Vector Frames#
The class VectorFrame
implements vector frames on differentiable
manifolds.
By vector frame, it is meant a field \(e\) on some
differentiable manifold \(U\) endowed with a differentiable map
\(\Phi: U \rightarrow M\) to a differentiable manifold \(M\) such that for each
\(p\in U\), \(e(p)\) is a vector basis of the tangent space \(T_{\Phi(p)}M\).
The standard case of a vector frame on \(U\) corresponds to \(U = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(M\) (\(U\) is then an open interval of \(\RR\)).
A derived class of VectorFrame
is CoordFrame
;
it regards the vector frames associated with a chart, i.e. the
so-called coordinate bases.
The vector frame duals, i.e. the coframes, are implemented via the class
CoFrame
. The derived class CoordCoFrame
is devoted to
coframes deriving from a chart.
AUTHORS:
Eric Gourgoulhon, Michal Bejger (2013-2015): initial version
Travis Scrimshaw (2016): review tweaks
Eric Gourgoulhon (2018): some refactoring and more functionalities in the choice of symbols for vector frame elements (github issue #24792)
REFERENCES:
EXAMPLES:
Introducing a chart on a manifold automatically endows it with a vector frame: the coordinate frame associated to the chart:
sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: M.frames()
[Coordinate frame (M, (∂/∂x,∂/∂y,∂/∂z))]
sage: M.frames()[0] is X.frame()
True
A new vector frame can be defined from a family of 3 linearly independent vector fields:
sage: e1 = M.vector_field(1, x, y)
sage: e2 = M.vector_field(z, -2, x*y)
sage: e3 = M.vector_field(1, 1, 0)
sage: e = M.vector_frame('e', (e1, e2, e3)); e
Vector frame (M, (e_0,e_1,e_2))
sage: latex(e)
\left(M, \left(e_{0},e_{1},e_{2}\right)\right)
The first frame defined on a manifold is its default frame; in the present
case it is the coordinate frame associated to the chart X
:
sage: M.default_frame()
Coordinate frame (M, (∂/∂x,∂/∂y,∂/∂z))
The default frame can be changed via the method
set_default_frame()
:
sage: M.set_default_frame(e)
sage: M.default_frame()
Vector frame (M, (e_0,e_1,e_2))
The elements of a vector frame are vector fields on the manifold:
sage: for vec in e:
....: print(vec)
....:
Vector field e_0 on the 3-dimensional differentiable manifold M
Vector field e_1 on the 3-dimensional differentiable manifold M
Vector field e_2 on the 3-dimensional differentiable manifold M
Each element of a vector frame can be accessed by its index:
sage: e[0]
Vector field e_0 on the 3-dimensional differentiable manifold M
sage: e[0].display(X.frame())
e_0 = ∂/∂x + x ∂/∂y + y ∂/∂z
sage: X.frame()[1]
Vector field ∂/∂y on the 3-dimensional differentiable manifold M
sage: X.frame()[1].display(e)
∂/∂y = x/(x^2 - x + z + 2) e_0 - 1/(x^2 - x + z + 2) e_1
- (x - z)/(x^2 - x + z + 2) e_2
The slice operator :
can be used to access to more than one element:
sage: e[0:2]
(Vector field e_0 on the 3-dimensional differentiable manifold M,
Vector field e_1 on the 3-dimensional differentiable manifold M)
sage: e[:]
(Vector field e_0 on the 3-dimensional differentiable manifold M,
Vector field e_1 on the 3-dimensional differentiable manifold M,
Vector field e_2 on the 3-dimensional differentiable manifold M)
Vector frames can be constructed from scratch, without any connection to
previously defined frames or vector fields (the connection can be performed
later via the method
set_change_of_frame()
):
sage: f = M.vector_frame('f'); f
Vector frame (M, (f_0,f_1,f_2))
sage: M.frames()
[Coordinate frame (M, (∂/∂x,∂/∂y,∂/∂z)),
Vector frame (M, (e_0,e_1,e_2)),
Vector frame (M, (f_0,f_1,f_2))]
The index range depends on the starting index defined on the manifold:
sage: M = Manifold(3, 'M', start_index=1)
sage: X.<x,y,z> = M.chart()
sage: e = M.vector_frame('e')
sage: [e[i] for i in M.irange()]
[Vector field e_1 on the 3-dimensional differentiable manifold M,
Vector field e_2 on the 3-dimensional differentiable manifold M,
Vector field e_3 on the 3-dimensional differentiable manifold M]
sage: e[1], e[2], e[3]
(Vector field e_1 on the 3-dimensional differentiable manifold M,
Vector field e_2 on the 3-dimensional differentiable manifold M,
Vector field e_3 on the 3-dimensional differentiable manifold M)
Let us check that the vector fields e[i]
are the frame vectors from
their components with respect to the frame \(e\):
sage: e[1].comp(e)[:]
[1, 0, 0]
sage: e[2].comp(e)[:]
[0, 1, 0]
sage: e[3].comp(e)[:]
[0, 0, 1]
Defining a vector frame on a manifold automatically creates the dual coframe, which, by default, bares the same name (here \(e\)):
sage: M.coframes()
[Coordinate coframe (M, (dx,dy,dz)), Coframe (M, (e^1,e^2,e^3))]
sage: f = M.coframes()[1] ; f
Coframe (M, (e^1,e^2,e^3))
sage: f is e.coframe()
True
Each element of the coframe is a 1-form:
sage: f[1], f[2], f[3]
(1-form e^1 on the 3-dimensional differentiable manifold M,
1-form e^2 on the 3-dimensional differentiable manifold M,
1-form e^3 on the 3-dimensional differentiable manifold M)
sage: latex(f[1]), latex(f[2]), latex(f[3])
(e^{1}, e^{2}, e^{3})
Let us check that the coframe \((e^i)\) is indeed the dual of the vector frame \((e_i)\):
sage: f[1](e[1]) # the 1-form e^1 applied to the vector field e_1
Scalar field e^1(e_1) on the 3-dimensional differentiable manifold M
sage: f[1](e[1]).expr() # the explicit expression of e^1(e_1)
1
sage: f[1](e[1]).expr(), f[1](e[2]).expr(), f[1](e[3]).expr()
(1, 0, 0)
sage: f[2](e[1]).expr(), f[2](e[2]).expr(), f[2](e[3]).expr()
(0, 1, 0)
sage: f[3](e[1]).expr(), f[3](e[2]).expr(), f[3](e[3]).expr()
(0, 0, 1)
The coordinate frame associated to spherical coordinates of the sphere \(S^2\):
sage: M = Manifold(2, 'S^2', start_index=1) # Part of S^2 covered by spherical coord.
