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

• Eric Gourgoulhon (2018): operators gradient, Laplacian and d’Alembertian

REFERENCES:

• [KN1963]

• [Lee2013]

• [ONe1983]

class sage.manifolds.differentiable.scalarfield.DiffScalarField(parent, coord_expression=None, chart=None, name=None, latex_name=None)

Bases: 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))
<class 'sage.symbolic.expression.Expression'>

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

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.chart_func.ChartFunctionRing_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 f+1 on the 2-dimensional differentiable manifold M
sage: s.display()
f+1: 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)
on W: (u, v) ↦ (u^4 + v^4 + u^3 + (2*u^2 + u)*v^2 + u)/(u^4 + v^4 + (2*u^2 + 1)*v^2 + u^2)
sage: s = f + u; s
Scalar field on the 2-dimensional differentiable manifold M
sage: s.display()
M → ℝ
on W: (x, y) ↦ (x^3 + (x + 1)*y^2 + x^2 + x)/(x^4 + y^4 + (2*x^2 + 1)*y^2 + x^2)
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 f+g 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()
f+g: 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)
on W: (u, v) ↦ u/(u^2 + v^2 + 1)
sage: s = u*f; s
Scalar field on the 2-dimensional differentiable manifold M
sage: s.display()
M → ℝ
on W: (x, y) ↦ x/(x^4 + y^4 + (2*x^2 + 1)*y^2 + x^2)
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 \(C^k(M)\rightarrow C^k(U)\) mentioned 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 f*g on the Open subset U of the 2-dimensional
 differentiable manifold M
sage: s.display()
f*g: 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 f/g on the Open subset U of the 2-dimensional
 differentiable manifold M
sage: s.display()
f/g: 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()

bracket(other)

Return the Schouten-Nijenhuis bracket of self, considered as a multivector field of degree 0, with a multivector field.

See bracket() for details.

INPUT:

• other – a multivector field of degree \(p\)

OUTPUT:

• if \(p=0\), a zero scalar field

• if \(p=1\), an instance of DiffScalarField representing the Schouten-Nijenhuis bracket [self,other]

• if \(p\geq 2\), an instance of MultivectorField representing the Schouten-Nijenhuis bracket [self,other]

EXAMPLES:

The Schouten-Nijenhuis bracket of two scalar fields is identically zero:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: f = M.scalar_field({X: x+y^2}, name='f')
sage: g = M.scalar_field({X: y-x}, name='g')
sage: s = f.bracket(g); s
Scalar field zero on the 2-dimensional differentiable manifold M
sage: s.display()
zero: M → ℝ
   (x, y) ↦ 0

while the Schouten-Nijenhuis bracket of a scalar field \(f\) with a multivector field \(a\) is equal to minus the interior product of the differential of \(f\) with \(a\):

sage: a = M.multivector_field(2, name='a')
sage: a[0,1] = x*y ; a.display()
a = x*y ∂/∂x∧∂/∂y
sage: s = f.bracket(a); s
Vector field -i_df a on the 2-dimensional differentiable manifold M
sage: s.display()
-i_df a = 2*x*y^2 ∂/∂x - x*y ∂/∂y

See bracket() for other examples.

dalembertian(metric=None)

Return the d’Alembertian of self with respect to a given Lorentzian metric.

The d’Alembertian of a scalar field \(f\) with respect to a Lorentzian metric \(g\) is nothing but the Laplacian (see laplacian()) of \(f\) with respect to that metric:

\[\Box f = g^{ij} \nabla_i \nabla_j f = \nabla_i \nabla^i f\]

where \(\nabla\) is the Levi-Civita connection of \(g\).

Note

If the metric \(g\) is not Lorentzian, the name d’Alembertian is not appropriate and one should use laplacian() instead.

