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 (20142015): initial version
REFERENCES:

class
sage.manifolds.differentiable.scalarfield_algebra.
DiffScalarFieldAlgebra
(domain)¶ Bases:
sage.manifolds.scalarfield_algebra.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 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 differentiable scalar fields on the 2dimensional 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 2dimensional 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() 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 \(C^k(M)\) are scalar fields on \(M\):
sage: CM.an_element() Scalar field on the 2dimensional 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
Those of \(C^k(W)\) are scalar fields on \(W\):
sage: CW.an_element() Scalar field on the Open subset W of the 2dimensional differentiable 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 differentiable 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 differentiable 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 differentiable 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 differentiable manifold M sage: 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 2dimensional 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 2dimensional differentiable 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 differentiable scalar fields on the 2dimensional differentiable manifold M sage: f1 == f True sage: M.scalar_field(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
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 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)
sage: 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 2dimensional 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)
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 2dimensional 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
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 OF THE ALGEBRA LAWS:
Ring laws:
sage: h = CM(pi*sqrt(2)) sage: s = f + h ; s Scalar field on the 2dimensional 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)
sage: s = f  h ; s Scalar field on the 2dimensional 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)
sage: s = f*h ; s Scalar field on the 2dimensional 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))
sage: s = f/h ; s Scalar field on the 2dimensional 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
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 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
Multiplication by a number:
sage: s = 2*f ; s Scalar field on the 2dimensional 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)
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 \(C^k(W)\):
sage: TestSuite(CW).run()

Element
¶ alias of
DiffScalarField