW = CW(f) ; fW

Scalar field on the Open subset W of the 2-dimensional differentiable

manifold M

sage: fW.display()

W --> R

(x, y) |--> arctan(x^2 + y^2)

(u, v) |--> 1/2*pi - arctan(u^2 + v^2)

sage: CW(CM.one()) == CW.one()

True

The coercion map allows for the addition of elements of `C^k(W)`

with elements of `C^k(M)`, the result being an element of

`C^k(W)`::

sage: s = fW + f

sage: s.parent()

Algebra of differentiable scalar fields on the Open subset W of the

2-dimensional differentiable manifold M

sage: s.display()

W --> R

(x, y) |--> 2*arctan(x^2 + y^2)

(u, v) |--> pi - 2*arctan(u^2 + v^2)

Another coercion is that from the Symbolic Ring, the parent of all

symbolic expressions (cf. :class:`~sage.symbolic.ring.SymbolicRing`).

Since the Symbolic Ring is the base ring for the algebra ``CM``, the

coercion of a symbolic expression ``s`` is performed by the operation

``s*CM.one()``, which invokes the reflected multiplication operator

:meth:`sage.manifolds.scalarfield.ScalarField._rmul_`. If the symbolic

expression does not involve any chart coordinate, the outcome is a

constant scalar field::

sage: h = CM(pi*sqrt(2)) ; h

Scalar field on the 2-dimensional differentiable manifold M

sage: h.display()

M --> R

on U: (x, y) |--> sqrt(2)*pi

on V: (u, v) |--> sqrt(2)*pi

sage: a = var('a')

sage: h = CM(a); h.display()

M --> R

on U: (x, y) |--> a

on V: (u, v) |--> a

If the symbolic expression involves some coordinate of one of the

manifold's charts, the outcome is initialized only on the chart domain::

sage: h = CM(a+x); h.display()

M --> R

on U: (x, y) |--> a + x

sage: h = CM(a+u); h.display()

M --> R

on V: (u, v) |--> a + u

If the symbolic expression involves coordinates of different charts,

the scalar field is created as a Python object, but is not initialized,

in order to avoid any ambiguity::

sage: h = CM(x+u); h.display()

M --> R

.. RUBRIC:: TESTS OF THE ALGEBRA LAWS:

Ring laws::

sage: h = CM(pi*sqrt(2))

sage: s = f + h ; s

Scalar field on the 2-dimensional differentiable manifold M

sage: s.display()

M --> R

on U: (x, y) |--> sqrt(2)*pi + arctan(x^2 + y^2)

on V: (u, v) |--> 1/2*pi*(2*sqrt(2) + 1) - arctan(u^2 + v^2)

sage: s = f - h ; s

Scalar field on the 2-dimensional differentiable manifold M

sage: s.display()

M --> R

on U: (x, y) |--> -sqrt(2)*pi + arctan(x^2 + y^2)

on V: (u, v) |--> -1/2*pi*(2*sqrt(2) - 1) - arctan(u^2 + v^2)

sage: s = f*h ; s

Scalar field on the 2-dimensional differentiable manifold M

sage: s.display()

M --> R

on U: (x, y) |--> sqrt(2)*pi*arctan(x^2 + y^2)

on V: (u, v) |--> 1/2*sqrt(2)*(pi^2 - 2*pi*arctan(u^2 + v^2))

sage: s = f/h ; s

Scalar field on the 2-dimensional differentiable manifold M

sage: s.display()

M --> R

on U: (x, y) |--> 1/2*sqrt(2)*arctan(x^2 + y^2)/pi

on V: (u, v) |--> 1/4*sqrt(2)*(pi - 2*arctan(u^2 + v^2))/pi

sage: f*(h+f) == f*h + f*f

True

Ring laws with coercion::

sage: f - fW == CW.zero()

True

sage: f/fW == CW.one()

True

sage: s = f*fW ; s

Scalar field on the Open subset W of the 2-dimensional differentiable

manifold M

sage: s.display()

W --> R

(x, y) |--> arctan(x^2 + y^2)^2

(u, v) |--> 1/4*pi^2 - pi*arctan(u^2 + v^2) + arctan(u^2 + v^2)^2

sage: s/f == fW

True

Multiplication by a number::

sage: s = 2*f ; s

Scalar field on the 2-dimensional differentiable manifold M

sage: s.display()

M --> R

on U: (x, y) |--> 2*arctan(x^2 + y^2)

on V: (u, v) |--> pi - 2*arctan(u^2 + v^2)

sage: 0*f == CM.zero()

True

sage: 1*f == f

True

sage: 2*(f/2) == f

True

sage: (f+2*f)/3 == f

True

sage: 1/3*(f+2*f) == f

True

The Sage test suite for algebras is passed::

sage: TestSuite(CM).run()

It is passed also for `C^k(W)`::

sage: TestSuite(CW).run()

"""

Element = DiffScalarField

def __init__(self, domain):

r"""

Construct an algebra of differentiable scalar fields.

TESTS::

sage: M = Manifold(2, 'M')

sage: X.<x,y> = M.chart()

sage: CM = M.scalar_field_algebra(); CM

Algebra of differentiable scalar fields on the 2-dimensional

differentiable manifold M

sage: type(CM)

sage: type(CM).__base__

sage: TestSuite(CM).run()

"""

ScalarFieldAlgebra.__init__(self, domain)

#### Methods required for any Parent

def _coerce_map_from_(self, other):

r"""

Determine whether coercion to self exists from other parent

TESTS::

sage: M = Manifold(2, 'M')

sage: X.<x,y> = M.chart()

sage: CM = M.scalar_field_algebra()

sage: CM._coerce_map_from_(SR)

True

sage: U = M.open_subset('U', coord_def={X: x>0})

sage: CU = U.scalar_field_algebra()

sage: CM._coerce_map_from_(CU)

False

sage: CU._coerce_map_from_(CM)

True

"""

if other is SR:

return True # coercion from the base ring (multiplication by the

# algebra unit, i.e. self.one())

# cf. ScalarField._lmul_() for the implementation of

# the coercion map

elif isinstance(other, DiffScalarFieldAlgebra):

return self._domain.is_subset(other._domain)

else:

return False

#### End of methods required for any Parent

def _repr_(self):

r"""

String representation of the object.

TESTS::

sage: M = Manifold(2, 'M')

sage: CM = M.scalar_field_algebra()

sage: CM._repr_()

'Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold M'

sage: repr(CM) # indirect doctest

'Algebra of differentiable scalar fields on the 2-dimensional differentiable manifold M'

"""

return "Algebra of differentiable scalar fields on " + \

"the {}".format(self._domain)

def _latex_(self):

r"""

LaTeX representation of the object.

TESTS::

sage: M = Manifold(2, 'M')

sage: CM = M.scalar_field_algebra()

sage: CM._latex_()

'C^{\\infty}\\left(M\\right)'

sage: latex(CM) # indirect doctest

C^{\infty}\left(M\right)

"""

degree = self._domain.diff_degree()

if degree == infinity:

latex_degree = r"\infty" # to skip the "+" in latex(infinity)

else:

latex_degree = "{}".format(degree)

return r"C^{" + latex_degree + r"}\left(" + self._domain._latex_() + \

r"\right)"

Coverage for local/lib/python2.7/site-packages/sage/manifolds/differentiable/scalarfield_algebra.py : 95%

22 statements 21 run 1 missing 0 excluded