INPUT:

• metric – (default: None) the Lorentzian metric \(g\) involved in the definition of the d’Alembertian; if none is provided, the domain of self is supposed to be endowed with a default Lorentzian metric (i.e. is supposed to be Lorentzian manifold, see PseudoRiemannianManifold) and the latter is used to define the d’Alembertian

OUTPUT:

• instance of DiffScalarField representing the d’Alembertian of self

EXAMPLES:

d’Alembertian of a scalar field in Minkowski spacetime:

sage: M = Manifold(4, 'M', structure='Lorentzian')
sage: X.<t,x,y,z> = M.chart()
sage: g = M.metric()
sage: g[0,0], g[1,1], g[2,2], g[3,3] = -1, 1, 1, 1
sage: f = M.scalar_field(t + x^2 + t^2*y^3 - x*z^4, name='f')
sage: s = f.dalembertian(); s
Scalar field Box(f) on the 4-dimensional Lorentzian manifold M
sage: s.display()
Box(f): M → ℝ
   (t, x, y, z) ↦ 6*t^2*y - 2*y^3 - 12*x*z^2 + 2

The function dalembertian() from the operators module can be used instead of the method dalembertian():

sage: from sage.manifolds.operators import dalembertian
sage: dalembertian(f) == s
True

degree()

Return the degree of self, considered as a differential form or a multivector field, i.e. zero.

This trivial method is provided for consistency with the exterior calculus scheme, cf. the methods degree() (differential forms) and degree() (multivector fields).

OUTPUT:

• 0

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: f = M.scalar_field({X: x+y^2})
sage: f.degree()
0

derivative()

Return the differential of self.

OUTPUT:

• a DiffForm (or of DiffFormParal if the scalar field’s domain is parallelizable) representing the 1-form that is the differential of the scalar field

EXAMPLES:

Differential of a scalar field on a 3-dimensional differentiable manifold:

sage: M = Manifold(3, 'M')
sage: c_xyz.<x,y,z> = M.chart()
sage: f = M.scalar_field(cos(x)*z^3 + exp(y)*z^2, name='f')
sage: df = f.differential() ; df
1-form df on the 3-dimensional differentiable manifold M
sage: df.display()
df = -z^3*sin(x) dx + z^2*e^y dy + (3*z^2*cos(x) + 2*z*e^y) dz
sage: latex(df)
\mathrm{d}f
sage: df.parent()
Free module Omega^1(M) of 1-forms on the 3-dimensional
 differentiable manifold M

The result is cached, i.e. is not recomputed unless f is changed:

sage: f.differential() is df
True

Instead of invoking the method differential(), one may apply the function diff to the scalar field:

sage: diff(f) is f.differential()
True

Since the exterior derivative of a scalar field (considered a 0-form) is nothing but its differential, exterior_derivative() is an alias of differential():

sage: df = f.exterior_derivative() ; df
1-form df on the 3-dimensional differentiable manifold M
sage: df.display()
df = -z^3*sin(x) dx + z^2*e^y dy + (3*z^2*cos(x) + 2*z*e^y) dz
sage: latex(df)
\mathrm{d}f

Differential computed on a chart that is not the default one:

sage: c_uvw.<u,v,w> = M.chart()
sage: g = M.scalar_field(u*v^2*w^3, c_uvw, name='g')
sage: dg = g.differential() ; dg
1-form dg on the 3-dimensional differentiable manifold M
sage: dg._components
{Coordinate frame (M, (∂/∂u,∂/∂v,∂/∂w)): 1-index components w.r.t.
 Coordinate frame (M, (∂/∂u,∂/∂v,∂/∂w))}
sage: dg.comp(c_uvw.frame())[:, c_uvw]
[v^2*w^3, 2*u*v*w^3, 3*u*v^2*w^2]
sage: dg.display(c_uvw)
dg = v^2*w^3 du + 2*u*v*w^3 dv + 3*u*v^2*w^2 dw

The exterior derivative is nilpotent:

sage: ddf = df.exterior_derivative() ; ddf
2-form ddf on the 3-dimensional differentiable manifold M
sage: ddf == 0
True
sage: ddf[:] # for the incredule
[0 0 0]
[0 0 0]
[0 0 0]
sage: ddg = dg.exterior_derivative() ; ddg
2-form ddg on the 3-dimensional differentiable manifold M
sage: ddg == 0
True

differential()

Return the differential of self.

