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 classDiffScalarFieldAlgebra
inherits 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 expressions
is 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