Coverage for local/lib/python2.7/site-packages/sage/manifolds/differentiable/chart.py : 97%

r""" 

Coordinate Charts on Differentiable Manifolds 

The class :class:`DiffChart` implements coordinate charts on a differentiable 

manifold over a topological field `K` (in most applications, `K = \RR` or 

`K = \CC`). 

The subclass :class:`RealDiffChart` is devoted 

to the case `K=\RR`, for which the concept of coordinate range is meaningful. 

Moreover, :class:`RealDiffChart` is endowed with some plotting 

capabilities (cf. method :meth:`~sage.manifolds.chart.RealChart.plot`). 

Transition maps between charts are implemented via the class 

:class:`DiffCoordChange`. 

AUTHORS: 

- Eric Gourgoulhon, Michal Bejger (2013-2015) : initial version 

REFERENCES: 

- Chap. 1 of [Lee2013]_ 

""" 

#***************************************************************************** 

# Copyright (C) 2015 Eric Gourgoulhon <eric.gourgoulhon@obspm.fr> 

# Copyright (C) 2015 Michal Bejger <bejger@camk.edu.pl> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# as published by the Free Software Foundation; either version 2 of 

# the License, or (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.misc.cachefunc import cached_method 

from sage.manifolds.chart import Chart, RealChart, CoordChange 

from sage.manifolds.differentiable.vectorframe import CoordFrame 

class DiffChart(Chart): 

r""" 

Chart on a differentiable manifold. 

Given a differentiable manifold `M` of dimension `n` over a topological 

field `K`, a *chart* is a member `(U,\varphi)` of the manifold's 

differentiable atlas; `U` is then an open subset of `M` and 

`\varphi: U \rightarrow V \subset K^n` is a homeomorphism from 

`U` to an open subset `V` of `K^n`. 

The components `(x^1,\ldots,x^n)` of `\varphi`, defined by 

`\varphi(p) = (x^1(p),\ldots,x^n(p))\in K^n` for any point `p\in U`, are 

called the *coordinates* of the chart `(U,\varphi)`. 

INPUT: 

- ``domain`` -- open subset `U` on which the chart is defined 

- ``coordinates`` -- (default: '' (empty string)) single string defining 

the coordinate symbols, with ' ' (whitespace) as a separator; each item 

has at most two fields, separated by ':': 

1. The coordinate symbol (a letter or a few letters) 

2. (optional) The LaTeX spelling of the coordinate; if not provided the 

coordinate symbol given in the first field will be used. 

If it contains any LaTeX expression, the string ``coordinates`` must be 

declared with the prefix 'r' (for "raw") to allow for a proper treatment 

of LaTeX's backslash character (see examples below). 

If no LaTeX spelling is to be set for any coordinate, the argument 

``coordinates`` can be omitted when the shortcut operator ``<,>`` is 

used via Sage preparser (see examples below) 

- ``names`` -- (default: ``None``) unused argument, except if 

``coordinates`` is not provided; it must then be a tuple containing 

the coordinate symbols (this is guaranteed if the shortcut operator 

``<,>`` is used). 

- ``calc_method`` -- (default: ``None``) string defining the calculus 

method for computations involving coordinates of the chart; must be 

one of 

- ``'SR'``: Sage's default symbolic engine (Symbolic Ring) 

- ``'sympy'``: SymPy 

- ``None``: the default of 

:class:`~sage.manifolds.calculus_method.CalculusMethod` will be 

used 

EXAMPLES: 

A chart on a complex 2-dimensional differentiable manifold:: 

sage: M = Manifold(2, 'M', field='complex') 

sage: X = M.chart('x y'); X 

Chart (M, (x, y)) 

sage: latex(X) 

\left(M,(x, y)\right) 

sage: type(X) 

<class 'sage.manifolds.differentiable.chart.DiffChart'> 

To manipulate the coordinates `(x,y)` as global variables, one has to set:: 

sage: x,y = X[:] 

However, a shortcut is to use the declarator ``<x,y>`` in the left-hand 

side of the chart declaration (there is then no need to pass the string 

``'x y'`` to ``chart()``):: 

sage: M = Manifold(2, 'M', field='complex') 

sage: X.<x,y> = M.chart(); X 

Chart (M, (x, y)) 