OUTPUT:

• a DiffForm (or of DiffFormParal if the scalar field’s domain is parallelizable) representing the 1-form that is the differential of the scalar field

EXAMPLES:

Differential of a scalar field on a 3-dimensional differentiable manifold:

sage: M = Manifold(3, 'M')
sage: c_xyz.<x,y,z> = M.chart()
sage: f = M.scalar_field(cos(x)*z^3 + exp(y)*z^2, name='f')
sage: df = f.differential() ; df
1-form df on the 3-dimensional differentiable manifold M
sage: df.display()
df = -z^3*sin(x) dx + z^2*e^y dy + (3*z^2*cos(x) + 2*z*e^y) dz
sage: latex(df)
\mathrm{d}f
sage: df.parent()
Free module Omega^1(M) of 1-forms on the 3-dimensional
 differentiable manifold M

The result is cached, i.e. is not recomputed unless f is changed:

sage: f.differential() is df
True

Instead of invoking the method differential(), one may apply the function diff to the scalar field:

sage: diff(f) is f.differential()
True

Since the exterior derivative of a scalar field (considered a 0-form) is nothing but its differential, exterior_derivative() is an alias of differential():

sage: df = f.exterior_derivative() ; df
1-form df on the 3-dimensional differentiable manifold M
sage: df.display()
df = -z^3*sin(x) dx + z^2*e^y dy + (3*z^2*cos(x) + 2*z*e^y) dz
sage: latex(df)
\mathrm{d}f

Differential computed on a chart that is not the default one:

sage: c_uvw.<u,v,w> = M.chart()
sage: g = M.scalar_field(u*v^2*w^3, c_uvw, name='g')
sage: dg = g.differential() ; dg
1-form dg on the 3-dimensional differentiable manifold M
sage: dg._components
{Coordinate frame (M, (∂/∂u,∂/∂v,∂/∂w)): 1-index components w.r.t.
 Coordinate frame (M, (∂/∂u,∂/∂v,∂/∂w))}
sage: dg.comp(c_uvw.frame())[:, c_uvw]
[v^2*w^3, 2*u*v*w^3, 3*u*v^2*w^2]
sage: dg.display(c_uvw)
dg = v^2*w^3 du + 2*u*v*w^3 dv + 3*u*v^2*w^2 dw

The exterior derivative is nilpotent:

sage: ddf = df.exterior_derivative() ; ddf
2-form ddf on the 3-dimensional differentiable manifold M
sage: ddf == 0
True
sage: ddf[:] # for the incredule
[0 0 0]
[0 0 0]
[0 0 0]
sage: ddg = dg.exterior_derivative() ; ddg
2-form ddg on the 3-dimensional differentiable manifold M
sage: ddg == 0
True

exterior_derivative()

Return the differential of self.

OUTPUT:

• a DiffForm (or of DiffFormParal if the scalar field’s domain is parallelizable) representing the 1-form that is the differential of the scalar field

EXAMPLES:

Differential of a scalar field on a 3-dimensional differentiable manifold:

sage: M = Manifold(3, 'M')
sage: c_xyz.<x,y,z> = M.chart()
sage: f = M.scalar_field(cos(x)*z^3 + exp(y)*z^2, name='f')
sage: df = f.differential() ; df
1-form df on the 3-dimensional differentiable manifold M
sage: df.display()
df = -z^3*sin(x) dx + z^2*e^y dy + (3*z^2*cos(x) + 2*z*e^y) dz
sage: latex(df)
\mathrm{d}f
sage: df.parent()
Free module Omega^1(M) of 1-forms on the 3-dimensional
 differentiable manifold M

The result is cached, i.e. is not recomputed unless f is changed:

sage: f.differential() is df
True

Instead of invoking the method differential(), one may apply the function diff to the scalar field:

sage: diff(f) is f.differential()
True

Since the exterior derivative of a scalar field (considered a 0-form) is nothing but its differential, exterior_derivative() is an alias of differential():

sage: df = f.exterior_derivative() ; df
1-form df on the 3-dimensional differentiable manifold M
sage: df.display()
df = -z^3*sin(x) dx + z^2*e^y dy + (3*z^2*cos(x) + 2*z*e^y) dz
sage: latex(df)
\mathrm{d}f

