Differentiable Scalar Fields

Given a differentiable manifold \(M\) of class \(C^k\) over a topological field \(K\) (in most applications, \(K = \RR\) or \(K = \CC\)), a differentiable scalar field on \(M\) is a map

\[f: M \longrightarrow K\]

of class \(C^k\).

Differentiable scalar fields are implemented by the class DiffScalarField.

AUTHORS:

  • Eric Gourgoulhon, Michal Bejger (2013-2015): initial version

REFERENCES:

[1]S. Kobayashi & K. Nomizu : Foundations of Differential Geometry, vol. 1, Interscience Publishers (New York) (1963)
[2]J.M. Lee : Introduction to Smooth Manifolds, 2nd ed., Springer (New York) (2013)
[3]B. O’Neill : Semi-Riemannian Geometry, Academic Press (San Diego) (1983)
class sage.manifolds.differentiable.scalarfield.DiffScalarField(parent, coord_expression=None, chart=None, name=None, latex_name=None)

Bases: sage.manifolds.scalarfield.ScalarField

Differentiable scalar field on a differentiable manifold.

Given a differentiable manifold \(M\) of class \(C^k\) over a topological field \(K\) (in most applications, \(K = \RR\) or \(K = \CC\)), a differentiable scalar field defined on \(M\) is a map

\[f: M \longrightarrow K\]

that is \(k\)-times continuously differentiable.

The class DiffScalarField is a Sage element class, whose parent class is DiffScalarFieldAlgebra. It inherits from the class ScalarField devoted to generic continuous scalar fields on topological manifolds.

INPUT:

  • parent – the algebra of scalar fields containing the scalar field (must be an instance of class DiffScalarFieldAlgebra)

  • coord_expression – (default: None) coordinate expression(s) of the scalar field; this can be either

    • a dictionary of coordinate expressions in various charts on the domain, with the charts as keys;
    • a single coordinate expression; if the argument chart is 'all', this expression is set to all the charts defined on the open set; otherwise, the expression is set in the specific chart provided by the argument chart

    NB: If coord_expression is None or incomplete, coordinate expressions can be added after the creation of the object, by means of the methods add_expr(), add_expr_by_continuation() and set_expr()

  • chart – (default: None) chart defining the coordinates used in coord_expression when the latter is a single coordinate expression; if none is provided (default), the default chart of the open set is assumed. If chart=='all', coord_expression is assumed to be independent of the chart (constant scalar field).

  • name – (default: None) string; name (symbol) given to the scalar field

  • latex_name – (default: None) string; LaTeX symbol to denote the scalar field; if none is provided, the LaTeX symbol is set to name

EXAMPLES:

A scalar field on the 2-sphere:

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: f = M.scalar_field({c_xy: 1/(1+x^2+y^2), c_uv: (u^2+v^2)/(1+u^2+v^2)},
....:                    name='f') ; f
Scalar field f on the 2-dimensional differentiable manifold M
sage: f.display()
f: M --> R
on U: (x, y) |--> 1/(x^2 + y^2 + 1)
on V: (u, v) |--> (u^2 + v^2)/(u^2 + v^2 + 1)

For scalar fields defined by a single coordinate expression, the latter can be passed instead of the dictionary over the charts:

sage: g = U.scalar_field(x*y, chart=c_xy, name='g') ; g
Scalar field g on the Open subset U of the 2-dimensional differentiable
 manifold M

The above is indeed equivalent to:

sage: g = U.scalar_field({c_xy: x*y}, name='g') ; g
Scalar field g on the Open subset U of the 2-dimensional differentiable
 manifold M

Since c_xy is the default chart of U, the argument chart can be skipped:

sage: g = U.scalar_field(x*y, name='g') ; g
Scalar field g on the Open subset U of the 2-dimensional differentiable
 manifold M

The scalar field \(g\) is defined on \(U\) and has an expression in terms of the coordinates \((u,v)\) on \(W=U\cap V\):

sage: g.display()
g: U --> R
   (x, y) |--> x*y
on W: (u, v) |--> u*v/(u^4 + 2*u^2*v^2 + v^4)