The coordinates are then immediately accessible:: 

sage: y 

y 

sage: x is X[0] and y is X[1] 

True 

The trick is performed by Sage preparser:: 

sage: preparse("X.<x,y> = M.chart()") 

"X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)" 

Note that ``x`` and ``y`` declared in ``<x,y>`` are mere Python variable 

names and do not have to coincide with the coordinate symbols; 

for instance, one may write:: 

sage: M = Manifold(2, 'M', field='complex') 

sage: X.<x1,y1> = M.chart('x y'); X 

Chart (M, (x, y)) 

Then ``y`` is not known as a global Python variable and the 

coordinate `y` is accessible only through the global variable ``y1``:: 

sage: y1 

y 

sage: latex(y1) 

y 

sage: y1 is X[1] 

True 

However, having the name of the Python variable coincide with the 

coordinate symbol is quite convenient; so it is recommended to declare:: 

sage: M = Manifold(2, 'M', field='complex') 

sage: X.<x,y> = M.chart() 

In the above example, the chart X covers entirely the manifold M:: 

sage: X.domain() 

2-dimensional complex manifold M 

Of course, one may declare a chart only on an open subset of M:: 

sage: U = M.open_subset('U') 

sage: Y.<z1, z2> = U.chart(r'z1:\zeta_1 z2:\zeta_2'); Y 

Chart (U, (z1, z2)) 

sage: Y.domain() 

Open subset U of the 2-dimensional complex manifold M 

In the above declaration, we have also specified some LaTeX writing 

of the coordinates different from the text one:: 

sage: latex(z1) 

{\zeta_1} 

Note the prefix ``r`` in front of the string ``r'z1:\zeta_1 z2:\zeta_2'``; 

it makes sure that the backslash character is treated as an ordinary 

character, to be passed to the LaTeX interpreter. 

Coordinates are Sage symbolic variables (see 

:mod:`sage.symbolic.expression`):: 

sage: type(z1) 

<type 'sage.symbolic.expression.Expression'> 

In addition to the Python variable name provided in the operator ``<.,.>``, 

the coordinates are accessible by their indices:: 

sage: Y[0], Y[1] 

(z1, z2) 

The index range is that declared during the creation of the manifold. By 

default, it starts at 0, but this can be changed via the parameter 

``start_index``:: 

sage: M1 = Manifold(2, 'M_1', field='complex', start_index=1) 

sage: Z.<u,v> = M1.chart() 

sage: Z[1], Z[2] 

(u, v) 

The full set of coordinates is obtained by means of the operator 

``[:]``:: 

sage: Y[:] 

(z1, z2) 

Each constructed chart is automatically added to the manifold's user 

atlas:: 

sage: M.atlas() 

[Chart (M, (x, y)), Chart (U, (z1, z2))] 

and to the atlas of the chart's domain:: 

sage: U.atlas() 

[Chart (U, (z1, z2))] 

Manifold subsets have a *default chart*, which, unless changed via the 

method 

:meth:`~sage.manifolds.manifold.TopologicalManifold.set_default_chart`, 

is the first defined chart on the subset (or on a open subset of it):: 

sage: M.default_chart() 

Chart (M, (x, y)) 

sage: U.default_chart() 

Chart (U, (z1, z2)) 

The default charts are not privileged charts on the manifold, but rather 

charts whose name can be skipped in the argument list of functions having 

an optional ``chart=`` argument. 

The action of the chart map `\varphi` on a point is obtained by means of 

the call operator, i.e. the operator ``()``:: 

sage: p = M.point((1+i, 2), chart=X); p 

Point on the 2-dimensional complex manifold M 

sage: X(p) 

(I + 1, 2) 

sage: X(p) == p.coord(X) 

True 

A vector frame is naturally associated to each chart:: 

sage: X.frame() 

Coordinate frame (M, (d/dx,d/dy)) 

sage: Y.frame() 

Coordinate frame (U, (d/dz1,d/dz2)) 

as well as a dual frame (basis of 1-forms):: 

sage: X.coframe() 

Coordinate coframe (M, (dx,dy)) 

sage: Y.coframe() 

Coordinate coframe (U, (dz1,dz2)) 

.. SEEALSO:: 

:class:`~sage.manifolds.differentiable.chart.RealDiffChart` for charts 

on differentiable manifolds over `\RR`. 

