Algebra of Scalar Fields

The class ScalarFieldAlgebra implements the commutative algebra \(C^0(M)\) of scalar fields on a topological manifold \(M\) over a topological field \(K\). By scalar field, it is meant a continuous function \(M \to K\). The set \(C^0(M)\) 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

  • Travis Scrimshaw (2016): review tweaks

REFERENCES:

class sage.manifolds.scalarfield_algebra.ScalarFieldAlgebra(domain)[source]

Bases: UniqueRepresentation, Parent

Commutative algebra of scalar fields on a topological manifold.

If \(M\) is a topological manifold over a topological field \(K\), the commutative algebra of scalar fields on \(M\) is the set \(C^0(M)\) of all continuous maps \(M \to K\). The set \(C^0(M)\) 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 \(C^0(M)\) is constructed is represented by the Symbolic Ring SR, since there is no exact representation of \(\RR\) nor \(\CC\).

INPUT:

  • domain – the topological manifold \(M\) on which the scalar fields are defined

EXAMPLES:

Algebras of scalar fields on the sphere \(S^2\) and on some open subsets of it:

sage: M = Manifold(2, 'M', structure='topological') # 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 scalar fields on the 2-dimensional topological manifold M
sage: W = U.intersection(V)  # S^2 minus the two poles
sage: CW = W.scalar_field_algebra(); CW
Algebra of scalar fields on the Open subset W of the
 2-dimensional topological manifold M

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological') # the 2-dimensional sphere S^2
>>> U = M.open_subset('U')  # complement of the North pole
>>> c_xy = U.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2)# stereographic coordinates from the North pole
>>> V = M.open_subset('V')  # complement of the South pole
>>> c_uv = V.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2)# stereographic coordinates from the South pole
>>> M.declare_union(U,V)    # S^2 is the union of U and V
>>> xy_to_uv = c_xy.transition_map(c_uv, (x/(x**Integer(2)+y**Integer(2)), y/(x**Integer(2)+y**Integer(2))),
...                                intersection_name='W',
...                                restrictions1= x**Integer(2)+y**Integer(2)!=Integer(0),
...                                restrictions2= u**Integer(2)+v**Integer(2)!=Integer(0))
>>> uv_to_xy = xy_to_uv.inverse()
>>> CM = M.scalar_field_algebra(); CM
Algebra of scalar fields on the 2-dimensional topological manifold M
>>> W = U.intersection(V)  # S^2 minus the two poles
>>> CW = W.scalar_field_algebra(); CW
Algebra of scalar fields on the Open subset W of the
 2-dimensional topological manifold M

\(C^0(M)\) and \(C^0(W)\) belong to the category of commutative algebras over \(\RR\) (represented here by SymbolicRing):

sage: CM.category()
Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces
sage: CM.base_ring()
Symbolic Ring
sage: CW.category()
Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces
sage: CW.base_ring()
Symbolic Ring

>>> from sage.all import *
>>> CM.category()
Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces
>>> CM.base_ring()
Symbolic Ring
>>> CW.category()
Join of Category of commutative algebras over Symbolic Ring and Category of homsets of topological spaces
>>> CW.base_ring()
Symbolic Ring

The elements of \(C^0(M)\) are scalar fields on \(M\):

sage: CM.an_element()
Scalar field on the 2-dimensional topological 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

>>> from sage.all import *
>>> CM.an_element()
Scalar field on the 2-dimensional topological manifold M
>>> 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 \(C^0(W)\) are scalar fields on \(W\):

sage: CW.an_element()
Scalar field on the Open subset W of the 2-dimensional topological
 manifold M
sage: CW.an_element().display()  # this sample element is a constant field
W → ℝ
(x, y) ↦ 2
(u, v) ↦ 2

>>> from sage.all import *
>>> CW.an_element()
Scalar field on the Open subset W of the 2-dimensional topological
 manifold M
>>> 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 topological manifold M
sage: CM.zero().display()
zero: M → ℝ
on U: (x, y) ↦ 0
on V: (u, v) ↦ 0

>>> from sage.all import *
>>> CM.zero()
Scalar field zero on the 2-dimensional topological manifold M
>>> 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
 topological manifold M
sage: CW.zero().display()
zero: W → ℝ
   (x, y) ↦ 0
   (u, v) ↦ 0

>>> from sage.all import *
>>> CW.zero()
Scalar field zero on the Open subset W of the 2-dimensional
 topological manifold M
>>> 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 topological manifold M
sage: CM.one().display()
1: M → ℝ
on U: (x, y) ↦ 1
on V: (u, v) ↦ 1

>>> from sage.all import *
>>> CM.one()
Scalar field 1 on the 2-dimensional topological manifold M
>>> 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 topological
 manifold M
sage: CW.one().display()
1: W → ℝ
  (x, y) ↦ 1
  (u, v) ↦ 1

>>> from sage.all import *
>>> CW.one()
Scalar field 1 on the Open subset W of the 2-dimensional topological
 manifold M
>>> CW.one().display()
1: W → ℝ
  (x, y) ↦ 1
  (u, v) ↦ 1

A generic element can be constructed by using a 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 topological 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 scalar fields on the 2-dimensional topological manifold M