sage: c_spher.<th,ph> = M.chart(r'th:[0,pi]:\theta ph:[0,2*pi):\phi')
sage: b = M.default_frame() ; b
Coordinate frame (S^2, (∂/∂th,∂/∂ph))
sage: b[1]
Vector field ∂/∂th on the 2-dimensional differentiable manifold S^2
sage: b[2]
Vector field ∂/∂ph on the 2-dimensional differentiable manifold S^2
The orthonormal frame constructed from the coordinate frame:
sage: e = M.vector_frame('e', (b[1], b[2]/sin(th))); e
Vector frame (S^2, (e_1,e_2))
sage: e[1].display()
e_1 = ∂/∂th
sage: e[2].display()
e_2 = 1/sin(th) ∂/∂ph
The change-of-frame automorphisms and their matrices:
sage: M.change_of_frame(c_spher.frame(), e)
Field of tangent-space automorphisms on the 2-dimensional
differentiable manifold S^2
sage: M.change_of_frame(c_spher.frame(), e)[:]
[ 1 0]
[ 0 1/sin(th)]
sage: M.change_of_frame(e, c_spher.frame())
Field of tangent-space automorphisms on the 2-dimensional
differentiable manifold S^2
sage: M.change_of_frame(e, c_spher.frame())[:]
[ 1 0]
[ 0 sin(th)]
- class sage.manifolds.differentiable.vectorframe.CoFrame(frame, symbol, latex_symbol=None, indices=None, latex_indices=None)#
Bases:
FreeModuleCoBasis
Coframe on a differentiable manifold.
By coframe, it is meant a field \(f\) on some differentiable manifold \(U\) endowed with a differentiable map \(\Phi: U \rightarrow M\) to a differentiable manifold \(M\) such that for each \(p\in U\), \(f(p)\) is a basis of the vector space \(T^*_{\Phi(p)}M\) (the dual to the tangent space \(T_{\Phi(p)}M\)).
The standard case of a coframe on \(U\) corresponds to \(U = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(M\) (\(U\) is then an open interval of \(\RR\)).
INPUT:
frame
– the vector frame dual to the coframesymbol
– either a string, to be used as a common base for the symbols of the 1-forms constituting the coframe, or a tuple of strings, representing the individual symbols of the 1-formslatex_symbol
– (default:None
) either a string, to be used as a common base for the LaTeX symbols of the 1-forms constituting the coframe, or a tuple of strings, representing the individual LaTeX symbols of the 1-forms; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the 1-forms of the coframe; ifNone
, the indices will be generated as integers within the range declared on the coframe’s domainlatex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the 1-forms of the coframe; ifNone
,indices
is used instead
EXAMPLES:
Coframe on a 3-dimensional manifold:
sage: M = Manifold(3, 'M', start_index=1) sage: X.<x,y,z> = M.chart() sage: v = M.vector_frame('v') sage: from sage.manifolds.differentiable.vectorframe import CoFrame sage: e = CoFrame(v, 'e') ; e Coframe (M, (e^1,e^2,e^3))
Instead of importing CoFrame in the global namespace, the coframe can be obtained by means of the method
dual_basis()
; the symbol is then the same as that of the frame:sage: a = v.dual_basis() ; a Coframe (M, (v^1,v^2,v^3)) sage: a[1] == e[1] True sage: a[1] is e[1] False sage: e[1].display(v) e^1 = v^1
The 1-forms composing the coframe are obtained via the operator
[]
:sage: e[1], e[2], e[3] (1-form e^1 on the 3-dimensional differentiable manifold M, 1-form e^2 on the 3-dimensional differentiable manifold M, 1-form e^3 on the 3-dimensional differentiable manifold M)
Checking that \(e\) is the dual of \(v\):
sage: e[1](v[1]).expr(), e[1](v[2]).expr(), e[1](v[3]).expr() (1, 0, 0) sage: e[2](v[1]).expr(), e[2](v[2]).expr(), e[2](v[3]).expr() (0, 1, 0) sage: e[3](v[1]).expr(), e[3](v[2]).expr(), e[3](v[3]).expr() (0, 0, 1)
- at(point)#
Return the value of
self
at a given point on the manifold, this value being a basis of the dual of the tangent space at the point.INPUT:
point
–ManifoldPoint
; point \(p\) in the domain \(U\) of the coframe (denoted \(f\) hereafter)
OUTPUT:
FreeModuleCoBasis
representing the basis \(f(p)\) of the vector space \(T^*_{\Phi(p)} M\), dual to the tangent space \(T_{\Phi(p)} M\), where \(\Phi: U \to M\) is the differentiable map associated with \(f\) (possibly \(\Phi = \mathrm{Id}_U\))
EXAMPLES:
Cobasis of a tangent space on a 2-dimensional manifold:
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: p = M.point((-1,2), name='p') sage: f = X.coframe() ; f Coordinate coframe (M, (dx,dy)) sage: fp = f.at(p) ; fp Dual basis (dx,dy) on the Tangent space at Point p on the 2-dimensional differentiable manifold M sage: type(fp) <class 'sage.tensor.modules.free_module_basis.FreeModuleCoBasis_with_category'> sage: fp[0] Linear form dx on the Tangent space at Point p on the 2-dimensional differentiable manifold M sage: fp[1] Linear form dy on the Tangent space at Point p on the 2-dimensional differentiable manifold M sage: fp is X.frame().at(p).dual_basis() True