""" 

def __init__(self, domain, coordinates='', names=None, calc_method=None): 

r""" 

Construct a chart. 

TESTS:: 

sage: M = Manifold(2, 'M', field='complex') 

sage: X.<x,y> = M.chart() 

sage: X 

Chart (M, (x, y)) 

sage: type(X) 

<class 'sage.manifolds.differentiable.chart.DiffChart'> 

sage: assumptions() # no assumptions on x,y set by X._init_coordinates 

[] 

sage: TestSuite(X).run() 

""" 

Chart.__init__(self, domain, coordinates=coordinates, names=names, 

calc_method=calc_method) 

# Construction of the coordinate frame associated to the chart: 

self._frame = CoordFrame(self) 

self._coframe = self._frame._coframe 

def transition_map(self, other, transformations, intersection_name=None, 

restrictions1=None, restrictions2=None): 

r""" 

Construct the transition map between the current chart, 

`(U,\varphi)` say, and another one, `(V,\psi)` say. 

If `n` is the manifold's dimension, the *transition map* is the 

map 

.. MATH:: 

\psi\circ\varphi^{-1}: \varphi(U\cap V) \subset K^n 

\rightarrow \psi(U\cap V) \subset K^n, 

where `K` is the manifold's base field. In other words, the 

transition map expresses the coordinates `(y^1,\ldots,y^n)` of 

`(V,\psi)` in terms of the coordinates `(x^1,\ldots,x^n)` of 

`(U,\varphi)` on the open subset where the two charts intersect, i.e. 

on `U\cap V`. 

By definition, the transition map `\psi\circ\varphi^{-1}` must be 

of classe `C^k`, where `k` is the degree of differentiability of the 

manifold (cf. 

:meth:`~sage.manifolds.differentiable.manifold.DifferentiableManifold.diff_degree`). 

INPUT: 

- ``other`` -- the chart `(V,\psi)` 

- ``transformations`` -- tuple (or list) `(Y_1,\ldots,Y_2)`, where 

`Y_i` is the symbolic expression of the coordinate `y^i` in terms 

of the coordinates `(x^1,\ldots,x^n)` 

- ``intersection_name`` -- (default: ``None``) name to be given to the 

subset `U\cap V` if the latter differs from `U` or `V` 

- ``restrictions1`` -- (default: ``None``) list of conditions on the 

coordinates of the current chart that define `U\cap V` if the 

latter differs from `U`. ``restrictions1`` must be a list of 

of symbolic equalities or inequalities involving the 

coordinates, such as x>y or x^2+y^2 != 0. The items of the list 

``restrictions1`` are combined with the ``and`` operator; if some 

restrictions are to be combined with the ``or`` operator instead, 

they have to be passed as a tuple in some single item of the list 

``restrictions1``. For example, ``restrictions1`` = [x>y, 

(x!=0, y!=0), z^2<x] means (x>y) and ((x!=0) or (y!=0)) and (z^2<x). 

If the list ``restrictions1`` contains only one item, this item can 

be passed as such, i.e. writing x>y instead of the single-element 

list [x>y]. 

- ``restrictions2`` -- (default: ``None``) list of conditions on the 

coordinates of the chart `(V,\psi)` that define `U\cap V` if the 

latter differs from `V` (see ``restrictions1`` for the syntax) 

OUTPUT: 

- The transition map `\psi\circ\varphi^{-1}` defined on `U\cap V`, as an 

instance of :class:`DiffCoordChange`. 