Differential computed on a chart that is not the default one:

sage: c_uvw.<u,v,w> = M.chart()
sage: g = M.scalar_field(u*v^2*w^3, c_uvw, name='g')
sage: dg = g.differential() ; dg
1-form dg on the 3-dimensional differentiable manifold M
sage: dg._components
{Coordinate frame (M, (∂/∂u,∂/∂v,∂/∂w)): 1-index components w.r.t.
 Coordinate frame (M, (∂/∂u,∂/∂v,∂/∂w))}
sage: dg.comp(c_uvw.frame())[:, c_uvw]
[v^2*w^3, 2*u*v*w^3, 3*u*v^2*w^2]
sage: dg.display(c_uvw)
dg = v^2*w^3 du + 2*u*v*w^3 dv + 3*u*v^2*w^2 dw

The exterior derivative is nilpotent:

sage: ddf = df.exterior_derivative() ; ddf
2-form ddf on the 3-dimensional differentiable manifold M
sage: ddf == 0
True
sage: ddf[:] # for the incredule
[0 0 0]
[0 0 0]
[0 0 0]
sage: ddg = dg.exterior_derivative() ; ddg
2-form ddg on the 3-dimensional differentiable manifold M
sage: ddg == 0
True

gradient(metric=None)

Return the gradient of self (with respect to a given metric).

The gradient of a scalar field \(f\) with respect to a metric \(g\) is the vector field \(\mathrm{grad}\, f\) whose components in any coordinate frame are

\[(\mathrm{grad}\, f)^i = g^{ij} \frac{\partial F}{\partial x^j}\]

where the \(x^j\)’s are the coordinates with respect to which the frame is defined and \(F\) is the chart function representing \(f\) in these coordinates: \(f(p) = F(x^1(p),\ldots,x^n(p))\) for any point \(p\) in the chart domain. In other words, the gradient of \(f\) is the vector field that is the \(g\)-dual of the differential of \(f\).

INPUT:

• metric – (default: None) the pseudo-Riemannian metric \(g\) involved in the definition of the gradient; if none is provided, the domain of self is supposed to be endowed with a default metric (i.e. is supposed to be pseudo-Riemannian manifold, see PseudoRiemannianManifold) and the latter is used to define the gradient

OUTPUT:

• instance of VectorField representing the gradient of self

EXAMPLES:

Gradient of a scalar field in the Euclidean plane:

sage: M.<x,y> = EuclideanSpace()
sage: f = M.scalar_field(cos(x*y), name='f')
sage: v = f.gradient(); v
Vector field grad(f) on the Euclidean plane E^2
sage: v.display()
grad(f) = -y*sin(x*y) e_x - x*sin(x*y) e_y
sage: v[:]
[-y*sin(x*y), -x*sin(x*y)]

Gradient in polar coordinates:

sage: M.<r,phi> = EuclideanSpace(coordinates='polar')
sage: f = M.scalar_field(r*cos(phi), name='f')
sage: f.gradient().display()
grad(f) = cos(phi) e_r - sin(phi) e_phi
sage: f.gradient()[:]
[cos(phi), -sin(phi)]

Note that (e_r, e_phi) is the orthonormal vector frame associated with polar coordinates (see polar_frame()); the gradient expressed in the coordinate frame is:

sage: f.gradient().display(M.polar_coordinates().frame())
grad(f) = cos(phi) ∂/∂r - sin(phi)/r ∂/∂phi

The function grad() from the operators module can be used instead of the method gradient():

sage: from sage.manifolds.operators import grad
sage: grad(f) == f.gradient()
True

The gradient can be taken with respect to a metric tensor that is not the default one:

sage: h = M.lorentzian_metric('h')
sage: h[1,1], h[2,2] = -1, 1/(1+r^2)
sage: h.display(M.polar_coordinates().frame())
h = -dr⊗dr + r^2/(r^2 + 1) dphi⊗dphi
sage: v = f.gradient(h); v
Vector field grad_h(f) on the Euclidean plane E^2
sage: v.display()
grad_h(f) = -cos(phi) e_r + (-r^2*sin(phi) - sin(phi)) e_phi

hodge_dual(nondegenerate_tensor)

Compute the Hodge dual of the scalar field with respect to some non-degenerate bilinear form (Riemannian metric or symplectic form).