- set_name(symbol, latex_symbol=None, indices=None, latex_indices=None, index_position='up', include_domain=True)#
Set (or change) the text name and LaTeX name of
self
.INPUT:
symbol
– either a string, to be used as a common base for the symbols of the 1-forms constituting the coframe, or a list/tuple of strings, representing the individual symbols of the 1-formslatex_symbol
– (default:None
) either a string, to be used as a common base for the LaTeX symbols of the 1-forms constituting the coframe, or a list/tuple of strings, representing the individual LaTeX symbols of the 1-forms; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the 1-forms of the coframe; ifNone
, the indices will be generated as integers within the range declared onself
latex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the 1-forms; ifNone
,indices
is used insteadindex_position
– (default:'up'
) determines the position of the indices labelling the 1-forms of the coframe; can be either'down'
or'up'
include_domain
– (default:True
) boolean determining whether the name of the domain is included in the beginning of the coframe name
EXAMPLES:
sage: M = Manifold(2, 'M') sage: e = M.vector_frame('e').coframe(); e Coframe (M, (e^0,e^1)) sage: e.set_name('f'); e Coframe (M, (f^0,f^1)) sage: e.set_name('e', latex_symbol=r'\epsilon') sage: latex(e) \left(M, \left(\epsilon^{0},\epsilon^{1}\right)\right) sage: e.set_name('e', include_domain=False); e Coframe (e^0,e^1) sage: e.set_name(['a', 'b'], latex_symbol=[r'\alpha', r'\beta']); e Coframe (M, (a,b)) sage: latex(e) \left(M, \left(\alpha,\beta\right)\right) sage: e.set_name('e', indices=['x','y'], ....: latex_indices=[r'\xi', r'\zeta']); e Coframe (M, (e^x,e^y)) sage: latex(e) \left(M, \left(e^{\xi},e^{\zeta}\right)\right)
- class sage.manifolds.differentiable.vectorframe.CoordCoFrame(coord_frame, symbol, latex_symbol=None, indices=None, latex_indices=None)#
Bases:
CoFrame
Coordinate coframe on a differentiable manifold.
By coordinate coframe, it is meant the \(n\)-tuple of the differentials of the coordinates of some chart on the manifold, with \(n\) being the manifold’s dimension.
INPUT:
coord_frame
– coordinate frame dual to the coordinate coframesymbol
– either a string, to be used as a common base for the symbols of the 1-forms constituting the coframe, or a tuple of strings, representing the individual symbols of the 1-formslatex_symbol
– (default:None
) either a string, to be used as a common base for the LaTeX symbols of the 1-forms constituting the coframe, or a tuple of strings, representing the individual LaTeX symbols of the 1-forms; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the 1-forms of the coframe; ifNone
, the indices will be generated as integers within the range declared on the vector frame’s domainlatex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the 1-forms of the coframe; ifNone
,indices
is used instead
EXAMPLES:
Coordinate coframe on a 3-dimensional manifold:
sage: M = Manifold(3, 'M', start_index=1) sage: X.<x,y,z> = M.chart() sage: M.frames() [Coordinate frame (M, (∂/∂x,∂/∂y,∂/∂z))] sage: M.coframes() [Coordinate coframe (M, (dx,dy,dz))] sage: dX = M.coframes()[0] ; dX Coordinate coframe (M, (dx,dy,dz))
The 1-forms composing the coframe are obtained via the operator
[]
:sage: dX[1] 1-form dx on the 3-dimensional differentiable manifold M sage: dX[2] 1-form dy on the 3-dimensional differentiable manifold M sage: dX[3] 1-form dz on the 3-dimensional differentiable manifold M sage: dX[1][:] [1, 0, 0] sage: dX[2][:] [0, 1, 0] sage: dX[3][:] [0, 0, 1]
The coframe is the dual of the coordinate frame:
sage: e = X.frame() ; e Coordinate frame (M, (∂/∂x,∂/∂y,∂/∂z)) sage: dX[1](e[1]).expr(), dX[1](e[2]).expr(), dX[1](e[3]).expr() (1, 0, 0) sage: dX[2](e[1]).expr(), dX[2](e[2]).expr(), dX[2](e[3]).expr() (0, 1, 0) sage: dX[3](e[1]).expr(), dX[3](e[2]).expr(), dX[3](e[3]).expr() (0, 0, 1)
Each 1-form of a coordinate coframe is closed:
sage: dX[1].exterior_derivative() 2-form ddx on the 3-dimensional differentiable manifold M sage: dX[1].exterior_derivative() == 0 True
- class sage.manifolds.differentiable.vectorframe.CoordFrame(chart)#
Bases:
VectorFrame
Coordinate frame on a differentiable manifold.
