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
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:
- 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 isDiffScalarFieldAlgebra
. It inherits from the classScalarField
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 classDiffScalarFieldAlgebra
)coord_expression
– (default:None
) coordinate expression(s) of the scalar field; this can be eithera 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 argumentchart
NB: If
coord_expression
isNone
or incomplete, coordinate expressions can be added after the creation of the object, by means of the methodsadd_expr()
,add_expr_by_continuation()
andset_expr()
chart
– (default:None
) chart defining the coordinates used incoord_expression
when the latter is a single coordinate expression; if none is provided (default), the default chart of the open set is assumed. Ifchart=='all'
,coord_expression
is assumed to be independent of the chart (constant scalar field).name
– (default:None
) string; name (symbol) given to the scalar fieldlatex_name
– (default:None
) string; LaTeX symbol to denote the scalar field; if none is provided, the LaTeX symbol is set toname
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 ofU
, the argumentchart
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 fromname
: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 ofChartFunction
: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 ofself
is supposed to be endowed with a default Lorentzian metric (i.e. is supposed to be Lorentzian manifold, seePseudoRiemannianManifold
) and the latter is used to define the d’Alembertian
OUTPUT:
instance of
DiffScalarField
representing the d’Alembertian ofself
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 theoperators
module can be used instead of the methoddalembertian()
: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) anddegree()
(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 ofDiffFormParal
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 functiondiff
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 ofdifferential()
: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 ofDiffFormParal
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 functiondiff
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 ofdifferential()
: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 ofDiffFormParal
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 functiondiff
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 ofdifferential()
: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 ofself
is supposed to be endowed with a default metric (i.e. is supposed to be pseudo-Riemannian manifold, seePseudoRiemannianManifold
) and the latter is used to define the gradient
OUTPUT:
instance of
VectorField
representing the gradient ofself
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 (seepolar_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 theoperators
module can be used instead of the methodgradient()
: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 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 ofPseudoRiemannianMetric
orSymplecticForm
.
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 methodhodge_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 ofself
is supposed to be endowed with a default metric (i.e. is supposed to be pseudo-Riemannian manifold, seePseudoRiemannianManifold
) and the latter is used to define the Laplacian
OUTPUT:
instance of
DiffScalarField
representing the Laplacian ofself
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 theoperators
module can be used instead of the methodlaplacian()
: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, withother
.See
wedge()
(exterior product of differential forms) orwedge()
(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