Scalar fields on \(M\) can also be declared with a single chart:

sage: f = M.scalar_field(1/(1+x^2+y^2), chart=c_xy, name='f') ; f
Scalar field f on the 2-dimensional differentiable manifold M

Their definition must then be completed by providing the expressions on other charts, via the method add_expr(), to get a global cover of the manifold:

sage: f.add_expr((u^2+v^2)/(1+u^2+v^2), chart=c_uv)
sage: f.display()
f: M --> R
on U: (x, y) |--> 1/(x^2 + y^2 + 1)
on V: (u, v) |--> (u^2 + v^2)/(u^2 + v^2 + 1)

We can even first declare the scalar field without any coordinate expression and provide them subsequently:

sage: f = M.scalar_field(name='f')
sage: f.add_expr(1/(1+x^2+y^2), chart=c_xy)
sage: f.add_expr((u^2+v^2)/(1+u^2+v^2), chart=c_uv)
sage: f.display()
f: M --> R
on U: (x, y) |--> 1/(x^2 + y^2 + 1)
on V: (u, v) |--> (u^2 + v^2)/(u^2 + v^2 + 1)

We may also use the method add_expr_by_continuation() to complete the coordinate definition using the analytic continuation from domains in which charts overlap:

sage: f = M.scalar_field(1/(1+x^2+y^2), chart=c_xy, name='f') ; f
Scalar field f on the 2-dimensional differentiable manifold M
sage: f.add_expr_by_continuation(c_uv, U.intersection(V))
sage: f.display()
f: M --> R
on U: (x, y) |--> 1/(x^2 + y^2 + 1)
on V: (u, v) |--> (u^2 + v^2)/(u^2 + v^2 + 1)

A scalar field can also be defined by some unspecified function of the coordinates:

sage: h = U.scalar_field(function('H')(x, y), name='h') ; h
Scalar field h on the Open subset U of the 2-dimensional differentiable
 manifold M
sage: h.display()
h: U --> R
   (x, y) |--> H(x, y)
on W: (u, v) |--> H(u/(u^2 + v^2), v/(u^2 + v^2))

We may use the argument latex_name to specify the LaTeX symbol denoting the scalar field if the latter is different from name:

sage: latex(f)
f
sage: f = M.scalar_field({c_xy: 1/(1+x^2+y^2), c_uv: (u^2+v^2)/(1+u^2+v^2)},
....:                    name='f', latex_name=r'\mathcal{F}')
sage: latex(f)
\mathcal{F}

The coordinate expression in a given chart is obtained via the method expr(), which returns a symbolic expression:

sage: f.expr(c_uv)
(u^2 + v^2)/(u^2 + v^2 + 1)
sage: type(f.expr(c_uv))
<type 'sage.symbolic.expression.Expression'>

The method coord_function() returns instead a function of the chart coordinates, i.e. an instance of CoordFunction:

sage: f.coord_function(c_uv)
(u^2 + v^2)/(u^2 + v^2 + 1)
sage: type(f.coord_function(c_uv))
<class 'sage.manifolds.coord_func_symb.CoordFunctionSymbRing_with_category.element_class'>
sage: f.coord_function(c_uv).display()
(u, v) |--> (u^2 + v^2)/(u^2 + v^2 + 1)

The value returned by the method expr() is actually the coordinate expression of the chart function:

sage: f.expr(c_uv) is f.coord_function(c_uv).expr()
True

A constant scalar field is declared by setting the argument chart to 'all':

sage: c = M.scalar_field(2, chart='all', name='c') ; c
Scalar field c on the 2-dimensional differentiable manifold M
sage: c.display()
c: M --> R
on U: (x, y) |--> 2
on V: (u, v) |--> 2

A shortcut is to use the method constant_scalar_field():

sage: c == M.constant_scalar_field(2)
True

The constant value can be some unspecified parameter:

sage: var('a')
a
sage: c = M.constant_scalar_field(a, name='c') ; c
Scalar field c on the 2-dimensional differentiable manifold M
sage: c.display()
c: M --> R
on U: (x, y) |--> a
on V: (u, v) |--> a