By coordinate frame, it is meant a vector frame on a differentiable manifold \(M\) that is associated to a coordinate chart on \(M\).
INPUT:
chart
– the chart defining the coordinates
EXAMPLES:
The coordinate frame associated to spherical coordinates of the sphere \(S^2\):
sage: M = Manifold(2, 'S^2', start_index=1) # Part of S^2 covered by spherical coord. sage: M.chart(r'th:[0,pi]:\theta ph:[0,2*pi):\phi') Chart (S^2, (th, ph)) sage: b = M.default_frame() sage: b Coordinate frame (S^2, (∂/∂th,∂/∂ph)) sage: b[1] Vector field ∂/∂th on the 2-dimensional differentiable manifold S^2 sage: b[2] Vector field ∂/∂ph on the 2-dimensional differentiable manifold S^2 sage: latex(b) \left(S^2, \left(\frac{\partial}{\partial {\theta} },\frac{\partial}{\partial {\phi} }\right)\right)
- chart()#
Return the chart defining this coordinate frame.
EXAMPLES:
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: e = X.frame() sage: e.chart() Chart (M, (x, y)) sage: U = M.open_subset('U', coord_def={X: x>0}) sage: e.restrict(U).chart() Chart (U, (x, y))
- structure_coeff()#
Return the structure coefficients associated to
self
.\(n\) being the manifold’s dimension, the structure coefficients of the frame \((e_i)\) are the \(n^3\) scalar fields \(C^k_{\ \, ij}\) defined by
\[[e_i, e_j] = C^k_{\ \, ij} e_k.\]In the present case, since \((e_i)\) is a coordinate frame, \(C^k_{\ \, ij}=0\).
OUTPUT:
the structure coefficients \(C^k_{\ \, ij}\), as a vanishing instance of
CompWithSym
with 3 indices ordered as \((k,i,j)\)
EXAMPLES:
Structure coefficients of the coordinate frame associated to spherical coordinates in the Euclidean space \(\RR^3\):
sage: M = Manifold(3, 'R^3', r'\RR^3', start_index=1) # Part of R^3 covered by spherical coord. sage: c_spher = M.chart(r'r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi') sage: b = M.default_frame() ; b Coordinate frame (R^3, (∂/∂r,∂/∂th,∂/∂ph)) sage: c = b.structure_coeff() ; c 3-indices components w.r.t. Coordinate frame (R^3, (∂/∂r,∂/∂th,∂/∂ph)), with antisymmetry on the index positions (1, 2) sage: c == 0 True
- class sage.manifolds.differentiable.vectorframe.VectorFrame(vector_field_module, symbol, latex_symbol=None, from_frame=None, indices=None, latex_indices=None, symbol_dual=None, latex_symbol_dual=None)#
Bases:
FreeModuleBasis
Vector frame on a differentiable manifold.
By vector frame, it is meant a field \(e\) on some differentiable manifold \(U\) endowed with a differentiable map \(\Phi: U\rightarrow M\) to a differentiable manifold \(M\) such that for each \(p\in U\), \(e(p)\) is a vector basis of the tangent space \(T_{\Phi(p)}M\).
The standard case of a vector frame on \(U\) corresponds to \(U=M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(M\) (\(U\) is then an open interval of \(\RR\)).
