Algebra of Differentiable Scalar Fields#

The 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:

class sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra(domain)[source]#

Bases: ScalarFieldAlgebra

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 ScalarFieldAlgebra, the class DiffScalarFieldAlgebra inherits from Parent, with the category set to CommutativeAlgebras. The corresponding element class is DiffScalarField.

INPUT:

domain – the differentiable manifold \(M\) on which the scalar fields are defined (must be an instance of class DifferentiableManifold)

EXAMPLES:

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

Sage

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

Python

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M') # 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 differentiable scalar fields on the 2-dimensional
 differentiable manifold M
>>> W = U.intersection(V)  # S^2 minus the two poles
>>> 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

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

Python

>>> 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 \(C^k(M)\) are scalar fields on \(M\):

Sage

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

Python

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

Sage

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

Python

>>> from sage.all import *
>>> CW.an_element()
Scalar field on the Open subset W of the 2-dimensional differentiable
 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

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

Python

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

Sage

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

Python

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

The unit element:

Sage

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

Python

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

Sage

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

Python

>>> from sage.all import *
>>> CW.one()
Scalar field 1 on the Open subset W of the 2-dimensional differentiable
 manifold M
>>> 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

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

Python

>>> 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 differentiable 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 differentiable scalar fields on the 2-dimensional
 differentiable manifold M

Specific elements can also be constructed in this way:

Sage

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

Python

>>> 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

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

Python

>>> 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

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

Python

>>> 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 differentiable scalar fields on the 2-dimensional
 differentiable manifold M
>>> f1 == f
True
>>> M.scalar_field(Integer(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

sage: CW.has_coerce_map_from(CM)
True

Python

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

The reverse is of course false:

Sage

sage: CM.has_coerce_map_from(CW)
False

Python

>>> 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

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)

Python

>>> from sage.all import *
>>> fW = CW(f) ; fW
Scalar field on the Open subset W of the 2-dimensional differentiable
 manifold M
>>> fW.display()
W → ℝ
(x, y) ↦ arctan(x^2 + y^2)
(u, v) ↦ 1/2*pi - arctan(u^2 + v^2)

Sage

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

Python

>>> from sage.all import *
>>> 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

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)

Python

>>> from sage.all import *
>>> s = fW + f
>>> s.parent()
Algebra of differentiable scalar fields on the Open subset W of the
 2-dimensional differentiable 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, the parent of all symbolic expressions (cf. 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 sage.manifolds.scalarfield.ScalarField._rmul_(). If the symbolic expression does not involve any chart coordinate, the outcome is a constant scalar field:

Sage

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

Python

>>> from sage.all import *
>>> h = CM(pi*sqrt(Integer(2))) ; h
Scalar field on the 2-dimensional differentiable 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

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

Python

>>> 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

sage: h = CM(x+u); h.display()
M → ℝ

Python

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

TESTS OF THE ALGEBRA LAWS:

Ring laws:

Sage

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)

Python

>>> from sage.all import *
>>> h = CM(pi*sqrt(Integer(2)))
>>> s = f + h ; s
Scalar field on the 2-dimensional differentiable manifold M
>>> 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

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)

Python

>>> from sage.all import *
>>> s = f - h ; s
Scalar field on the 2-dimensional differentiable manifold M
>>> 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

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))

Python

>>> from sage.all import *
>>> s = f*h ; s
Scalar field on the 2-dimensional differentiable manifold M
>>> 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

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

Python

>>> from sage.all import *
>>> s = f/h ; s
Scalar field on the 2-dimensional differentiable manifold M
>>> 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

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

Python

>>> from sage.all import *
>>> f*(h+f) == f*h + f*f
True

Ring laws with coercion:

Sage

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

Python

>>> from sage.all import *
>>> f - fW == CW.zero()
True
>>> f/fW == CW.one()
True
>>> s = f*fW ; s
Scalar field on the Open subset W of the 2-dimensional differentiable
 manifold M
>>> 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
>>> s/f == fW
True

Multiplication by a number:

Sage

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)

Python

>>> from sage.all import *
>>> s = Integer(2)*f ; s
Scalar field on the 2-dimensional differentiable manifold M
>>> 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

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

Python

>>> from sage.all import *
>>> Integer(0)*f == CM.zero()
True
>>> Integer(1)*f == f
True
>>> Integer(2)*(f/Integer(2)) == f
True
>>> (f+Integer(2)*f)/Integer(3) == f
True
>>> Integer(1)/Integer(3)*(f+Integer(2)*f) == f
True

The Sage test suite for algebras is passed:

Sage

sage: TestSuite(CM).run()

Python

>>> from sage.all import *
>>> TestSuite(CM).run()

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

Sage

sage: TestSuite(CW).run()

Python

>>> from sage.all import *
>>> TestSuite(CW).run()

Element[source]#: alias of DiffScalarField