EXAMPLES: 

Transition map between two stereographic charts on the circle `S^1`:: 

sage: M = Manifold(1, 'S^1') 

sage: U = M.open_subset('U') # Complement of the North pole 

sage: cU.<x> = U.chart() # Stereographic chart from the North pole 

sage: V = M.open_subset('V') # Complement of the South pole 

sage: cV.<y> = V.chart() # Stereographic chart from the South pole 

sage: M.declare_union(U,V) # S^1 is the union of U and V 

sage: trans = cU.transition_map(cV, 1/x, intersection_name='W', 

....: restrictions1= x!=0, restrictions2 = y!=0) 

sage: trans 

Change of coordinates from Chart (W, (x,)) to Chart (W, (y,)) 

sage: trans.display() 

y = 1/x 

The subset `W`, intersection of `U` and `V`, has been created by 

``transition_map()``:: 

sage: M.list_of_subsets() 

[1-dimensional differentiable manifold S^1, 

Open subset U of the 1-dimensional differentiable manifold S^1, 

Open subset V of the 1-dimensional differentiable manifold S^1, 

Open subset W of the 1-dimensional differentiable manifold S^1] 

sage: W = M.list_of_subsets()[3] 

sage: W is U.intersection(V) 

True 

sage: M.atlas() 

[Chart (U, (x,)), Chart (V, (y,)), Chart (W, (x,)), Chart (W, (y,))] 

Transition map between the polar chart and the Cartesian one on 

`\RR^2`:: 

sage: M = Manifold(2, 'R^2') 

sage: c_cart.<x,y> = M.chart() 

sage: U = M.open_subset('U') # the complement of the half line {y=0, x >= 0} 

sage: c_spher.<r,phi> = U.chart(r'r:(0,+oo) phi:(0,2*pi):\phi') 

sage: trans = c_spher.transition_map(c_cart, (r*cos(phi), r*sin(phi)), 

....: restrictions2=(y!=0, x<0)) 

sage: trans 

Change of coordinates from Chart (U, (r, phi)) to Chart (U, (x, y)) 

sage: trans.display() 

x = r*cos(phi) 

y = r*sin(phi) 

In this case, no new subset has been created since `U\cap M = U`:: 

sage: M.list_of_subsets() 

[2-dimensional differentiable manifold R^2, 

Open subset U of the 2-dimensional differentiable manifold R^2] 

but a new chart has been created: `(U, (x, y))`:: 

sage: M.atlas() 

[Chart (R^2, (x, y)), Chart (U, (r, phi)), Chart (U, (x, y))] 

""" 

dom1 = self._domain 

dom2 = other._domain 

dom = dom1.intersection(dom2, name=intersection_name) 

if dom is dom1: 

chart1 = self 

else: 

chart1 = self.restrict(dom, restrictions1) 

if dom is dom2: 

chart2 = other 

else: 

chart2 = other.restrict(dom, restrictions2) 

if not isinstance(transformations, (tuple, list)): 

transformations = [transformations] 

return DiffCoordChange(chart1, chart2, *transformations) 

def frame(self): 

r""" 

Return the vector frame (coordinate frame) associated with ``self``. 

OUTPUT: 

- a :class:`~sage.manifolds.differentiable.vectorframe.CoordFrame` 

representing the coordinate frame 

EXAMPLES: 

Coordinate frame associated with some chart on a 2-dimensional 

manifold:: 

sage: M = Manifold(2, 'M') 

sage: c_xy.<x,y> = M.chart() 

sage: c_xy.frame() 

Coordinate frame (M, (d/dx,d/dy)) 

sage: type(c_xy.frame()) 

<class 'sage.manifolds.differentiable.vectorframe.CoordFrame'> 

Check that ``c_xy.frame()`` is indeed the coordinate frame associated 

with the coordinates `(x,y)`:: 

sage: ex = c_xy.frame()[0] ; ex 

Vector field d/dx on the 2-dimensional differentiable manifold M 

sage: ey = c_xy.frame()[1] ; ey 

Vector field d/dy on the 2-dimensional differentiable manifold M 

sage: ex(M.scalar_field(x)).display() 

M --> R 

(x, y) |--> 1 

sage: ex(M.scalar_field(y)).display() 

M --> R 

(x, y) |--> 0 

sage: ey(M.scalar_field(x)).display() 

M --> R 

(x, y) |--> 0 

sage: ey(M.scalar_field(y)).display() 

M --> R 

(x, y) |--> 1 

""" 

return self._frame 

def coframe(self): 

r""" 

Return the coframe (basis of coordinate differentials) associated 

with ``self``. 