A special case of constant field is the zero scalar field:

sage: zer = M.constant_scalar_field(0) ; zer
Scalar field zero on the 2-dimensional differentiable manifold M
sage: zer.display()
zero: M --> R
on U: (x, y) |--> 0
on V: (u, v) |--> 0

It can be obtained directly by means of the function zero_scalar_field():

sage: zer is M.zero_scalar_field()
True

A third way is to get it as the zero element of the algebra \(C^k(M)\) of scalar fields on \(M\) (see below):

sage: zer is M.scalar_field_algebra().zero()
True

By definition, a scalar field acts on the manifold’s points, sending them to elements of the manifold’s base field (real numbers in the present case):

sage: N = M.point((0,0), chart=c_uv) # the North pole
sage: S = M.point((0,0), chart=c_xy) # the South pole
sage: E = M.point((1,0), chart=c_xy) # a point at the equator
sage: f(N)
0
sage: f(S)
1
sage: f(E)
1/2
sage: h(E)
H(1, 0)
sage: c(E)
a
sage: zer(E)
0

A scalar field can be compared to another scalar field:

sage: f == g
False

...to a symbolic expression:

sage: f == x*y
False
sage: g == x*y
True
sage: c == a
True

...to a number:

sage: f == 2
False
sage: zer == 0
True

...to anything else:

sage: f == M
False

Standard mathematical functions are implemented:

sage: sqrt(f)
Scalar field sqrt(f) on the 2-dimensional differentiable manifold M
sage: sqrt(f).display()
sqrt(f): M --> R
on U: (x, y) |--> 1/sqrt(x^2 + y^2 + 1)
on V: (u, v) |--> sqrt(u^2 + v^2)/sqrt(u^2 + v^2 + 1)
sage: tan(f)
Scalar field tan(f) on the 2-dimensional differentiable manifold M
sage: tan(f).display()
tan(f): M --> R
on U: (x, y) |--> sin(1/(x^2 + y^2 + 1))/cos(1/(x^2 + y^2 + 1))
on V: (u, v) |--> sin((u^2 + v^2)/(u^2 + v^2 + 1))/cos((u^2 + v^2)/(u^2 + v^2 + 1))

Arithmetics of scalar fields

Scalar fields on \(M\) (resp. \(U\)) belong to the algebra \(C^k(M)\) (resp. \(C^k(U)\)):

sage: f.parent()
Algebra of differentiable scalar fields on the 2-dimensional
 differentiable manifold M
sage: f.parent() is M.scalar_field_algebra()
True
sage: g.parent()
Algebra of differentiable scalar fields on the Open subset U of the
 2-dimensional differentiable manifold M
sage: g.parent() is U.scalar_field_algebra()
True

Consequently, scalar fields can be added:

sage: s = f + c ; s
Scalar field f+c on the 2-dimensional differentiable manifold M
sage: s.display()
f+c: M --> R
on U: (x, y) |--> (a*x^2 + a*y^2 + a + 1)/(x^2 + y^2 + 1)
on V: (u, v) |--> ((a + 1)*u^2 + (a + 1)*v^2 + a)/(u^2 + v^2 + 1)

and subtracted:

sage: s = f - c ; s
Scalar field f-c on the 2-dimensional differentiable manifold M
sage: s.display()
f-c: M --> R
on U: (x, y) |--> -(a*x^2 + a*y^2 + a - 1)/(x^2 + y^2 + 1)
on V: (u, v) |--> -((a - 1)*u^2 + (a - 1)*v^2 + a)/(u^2 + v^2 + 1)

Some tests:

sage: f + zer == f
True
sage: f - f == zer
True
sage: f + (-f) == zer
True
sage: (f+c)-f == c
True
sage: (f-c)+c == f
True