If \(M\) is the domain of the scalar field (denoted by \(f\)), \(n\) is the dimension of \(M\) and \(g\) is a non-degenerate bilinear form on \(M\), the Hodge dual of \(f\) w.r.t. \(g\) is the \(n\)-form \(*f\) defined by

\[*f = f \epsilon,\]

where \(\epsilon\) is the volume \(n\)-form associated with \(g\) (see volume_form()).

INPUT:

• nondegenerate_tensor: a non-degenerate bilinear form defined on the same manifold as the current differential form; must be an instance of PseudoRiemannianMetric or SymplecticForm.

OUTPUT:

• the \(n\)-form \(*f\)

EXAMPLES:

Hodge dual of a scalar field in the Euclidean space \(R^3\):

sage: M = Manifold(3, 'M', start_index=1)
sage: X.<x,y,z> = M.chart()
sage: g = M.metric('g')
sage: g[1,1], g[2,2], g[3,3] = 1, 1, 1
sage: f = M.scalar_field(function('F')(x,y,z), name='f')
sage: sf = f.hodge_dual(g) ; sf
3-form *f on the 3-dimensional differentiable manifold M
sage: sf.display()
*f = F(x, y, z) dx∧dy∧dz
sage: ssf = sf.hodge_dual(g) ; ssf
Scalar field **f on the 3-dimensional differentiable manifold M
sage: ssf.display()
**f: M → ℝ
   (x, y, z) ↦ F(x, y, z)
sage: ssf == f # must hold for a Riemannian metric
True

Instead of calling the method hodge_dual() on the scalar field, one can invoke the method hodge_star() of the metric:

sage: f.hodge_dual(g) == g.hodge_star(f)
True

laplacian(metric=None)

Return the Laplacian of self with respect to a given metric (Laplace-Beltrami operator).

The Laplacian of a scalar field \(f\) with respect to a metric \(g\) is the scalar field

\[\Delta f = g^{ij} \nabla_i \nabla_j f = \nabla_i \nabla^i f\]

where \(\nabla\) is the Levi-Civita connection of \(g\). \(\Delta\) is also called the Laplace-Beltrami operator.

INPUT:

• metric – (default: None) the pseudo-Riemannian metric \(g\) involved in the definition of the Laplacian; if none is provided, the domain of self is supposed to be endowed with a default metric (i.e. is supposed to be pseudo-Riemannian manifold, see PseudoRiemannianManifold) and the latter is used to define the Laplacian

OUTPUT:

• instance of DiffScalarField representing the Laplacian of self

EXAMPLES:

Laplacian of a scalar field on the Euclidean plane:

sage: M.<x,y> = EuclideanSpace()
sage: f = M.scalar_field(function('F')(x,y), name='f')
sage: s = f.laplacian(); s
Scalar field Delta(f) on the Euclidean plane E^2
sage: s.display()
Delta(f): E^2 → ℝ
   (x, y) ↦ d^2(F)/dx^2 + d^2(F)/dy^2

The function laplacian() from the operators module can be used instead of the method laplacian():

sage: from sage.manifolds.operators import laplacian
sage: laplacian(f) == s
True

The Laplacian can be taken with respect to a metric tensor that is not the default one:

sage: h = M.lorentzian_metric('h')
sage: h[1,1], h[2,2] = -1, 1/(1+x^2+y^2)
sage: s = f.laplacian(h); s
Scalar field Delta_h(f) on the Euclidean plane E^2
sage: s.display()
Delta_h(f): E^2 → ℝ
   (x, y) ↦ (y^4*d^2(F)/dy^2 + y^3*d(F)/dy
   + (2*(x^2 + 1)*d^2(F)/dy^2 - d^2(F)/dx^2)*y^2
   + (x^2 + 1)*y*d(F)/dy + x*d(F)/dx - (x^2 + 1)*d^2(F)/dx^2
   + (x^4 + 2*x^2 + 1)*d^2(F)/dy^2)/(x^2 + y^2 + 1)

The Laplacian of \(f\) is equal to the divergence of the gradient of \(f\):

\[\Delta f = \mathrm{div}( \mathrm{grad}\, f )\]

Let us check this formula:

sage: s == f.gradient(h).div(h)
True

lie_der(vector)

Compute the Lie derivative with respect to a vector field.

