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:
ScalarFieldAlgebraCommutative 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 classDiffScalarFieldAlgebrainherits fromParent, with the category set toCommutativeAlgebras. The corresponding element class isDiffScalarField.INPUT:
domain– the differentiable manifold \(M\) on which the scalar fields are defined (must be an instance of classDifferentiableManifold)
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
>>> 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: 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 \(C^k(M)\) 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
>>> 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: 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
>>> 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: 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
>>> 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: 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
>>> 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: 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
>>> 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: 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
>>> 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: 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
>>> 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: 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 differentiable scalar fields on the 2-dimensional differentiable 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 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: 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 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)
>>> 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: 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 \(C^k(W)\) 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)
>>> 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 algebraCM, the coercion of a symbolic expressionsis performed by the operations*CM.one(), which invokes the reflected multiplication operatorsage.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
>>> 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: 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
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)
>>> 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: 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)
>>> 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: 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))
>>> 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: 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
>>> 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: f*(h+f) == f*h + f*f True
>>> from sage.all import * >>> 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
>>> 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: 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)
>>> 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: 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
>>> 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: TestSuite(CM).run()
>>> from sage.all import * >>> TestSuite(CM).run()
It is passed also for \(C^k(W)\):
sage: TestSuite(CW).run()
>>> from sage.all import * >>> TestSuite(CW).run()
- Element[source]¶
alias of
DiffScalarField