We may add a number (interpreted as a constant scalar field) to a scalar field:

sage: s = f + 1 ; s
Scalar field on the 2-dimensional differentiable manifold M
sage: s.display()
M --> R
on U: (x, y) |--> (x^2 + y^2 + 2)/(x^2 + y^2 + 1)
on V: (u, v) |--> (2*u^2 + 2*v^2 + 1)/(u^2 + v^2 + 1)
sage: (f+1)-1 == f
True

The number can represented by a symbolic variable:

sage: s = a + f ; s
Scalar field on the 2-dimensional differentiable manifold M
sage: s == c + f
True

However if the symbolic variable is a chart coordinate, the addition is performed only on the chart domain:

sage: s = f + x; s
Scalar field on the 2-dimensional differentiable manifold M
sage: s.display()
M --> R
on U: (x, y) |--> (x^3 + x*y^2 + x + 1)/(x^2 + y^2 + 1)
sage: s = f + u; s
Scalar field on the 2-dimensional differentiable manifold M
sage: s.display()
M --> R
on V: (u, v) |--> (u^3 + (u + 1)*v^2 + u^2 + u)/(u^2 + v^2 + 1)

The addition of two scalar fields with different domains is possible if the domain of one of them is a subset of the domain of the other; the domain of the result is then this subset:

sage: f.domain()
2-dimensional differentiable manifold M
sage: g.domain()
Open subset U of the 2-dimensional differentiable manifold M
sage: s = f + g ; s
Scalar field on the Open subset U of the 2-dimensional differentiable
 manifold M
sage: s.domain()
Open subset U of the 2-dimensional differentiable manifold M
sage: s.display()
U --> R
(x, y) |--> (x*y^3 + (x^3 + x)*y + 1)/(x^2 + y^2 + 1)
on W: (u, v) |--> (u^6 + 3*u^4*v^2 + 3*u^2*v^4 + v^6 + u*v^3
 + (u^3 + u)*v)/(u^6 + v^6 + (3*u^2 + 1)*v^4 + u^4 + (3*u^4 + 2*u^2)*v^2)

The operation actually performed is \(f|_U + g\):

sage: s == f.restrict(U) + g
True

In Sage framework, the addition of \(f\) and \(g\) is permitted because there is a coercion of the parent of \(f\), namely \(C^k(M)\), to the parent of \(g\), namely \(C^k(U)\) (see DiffScalarFieldAlgebra):

sage: CM = M.scalar_field_algebra()
sage: CU = U.scalar_field_algebra()
sage: CU.has_coerce_map_from(CM)
True

The coercion map is nothing but the restriction to domain \(U\):

sage: CU.coerce(f) == f.restrict(U)
True

Since the algebra \(C^k(M)\) is a vector space over \(\RR\), scalar fields can be multiplied by a number, either an explicit one:

sage: s = 2*f ; s
Scalar field on the 2-dimensional differentiable manifold M
sage: s.display()
M --> R
on U: (x, y) |--> 2/(x^2 + y^2 + 1)
on V: (u, v) |--> 2*(u^2 + v^2)/(u^2 + v^2 + 1)

or a symbolic one:

sage: s = a*f ; s
Scalar field on the 2-dimensional differentiable manifold M
sage: s.display()
M --> R
on U: (x, y) |--> a/(x^2 + y^2 + 1)
on V: (u, v) |--> (u^2 + v^2)*a/(u^2 + v^2 + 1)

However, if the symbolic variable is a chart coordinate, the multiplication is performed only in the corresponding chart:

sage: s = x*f; s
Scalar field on the 2-dimensional differentiable manifold M
sage: s.display()
M --> R
on U: (x, y) |--> x/(x^2 + y^2 + 1)
sage: s = u*f; s
Scalar field on the 2-dimensional differentiable manifold M
sage: s.display()
M --> R
on V: (u, v) |--> (u^2 + v^2)*u/(u^2 + v^2 + 1)

Some tests:

sage: 0*f == 0
True
sage: 0*f == zer
True
sage: 1*f == f
True
sage: (-2)*f == - f - f
True