For each instantiation of a vector frame, a coframe is automatically created, as an instance of the class
CoFrame
. It is returned by the methodcoframe()
.INPUT:
vector_field_module
– free module \(\mathfrak{X}(U, \Phi)\) of vector fields along \(U\) with values on \(M \supset \Phi(U)\)symbol
– either a string, to be used as a common base for the symbols of the vector fields constituting the vector frame, or a tuple of strings, representing the individual symbols of the vector fieldslatex_symbol
– (default:None
) either a string, to be used as a common base for the LaTeX symbols of the vector fields constituting the vector frame, or a tuple of strings, representing the individual LaTeX symbols of the vector fields; ifNone
,symbol
is used in place oflatex_symbol
from_frame
– (default:None
) vector frame \(\tilde e\) on the codomain \(M\) of the destination map \(\Phi\); the constructed frame \(e\) is then such that \(\forall p \in U, e(p) = \tilde{e}(\Phi(p))\)indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the vector fields of the frame; ifNone
, the indices will be generated as integers within the range declared on the vector frame’s domainlatex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the vector fields; ifNone
,indices
is used insteadsymbol_dual
– (default:None
) same assymbol
but for the dual coframe; ifNone
,symbol
must be a string and is used for the common base of the symbols of the elements of the dual coframelatex_symbol_dual
– (default:None
) same aslatex_symbol
but for the dual coframe
EXAMPLES:
Defining a vector frame on a 3-dimensional manifold:
sage: M = Manifold(3, 'M', start_index=1) sage: X.<x,y,z> = M.chart() sage: e = M.vector_frame('e') ; e Vector frame (M, (e_1,e_2,e_3)) sage: latex(e) \left(M, \left(e_{1},e_{2},e_{3}\right)\right)
The individual elements of the vector frame are accessed via square brackets, with the possibility to invoke the slice operator ‘
:
’ to get more than a single element:sage: e[2] Vector field e_2 on the 3-dimensional differentiable manifold M sage: e[1:3] (Vector field e_1 on the 3-dimensional differentiable manifold M, Vector field e_2 on the 3-dimensional differentiable manifold M) sage: e[:] (Vector field e_1 on the 3-dimensional differentiable manifold M, Vector field e_2 on the 3-dimensional differentiable manifold M, Vector field e_3 on the 3-dimensional differentiable manifold M)
The LaTeX symbol can be specified:
sage: E = M.vector_frame('E', latex_symbol=r"\epsilon") sage: latex(E) \left(M, \left(\epsilon_{1},\epsilon_{2},\epsilon_{3}\right)\right)
By default, the elements of the vector frame are labelled by integers within the range specified at the manifold declaration. It is however possible to fully customize the labels, via the argument
indices
:sage: u = M.vector_frame('u', indices=('x', 'y', 'z')) ; u Vector frame (M, (u_x,u_y,u_z)) sage: u[1] Vector field u_x on the 3-dimensional differentiable manifold M sage: u.coframe() Coframe (M, (u^x,u^y,u^z))
The LaTeX format of the indices can be adjusted:
sage: v = M.vector_frame('v', indices=('a', 'b', 'c'), ....: latex_indices=(r'\alpha', r'\beta', r'\gamma')) sage: v Vector frame (M, (v_a,v_b,v_c)) sage: latex(v) \left(M, \left(v_{\alpha},v_{\beta},v_{\gamma}\right)\right) sage: latex(v.coframe()) \left(M, \left(v^{\alpha},v^{\beta},v^{\gamma}\right)\right)
The symbol of each element of the vector frame can also be freely chosen, by providing a tuple of symbols as the first argument of
vector_frame
; it is then mandatory to specify as well some symbols for the dual coframe:sage: h = M.vector_frame(('a', 'b', 'c'), symbol_dual=('A', 'B', 'C')) sage: h Vector frame (M, (a,b,c)) sage: h[1] Vector field a on the 3-dimensional differentiable manifold M sage: h.coframe() Coframe (M, (A,B,C)) sage: h.coframe()[1] 1-form A on the 3-dimensional differentiable manifold M
Example with a non-trivial map \(\Phi\) (see above); a vector frame along a curve:
sage: U = Manifold(1, 'U') # open interval (-1,1) as a 1-dimensional manifold sage: T.<t> = U.chart('t:(-1,1)') # canonical chart on U sage: Phi = U.diff_map(M, [cos(t), sin(t), t], name='Phi', ....: latex_name=r'\Phi') sage: Phi Differentiable map Phi from the 1-dimensional differentiable manifold U to the 3-dimensional differentiable manifold M sage: f = U.vector_frame('f', dest_map=Phi) ; f Vector frame (U, (f_1,f_2,f_3)) with values on the 3-dimensional differentiable manifold M sage: f.domain() 1-dimensional differentiable manifold U sage: f.ambient_domain() 3-dimensional differentiable manifold M
The value of the vector frame at a given point is a basis of the corresponding tangent space:
sage: p = U((0,), name='p') ; p Point p on the 1-dimensional differentiable manifold U sage: f.at(p) Basis (f_1,f_2,f_3) on the Tangent space at Point Phi(p) on the 3-dimensional differentiable manifold M
Vector frames are bases of free modules formed by vector fields:
sage: e.module() Free module X(M) of vector fields on the 3-dimensional differentiable manifold M sage: e.module().base_ring() Algebra of differentiable scalar fields on the 3-dimensional differentiable manifold M sage: e.module() is M.vector_field_module() True sage: e in M.vector_field_module().bases() True
sage: f.module() Free module X(U,Phi) of vector fields along the 1-dimensional differentiable manifold U mapped into the 3-dimensional differentiable manifold M sage: f.module().base_ring() Algebra of differentiable scalar fields on the 1-dimensional differentiable manifold U sage: f.module() is U.vector_field_module(dest_map=Phi) True sage: f in U.vector_field_module(dest_map=Phi).bases() True
- along(mapping)#
Return the vector frame deduced from the current frame via a differentiable map, the codomain of which is included in the domain of of the current frame.
If \(e\) is the current vector frame, \(V\) its domain and if \(\Phi: U \rightarrow V\) is a differentiable map from some differentiable manifold \(U\) to \(V\), the returned object is a vector frame \(\tilde e\) along \(U\) with values on \(V\) such that
\[\forall p \in U,\ \tilde e(p) = e(\Phi(p)).\]INPUT:
mapping
– differentiable map \(\Phi: U \rightarrow V\)
OUTPUT:
vector frame \(\tilde e\) along \(U\) defined above.
