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:

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()
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 /\^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

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

One may also use the global function exterior_derivative() or its alias xder() instead of the method exterior_derivative():

sage: from sage.manifolds.utilities import xder
sage: xder(f) is f.exterior_derivative()
True

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, (d/du,d/dv,d/dw)): 1-index components w.r.t.
 Coordinate frame (M, (d/du,d/dv,d/dw))}
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.frame(), 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 /\^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

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

One may also use the global function exterior_derivative() or its alias xder() instead of the method exterior_derivative():

sage: from sage.manifolds.utilities import xder
sage: xder(f) is f.exterior_derivative()
True

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, (d/du,d/dv,d/dw)): 1-index components w.r.t.
 Coordinate frame (M, (d/du,d/dv,d/dw))}
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.frame(), 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
hodge_dual(metric)

Compute the Hodge dual of the scalar field with respect to some metric.

If \(M\) is the domain of the scalar field (denoted by \(f\)), \(n\) is the dimension of \(M\) and \(g\) is a pseudo-Riemannian metric 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:

  • metric: a pseudo-Riemannian metric defined on the same manifold as the current scalar field; must be an instance of PseudoRiemannianMetric

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
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)