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:

[KN1963]
[Lee2013]
[ONe1983]

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 $(0, 0)$

EXAMPLE:

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)

Differentiable Scalar Fields¶

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