EXAMPLES:
Vector frame along a curve:
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: R = Manifold(1, 'R') # R as a 1-dimensional manifold sage: T.<t> = R.chart() # canonical chart on R sage: Phi = R.diff_map(M, {(T,X): [cos(t), t]}, name='Phi', ....: latex_name=r'\Phi') ; Phi Differentiable map Phi from the 1-dimensional differentiable manifold R to the 2-dimensional differentiable manifold M sage: e = X.frame() ; e Coordinate frame (M, (∂/∂x,∂/∂y)) sage: te = e.along(Phi) ; te Vector frame (R, (∂/∂x,∂/∂y)) with values on the 2-dimensional differentiable manifold M
Check of the formula \(\tilde e(p) = e(\Phi(p))\):
sage: p = R((pi,)) ; p Point on the 1-dimensional differentiable manifold R sage: te[0].at(p) == e[0].at(Phi(p)) True sage: te[1].at(p) == e[1].at(Phi(p)) True
The result is cached:
sage: te is e.along(Phi) True
- ambient_domain()#
Return the differentiable manifold in which
self
takes its values.The ambient domain is the codomain \(M\) of the differentiable map \(\Phi: U \rightarrow M\) associated with the frame.
OUTPUT:
a
DifferentiableManifold
representing \(M\)
EXAMPLES:
sage: M = Manifold(2, 'M') sage: e = M.vector_frame('e') sage: e.ambient_domain() 2-dimensional differentiable manifold M
In the present case, since \(\Phi\) is the identity map:
sage: e.ambient_domain() == e.domain() True
An example with a non trivial map \(\Phi\):
sage: U = Manifold(1, 'U') sage: T.<t> = U.chart() sage: X.<x,y> = M.chart() sage: Phi = U.diff_map(M, {(T,X): [cos(t), t]}, name='Phi', ....: latex_name=r'\Phi') ; Phi Differentiable map Phi from the 1-dimensional differentiable manifold U to the 2-dimensional differentiable manifold M sage: f = U.vector_frame('f', dest_map=Phi); f Vector frame (U, (f_0,f_1)) with values on the 2-dimensional differentiable manifold M sage: f.ambient_domain() 2-dimensional differentiable manifold M sage: f.domain() 1-dimensional differentiable manifold U
- at(point)#
Return the value of
self
at a given point, this value being a basis of the tangent vector space at the point.INPUT:
point
–ManifoldPoint
; point \(p\) in the domain \(U\) of the vector frame (denoted \(e\) hereafter)
OUTPUT:
FreeModuleBasis
representing the basis \(e(p)\) of the tangent vector space \(T_{\Phi(p)} M\), where \(\Phi: U \to M\) is the differentiable map associated with \(e\) (possibly \(\Phi = \mathrm{Id}_U\))
EXAMPLES:
Basis of a tangent space to a 2-dimensional manifold:
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: p = M.point((-1,2), name='p') sage: e = X.frame() ; e Coordinate frame (M, (∂/∂x,∂/∂y)) sage: ep = e.at(p) ; ep Basis (∂/∂x,∂/∂y) on the Tangent space at Point p on the 2-dimensional differentiable manifold M sage: type(ep) <class 'sage.tensor.modules.free_module_basis.FreeModuleBasis_with_category'> sage: ep[0] Tangent vector ∂/∂x at Point p on the 2-dimensional differentiable manifold M sage: ep[1] Tangent vector ∂/∂y at Point p on the 2-dimensional differentiable manifold M
Note that the symbols used to denote the vectors are same as those for the vector fields of the frame. At this stage,
ep
is the unique basis on the tangent space atp
:sage: Tp = M.tangent_space(p) sage: Tp.bases() [Basis (∂/∂x,∂/∂y) on the Tangent space at Point p on the 2-dimensional differentiable manifold M]
Let us consider a vector frame that is a not a coordinate one:
sage: aut = M.automorphism_field() sage: aut[:] = [[1+y^2, 0], [0, 2]] sage: f = e.new_frame(aut, 'f') ; f Vector frame (M, (f_0,f_1)) sage: fp = f.at(p) ; fp Basis (f_0,f_1) on the Tangent space at Point p on the 2-dimensional differentiable manifold M
There are now two bases on the tangent space:
sage: Tp.bases() [Basis (∂/∂x,∂/∂y) on the Tangent space at Point p on the 2-dimensional differentiable manifold M, Basis (f_0,f_1) on the Tangent space at Point p on the 2-dimensional differentiable manifold M]
Moreover, the changes of bases in the tangent space have been computed from the known relation between the frames
e
andf
(field of automorphismsaut
defined above):sage: Tp.change_of_basis(ep, fp) Automorphism of the Tangent space at Point p on the 2-dimensional differentiable manifold M sage: Tp.change_of_basis(ep, fp).display() 5 ∂/∂x⊗dx + 2 ∂/∂y⊗dy sage: Tp.change_of_basis(fp, ep) Automorphism of the Tangent space at Point p on the 2-dimensional differentiable manifold M sage: Tp.change_of_basis(fp, ep).display() 1/5 ∂/∂x⊗dx + 1/2 ∂/∂y⊗dy
The dual bases:
sage: e.coframe() Coordinate coframe (M, (dx,dy)) sage: ep.dual_basis() Dual basis (dx,dy) on the Tangent space at Point p on the 2-dimensional differentiable manifold M sage: ep.dual_basis() is e.coframe().at(p) True sage: f.coframe() Coframe (M, (f^0,f^1)) sage: fp.dual_basis() Dual basis (f^0,f^1) on the Tangent space at Point p on the 2-dimensional differentiable manifold M sage: fp.dual_basis() is f.coframe().at(p) True
- coframe()#
Return the coframe of
self
.EXAMPLES:
sage: M = Manifold(2, 'M') sage: e = M.vector_frame('e') sage: e.coframe() Coframe (M, (e^0,e^1)) sage: X.<x,y> = M.chart() sage: X.frame().coframe() Coordinate coframe (M, (dx,dy))
- destination_map()#
Return the differential map associated to this vector frame.