The ring multiplication of the algebras \(C^k(M)\) and \(C^k(U)\) is the pointwise multiplication of functions:

sage: s = f*f ; s
Scalar field f*f on the 2-dimensional differentiable manifold M
sage: s.display()
f*f: M --> R
on U: (x, y) |--> 1/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1)
on V: (u, v) |--> (u^4 + 2*u^2*v^2 + v^4)/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1)
sage: s = g*h ; s
Scalar field g*h on the Open subset U of the 2-dimensional
 differentiable manifold M
sage: s.display()
g*h: U --> R
   (x, y) |--> x*y*H(x, y)
on W: (u, v) |--> u*v*H(u/(u^2 + v^2), v/(u^2 + v^2))/(u^4 + 2*u^2*v^2 + v^4)

Thanks to the coercion \(C^k(M)\rightarrow C^k(U)\) mentionned above, it is possible to multiply a scalar field defined on \(M\) by a scalar field defined on \(U\), the result being a scalar field defined on \(U\):

sage: f.domain(), g.domain()
(2-dimensional differentiable manifold M,
 Open subset U of the 2-dimensional differentiable manifold M)
sage: s = f*g ; s
Scalar field on the Open subset U of the 2-dimensional differentiable
 manifold M
sage: s.display()
U --> R
(x, y) |--> x*y/(x^2 + y^2 + 1)
on W: (u, v) |--> u*v/(u^4 + v^4 + (2*u^2 + 1)*v^2 + u^2)
sage: s == f.restrict(U)*g
True

Scalar fields can be divided (pointwise division):

sage: s = f/c ; s
Scalar field f/c on the 2-dimensional differentiable manifold M
sage: s.display()
f/c: M --> R
on U: (x, y) |--> 1/(a*x^2 + a*y^2 + a)
on V: (u, v) |--> (u^2 + v^2)/(a*u^2 + a*v^2 + a)
sage: s = g/h ; s
Scalar field g/h on the Open subset U of the 2-dimensional
 differentiable manifold M
sage: s.display()
g/h: U --> R
   (x, y) |--> x*y/H(x, y)
on W: (u, v) |--> u*v/((u^4 + 2*u^2*v^2 + v^4)*H(u/(u^2 + v^2), v/(u^2 + v^2)))
sage: s = f/g ; s
Scalar field on the Open subset U of the 2-dimensional differentiable
 manifold M
sage: s.display()
U --> R
(x, y) |--> 1/(x*y^3 + (x^3 + x)*y)
on W: (u, v) |--> (u^6 + 3*u^4*v^2 + 3*u^2*v^4 + v^6)/(u*v^3 + (u^3 + u)*v)
sage: s == f.restrict(U)/g
True

For scalar fields defined on a single chart domain, we may perform some arithmetics with symbolic expressions involving the chart coordinates:

sage: s = g + x^2 - y ; s
Scalar field on the Open subset U of the 2-dimensional differentiable
 manifold M
sage: s.display()
U --> R
(x, y) |--> x^2 + (x - 1)*y
on W: (u, v) |--> -(v^3 - u^2 + (u^2 - u)*v)/(u^4 + 2*u^2*v^2 + v^4)
sage: s = g*x ; s
Scalar field on the Open subset U of the 2-dimensional differentiable
 manifold M
sage: s.display()
U --> R
(x, y) |--> x^2*y
on W: (u, v) |--> u^2*v/(u^6 + 3*u^4*v^2 + 3*u^2*v^4 + v^6)
sage: s = g/x ; s
Scalar field on the Open subset U of the 2-dimensional differentiable
 manifold M
sage: s.display()
U --> R
(x, y) |--> y
on W: (u, v) |--> v/(u^2 + v^2)
sage: s = x/g ; s
Scalar field on the Open subset U of the 2-dimensional differentiable
 manifold M
sage: s.display()
U --> R
(x, y) |--> 1/y
on W: (u, v) |--> (u^2 + v^2)/v

The test suite is passed:

sage: TestSuite(f).run()
sage: TestSuite(zer).run()