>>> from sage.all import *
>>> f = CM({c_xy: atan(x**Integer(2)+y**Integer(2)), c_uv: pi/Integer(2) - atan(u**Integer(2)+v**Integer(2))}); f
Scalar field on the 2-dimensional topological manifold M
>>> 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)
>>> f.parent()
Algebra of scalar fields on the 2-dimensional topological manifold M

Specific elements can also be constructed in this way:

sage: CM(0) == CM.zero()
True
sage: CM(1) == CM.one()
True

>>> from sage.all import *
>>> CM(Integer(0)) == CM.zero()
True
>>> CM(Integer(1)) == CM.one()
True

Note that the zero scalar field is cached:

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

>>> from sage.all import *
>>> CM(Integer(0)) is CM.zero()
True

Elements can also be constructed by means of the method 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 scalar fields on the 2-dimensional topological manifold M
sage: f1 == f
True
sage: M.scalar_field(0, chart='all') == CM.zero()
True

>>> from sage.all import *
>>> f1 = M.scalar_field({c_xy: atan(x**Integer(2)+y**Integer(2)), c_uv: pi/Integer(2) - atan(u**Integer(2)+v**Integer(2))},
...                     name='f')
>>> f1.parent()
Algebra of scalar fields on the 2-dimensional topological manifold M
>>> f1 == f
True
>>> M.scalar_field(Integer(0), chart='all') == CM.zero()
True

The algebra \(C^0(M)\) coerces to \(C^0(W)\) since \(W\) is an open subset of \(M\):

sage: CW.has_coerce_map_from(CM)
True

>>> from sage.all import *
>>> CW.has_coerce_map_from(CM)
True

The reverse is of course false:

sage: CM.has_coerce_map_from(CW)
False

>>> from sage.all import *
>>> 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 topological manifold M
sage: fW.display()
W → ℝ
  (x, y) ↦ arctan(x^2 + y^2)
  (u, v) ↦ 1/2*pi - arctan(u^2 + v^2)

>>> from sage.all import *
>>> fW = CW(f) ; fW
Scalar field on the Open subset W of the
 2-dimensional topological manifold M
>>> 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

>>> from sage.all import *
>>> CW(CM.one()) == CW.one()
True

The coercion map allows for the addition of elements of \(C^0(W)\) with elements of \(C^0(M)\), the result being an element of \(C^0(W)\):

sage: s = fW + f
sage: s.parent()
Algebra of scalar fields on the Open subset W of the
 2-dimensional topological manifold M
sage: s.display()
W → ℝ
  (x, y) ↦ 2*arctan(x^2 + y^2)
  (u, v) ↦ pi - 2*arctan(u^2 + v^2)

>>> from sage.all import *
>>> s = fW + f
>>> s.parent()
Algebra of scalar fields on the Open subset W of the
 2-dimensional topological manifold M
>>> 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. 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. 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 topological 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

>>> from sage.all import *
>>> h = CM(pi*sqrt(Integer(2))) ; h
Scalar field on the 2-dimensional topological manifold M
>>> h.display()
M → ℝ
on U: (x, y) ↦ sqrt(2)*pi
on V: (u, v) ↦ sqrt(2)*pi
>>> a = var('a')
>>> 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
on W: (u, v) ↦ (a*u^2 + a*v^2 + u)/(u^2 + v^2)
sage: h = CM(a+u); h.display()
M → ℝ
on W: (x, y) ↦ (a*x^2 + a*y^2 + x)/(x^2 + y^2)
on V: (u, v) ↦ a + u

>>> from sage.all import *
>>> h = CM(a+x); h.display()
M → ℝ
on U: (x, y) ↦ a + x
on W: (u, v) ↦ (a*u^2 + a*v^2 + u)/(u^2 + v^2)
>>> h = CM(a+u); h.display()
M → ℝ
on W: (x, y) ↦ (a*x^2 + a*y^2 + x)/(x^2 + y^2)
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

>>> from sage.all import *
>>> h = CM(x+u); h.display()
M → ℝ
Element[source]

alias of ScalarField

one()[source]

Return the unit element of the algebra.

This is nothing but the constant scalar field \(1\) on the manifold, where \(1\) is the unit element of the base field.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: CM = M.scalar_field_algebra()
sage: h = CM.one(); h
Scalar field 1 on the 2-dimensional topological manifold M
sage: h.display()
1: M → ℝ
   (x, y) ↦ 1

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> CM = M.scalar_field_algebra()
>>> h = CM.one(); h
Scalar field 1 on the 2-dimensional topological manifold M
>>> h.display()
1: M → ℝ
   (x, y) ↦ 1

The result is cached:

sage: CM.one() is h
True

>>> from sage.all import *
>>> CM.one() is h
True
zero()[source]

Return the zero element of the algebra.

This is nothing but the constant scalar field \(0\) on the manifold, where \(0\) is the zero element of the base field.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='topological')
sage: X.<x,y> = M.chart()
sage: CM = M.scalar_field_algebra()
sage: z = CM.zero(); z
Scalar field zero on the 2-dimensional topological manifold M
sage: z.display()
zero: M → ℝ
   (x, y) ↦ 0

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='topological')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> CM = M.scalar_field_algebra()
>>> z = CM.zero(); z
Scalar field zero on the 2-dimensional topological manifold M
>>> z.display()
zero: M → ℝ
   (x, y) ↦ 0

The result is cached:

sage: CM.zero() is z
True

>>> from sage.all import *
>>> CM.zero() is z
True