Let \(e\) denote the vector frame; the differential map associated to it is the map \(\Phi: U\rightarrow M\) such that for each \(p \in U\), \(e(p)\) is a vector basis of the tangent space \(T_{\Phi(p)}M\).
OUTPUT:
a
DiffMap
representing the differential map \(\Phi\)
EXAMPLES:
sage: M = Manifold(2, 'M') sage: e = M.vector_frame('e') sage: e.destination_map() Identity map Id_M of the 2-dimensional differentiable manifold M
An example with a non trivial map \(\Phi\):
sage: U = Manifold(1, 'U') sage: T.<t> = U.chart() sage: X.<x,y> = M.chart() sage: Phi = U.diff_map(M, {(T,X): [cos(t), t]}, name='Phi', ....: latex_name=r'\Phi') ; Phi Differentiable map Phi from the 1-dimensional differentiable manifold U to the 2-dimensional differentiable manifold M sage: f = U.vector_frame('f', dest_map=Phi); f Vector frame (U, (f_0,f_1)) with values on the 2-dimensional differentiable manifold M sage: f.destination_map() Differentiable map Phi from the 1-dimensional differentiable manifold U to the 2-dimensional differentiable manifold M
- domain()#
Return the domain on which
self
is defined.OUTPUT:
a
DifferentiableManifold
; representing the domain of the vector frame
EXAMPLES:
sage: M = Manifold(2, 'M') sage: e = M.vector_frame('e') sage: e.domain() 2-dimensional differentiable manifold M sage: U = M.open_subset('U') sage: f = e.restrict(U) sage: f.domain() Open subset U of the 2-dimensional differentiable manifold M
- new_frame(change_of_frame, symbol, latex_symbol=None, indices=None, latex_indices=None, symbol_dual=None, latex_symbol_dual=None)#
Define a new vector frame from
self
.The new vector frame is defined from a field of tangent-space automorphisms; its domain is the same as that of the current frame.
INPUT:
change_of_frame
–AutomorphismFieldParal
; the field of tangent space automorphisms \(P\) that relates the current frame \((e_i)\) to the new frame \((n_i)\) according to \(n_i = P(e_i)\)symbol
– either a string, to be used as a common base for the symbols of the vector fields constituting the vector frame, or a list/tuple of strings, representing the individual symbols of the vector fieldslatex_symbol
– (default:None
) either a string, to be used as a common base for the LaTeX symbols of the vector fields constituting the vector frame, or a list/tuple of strings, representing the individual LaTeX symbols of the vector fields; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the vector fields of the frame; ifNone
, the indices will be generated as integers within the range declared onself
latex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the vector fields; ifNone
,indices
is used insteadsymbol_dual
– (default:None
) same assymbol
but for the dual coframe; ifNone
,symbol
must be a string and is used for the common base of the symbols of the elements of the dual coframelatex_symbol_dual
– (default:None
) same aslatex_symbol
but for the dual coframe
OUTPUT:
the new frame \((n_i)\), as an instance of
VectorFrame
EXAMPLES:
Frame resulting from a \(\pi/3\)-rotation in the Euclidean plane:
sage: M = Manifold(2, 'R^2') sage: X.<x,y> = M.chart() sage: e = M.vector_frame('e') ; M.set_default_frame(e) sage: M._frame_changes {} sage: rot = M.automorphism_field() sage: rot[:] = [[sqrt(3)/2, -1/2], [1/2, sqrt(3)/2]] sage: n = e.new_frame(rot, 'n') sage: n[0][:] [1/2*sqrt(3), 1/2] sage: n[1][:] [-1/2, 1/2*sqrt(3)] sage: a = M.change_of_frame(e,n) sage: a[:] [1/2*sqrt(3) -1/2] [ 1/2 1/2*sqrt(3)] sage: a == rot True sage: a is rot False sage: a._components # random (dictionary output) {Vector frame (R^2, (e_0,e_1)): 2-indices components w.r.t. Vector frame (R^2, (e_0,e_1)), Vector frame (R^2, (n_0,n_1)): 2-indices components w.r.t. Vector frame (R^2, (n_0,n_1))} sage: a.comp(n)[:] [1/2*sqrt(3) -1/2] [ 1/2 1/2*sqrt(3)] sage: a1 = M.change_of_frame(n,e) sage: a1[:] [1/2*sqrt(3) 1/2] [ -1/2 1/2*sqrt(3)] sage: a1 == rot.inverse() True sage: a1 is rot.inverse() False sage: e[0].comp(n)[:] [1/2*sqrt(3), -1/2] sage: e[1].comp(n)[:] [1/2, 1/2*sqrt(3)]
- restrict(subdomain)#
Return the restriction of
self
to some open subset of its domain.If the restriction has not been defined yet, it is constructed here.
