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 (20142015): initial version
 Travis Scrimshaw (2016): review tweaks
REFERENCES:

class
sage.manifolds.scalarfield_algebra.
ScalarFieldAlgebra
(domain)¶ Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.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 2dimensional 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 2dimensional 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 2dimensional 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() 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 \(C^0(M)\) are scalar fields on \(M\):
sage: CM.an_element() Scalar field on the 2dimensional 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
Those of \(C^0(W)\) are scalar fields on \(W\):
sage: CW.an_element() Scalar field on the Open subset W of the 2dimensional topological 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 2dimensional topological 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 2dimensional topological 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 2dimensional topological 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 2dimensional topological manifold M sage: 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 2dimensional 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 2dimensional topological 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
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 2dimensional topological manifold M sage: f1 == f True sage: M.scalar_field(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
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 2dimensional 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)
sage: 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 2dimensional 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)
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 expressions
is performed by the operations*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 2dimensional 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
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
TESTS:
Ring laws:
sage: h = CM(pi*sqrt(2)) sage: s = f + h ; s Scalar field on the 2dimensional topological 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 2dimensional topological 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 2dimensional topological 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 2dimensional topological 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*sqrt(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 2dimensional topological 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 real number:
sage: s = 2*f ; s Scalar field on the 2dimensional topological 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 \(C^0(W)\):
sage: TestSuite(CW).run()

Element
¶ alias of
ScalarField

one
()¶ 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 2dimensional topological manifold M sage: h.display() 1: M > R (x, y) > 1
The result is cached:
sage: CM.one() is h True

zero
()¶ 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.
EXAMPLE:
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 2dimensional topological manifold M sage: z.display() zero: M > R (x, y) > 0
The result is cached:
sage: CM.zero() is z True