In the present case (scalar field), the Lie derivative is equal to the scalar field resulting from the action of the vector field on the scalar field.

INPUT:

• vector – vector field with respect to which the Lie derivative is to be taken

OUTPUT:

• the scalar field that is the Lie derivative of the scalar field with respect to vector

EXAMPLES:

Lie derivative on a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: f = M.scalar_field(x^2*cos(y))
sage: v = M.vector_field(name='v')
sage: v[:] = (-y, x)
sage: f.lie_derivative(v)
Scalar field on the 2-dimensional differentiable manifold M
sage: f.lie_derivative(v).expr()
-x^3*sin(y) - 2*x*y*cos(y)

The result is cached:

sage: f.lie_derivative(v) is f.lie_derivative(v)
True

An alias is lie_der:

sage: f.lie_der(v) is f.lie_derivative(v)
True

Alternative expressions of the Lie derivative of a scalar field:

sage: f.lie_der(v) == v(f)  # the vector acting on f
True
sage: f.lie_der(v) == f.differential()(v)  # the differential of f acting on the vector
True

A vanishing Lie derivative:

sage: f.set_expr(x^2 + y^2)
sage: f.lie_der(v).display()
M → ℝ
(x, y) ↦ 0

lie_derivative(vector)

Compute the Lie derivative with respect to a vector field.

In the present case (scalar field), the Lie derivative is equal to the scalar field resulting from the action of the vector field on the scalar field.

INPUT:

• vector – vector field with respect to which the Lie derivative is to be taken

OUTPUT:

• the scalar field that is the Lie derivative of the scalar field with respect to vector

EXAMPLES:

Lie derivative on a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: f = M.scalar_field(x^2*cos(y))
sage: v = M.vector_field(name='v')
sage: v[:] = (-y, x)
sage: f.lie_derivative(v)
Scalar field on the 2-dimensional differentiable manifold M
sage: f.lie_derivative(v).expr()
-x^3*sin(y) - 2*x*y*cos(y)

The result is cached:

sage: f.lie_derivative(v) is f.lie_derivative(v)
True

An alias is lie_der:

sage: f.lie_der(v) is f.lie_derivative(v)
True

Alternative expressions of the Lie derivative of a scalar field:

sage: f.lie_der(v) == v(f)  # the vector acting on f
True
sage: f.lie_der(v) == f.differential()(v)  # the differential of f acting on the vector
True

A vanishing Lie derivative:

sage: f.set_expr(x^2 + y^2)
sage: f.lie_der(v).display()
M → ℝ
(x, y) ↦ 0

tensor_type()

Return the tensor type of self, when the latter is considered as a tensor field on the manifold. This is always \((0, 0)\).

OUTPUT:

• always \((0, 0)\)

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: f = M.scalar_field(x+2*y)
sage: f.tensor_type()
(0, 0)

wedge(other)

Return the exterior product of self, considered as a differential form of degree 0 or a multivector field of degree 0, with other.

See wedge() (exterior product of differential forms) or wedge() (exterior product of multivector fields) for details.

For a scalar field \(f\) and a \(p\)-form (or \(p\)-vector field) \(a\), the exterior product reduces to the standard product on the left by an element of the base ring of the module of \(p\)-forms (or \(p\)-vector fields): \(f\wedge a = f a\).

INPUT:

• other – a differential form or a multivector field \(a\)

OUTPUT:

• the product \(f a\), where \(f\) is self

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: f = M.scalar_field({X: x+y^2}, name='f')
sage: a = M.diff_form(2, name='a')
sage: a[0,1] = x*y
sage: s = f.wedge(a); s
2-form f*a on the 2-dimensional differentiable manifold M
sage: s.display()
f*a = (x*y^3 + x^2*y) dx∧dy