INPUT:
subdomain
– open subset \(V\) of the current frame domain \(U\)
OUTPUT:
the restriction of the current frame to \(V\) as a
VectorFrame
EXAMPLES:
Restriction of a frame defined on \(\RR^2\) to the unit disk:
sage: M = Manifold(2, 'R^2', start_index=1) sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2 sage: a = M.automorphism_field() sage: a[:] = [[1-y^2,0], [1+x^2, 2]] sage: e = c_cart.frame().new_frame(a, 'e') ; e Vector frame (R^2, (e_1,e_2)) sage: U = M.open_subset('U', coord_def={c_cart: x^2+y^2<1}) sage: e_U = e.restrict(U) ; e_U Vector frame (U, (e_1,e_2))
The vectors of the restriction have the same symbols as those of the original frame:
sage: e_U[1].display() e_1 = (-y^2 + 1) ∂/∂x + (x^2 + 1) ∂/∂y sage: e_U[2].display() e_2 = 2 ∂/∂y
They are actually the restrictions of the original frame vectors:
sage: e_U[1] is e[1].restrict(U) True sage: e_U[2] is e[2].restrict(U) True
- set_name(symbol, latex_symbol=None, indices=None, latex_indices=None, index_position='down', include_domain=True)#
Set (or change) the text name and LaTeX name of
self
.INPUT:
symbol
– either a string, to be used as a common base for the symbols of the vector fields constituting the vector frame, or a list/tuple of strings, representing the individual symbols of the vector fieldslatex_symbol
– (default:None
) either a string, to be used as a common base for the LaTeX symbols of the vector fields constituting the vector frame, or a list/tuple of strings, representing the individual LaTeX symbols of the vector fields; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the vector fields of the frame; ifNone
, the indices will be generated as integers within the range declared onself
latex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the vector fields; ifNone
,indices
is used insteadindex_position
– (default:'down'
) determines the position of the indices labelling the vector fields of the frame; can be either'down'
or'up'
include_domain
– (default:True
) boolean determining whether the name of the domain is included in the beginning of the vector frame name
EXAMPLES:
sage: M = Manifold(2, 'M') sage: e = M.vector_frame('e'); e Vector frame (M, (e_0,e_1)) sage: e.set_name('f'); e Vector frame (M, (f_0,f_1)) sage: e.set_name('e', include_domain=False); e Vector frame (e_0,e_1) sage: e.set_name(['a', 'b']); e Vector frame (M, (a,b)) sage: e.set_name('e', indices=['x', 'y']); e Vector frame (M, (e_x,e_y)) sage: e.set_name('e', latex_symbol=r'\epsilon') sage: latex(e) \left(M, \left(\epsilon_{0},\epsilon_{1}\right)\right) sage: e.set_name('e', latex_symbol=[r'\alpha', r'\beta']) sage: latex(e) \left(M, \left(\alpha,\beta\right)\right) sage: e.set_name('e', latex_symbol='E', ....: latex_indices=[r'\alpha', r'\beta']) sage: latex(e) \left(M, \left(E_{\alpha},E_{\beta}\right)\right)
- structure_coeff()#
Evaluate the structure coefficients associated to
self
.\(n\) being the manifold’s dimension, the structure coefficients of the vector frame \((e_i)\) are the \(n^3\) scalar fields \(C^k_{\ \, ij}\) defined by
\[[e_i, e_j] = C^k_{\ \, ij} e_k\]OUTPUT:
the structure coefficients \(C^k_{\ \, ij}\), as an instance of
CompWithSym
with 3 indices ordered as \((k,i,j)\).
EXAMPLES:
Structure coefficients of the orthonormal frame associated to spherical coordinates in the Euclidean space \(\RR^3\):
sage: M = Manifold(3, 'R^3', r'\RR^3', start_index=1) # Part of R^3 covered by spherical coordinates sage: c_spher.<r,th,ph> = M.chart(r'r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi') sage: ch_frame = M.automorphism_field() sage: ch_frame[1,1], ch_frame[2,2], ch_frame[3,3] = 1, 1/r, 1/(r*sin(th)) sage: M.frames() [Coordinate frame (R^3, (∂/∂r,∂/∂th,∂/∂ph))] sage: e = c_spher.frame().new_frame(ch_frame, 'e') sage: e[1][:] # components of e_1 in the manifold's default frame (∂/∂r, ∂/∂th, ∂/∂th) [1, 0, 0] sage: e[2][:] [0, 1/r, 0] sage: e[3][:] [0, 0, 1/(r*sin(th))] sage: c = e.structure_coeff() ; c 3-indices components w.r.t. Vector frame (R^3, (e_1,e_2,e_3)), with antisymmetry on the index positions (1, 2) sage: c[:] [[[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, -1/r, 0], [1/r, 0, 0], [0, 0, 0]], [[0, 0, -1/r], [0, 0, -cos(th)/(r*sin(th))], [1/r, cos(th)/(r*sin(th)), 0]]] sage: c[2,1,2] # C^2_{12} -1/r sage: c[3,1,3] # C^3_{13} -1/r sage: c[3,2,3] # C^3_{23} -cos(th)/(r*sin(th))