OUTPUT: 

- a :class:`~sage.manifolds.differentiable.vectorframe.CoordCoFrame` 

representing the coframe 

EXAMPLES: 

Coordinate coframe associated with some chart on a 2-dimensional 

manifold:: 

sage: M = Manifold(2, 'M') 

sage: c_xy.<x,y> = M.chart() 

sage: c_xy.coframe() 

Coordinate coframe (M, (dx,dy)) 

sage: type(c_xy.coframe()) 

<class 'sage.manifolds.differentiable.vectorframe.CoordCoFrame'> 

Check that ``c_xy.coframe()`` is indeed the coordinate coframe 

associated with the coordinates `(x, y)`:: 

sage: dx = c_xy.coframe()[0] ; dx 

1-form dx on the 2-dimensional differentiable manifold M 

sage: dy = c_xy.coframe()[1] ; dy 

1-form dy on the 2-dimensional differentiable manifold M 

sage: ex = c_xy.frame()[0] ; ex 

Vector field d/dx on the 2-dimensional differentiable manifold M 

sage: ey = c_xy.frame()[1] ; ey 

Vector field d/dy on the 2-dimensional differentiable manifold M 

sage: dx(ex).display() 

dx(d/dx): M --> R 

(x, y) |--> 1 

sage: dx(ey).display() 

dx(d/dy): M --> R 

(x, y) |--> 0 

sage: dy(ex).display() 

dy(d/dx): M --> R 

(x, y) |--> 0 

sage: dy(ey).display() 

dy(d/dy): M --> R 

(x, y) |--> 1 

""" 

return self._coframe 

def restrict(self, subset, restrictions=None): 

r""" 

Return the restriction of ``self`` to some subset. 

If the current chart is `(U, \varphi)`, a *restriction* (or 

*subchart*) is a chart `(V, \psi)` such that `V \subset U` 

and `\psi = \varphi |_V`. 

If such subchart has not been defined yet, it is constructed here. 

The coordinates of the subchart bare the same names as the 

coordinates of the original chart. 

INPUT: 

- ``subset`` -- open subset `V` of the chart domain `U` 

- ``restrictions`` -- (default: ``None``) list of coordinate 

restrictions defining the subset `V` 

A restriction can be any symbolic equality or inequality involving 

the coordinates, such as ``x > y`` or ``x^2 + y^2 != 0``. The items 

of the list ``restrictions`` are combined with the ``and`` operator; 

if some restrictions are to be combined with the ``or`` operator 

instead, they have to be passed as a tuple in some single item 

of the list ``restrictions``. For example:: 

restrictions = [x > y, (x != 0, y != 0), z^2 < x] 

means ``(x > y) and ((x != 0) or (y != 0)) and (z^2 < x)``. 

If the list ``restrictions`` contains only one item, this 

item can be passed as such, i.e. writing ``x > y`` instead 

of the single element list ``[x > y]``. 

OUTPUT: 

- a :class:`DiffChart` `(V, \psi)` 

EXAMPLES: 

Coordinates on the unit open ball of `\CC^2` as a subchart 

of the global coordinates of `\CC^2`:: 

sage: M = Manifold(2, 'C^2', field='complex') 

sage: X.<z1, z2> = M.chart() 

sage: B = M.open_subset('B') 

sage: X_B = X.restrict(B, abs(z1)^2 + abs(z2)^2 < 1); X_B 

Chart (B, (z1, z2)) 

""" 

if subset == self._domain: 

return self 

if subset not in self._dom_restrict: 

resu = Chart.restrict(self, subset, restrictions=restrictions) 

# Update of superframes and subframes: 

resu._frame._superframes.update(self._frame._superframes) 

for sframe in self._frame._superframes: 

sframe._subframes.add(resu._frame) 

sframe._restrictions[subset] = resu._frame 

# The subchart frame is not a "top frame" in the supersets 

# (including self._domain): 

for dom in self._domain._supersets: 

if resu._frame in dom._top_frames: 

# it was added by the Chart constructor invoked in 

# Chart.restrict above 

dom._top_frames.remove(resu._frame) 

return self._dom_restrict[subset] 

def symbolic_velocities(self, left='D', right=None): 

r""" 

Return a list of symbolic variables ready to be used by the 

user as the derivatives of the coordinate functions with respect 

to a curve parameter (i.e. the velocities along the curve). 

It may actually serve to denote anything else than velocities, 

with a name including the coordinate functions. 

The choice of strings provided as 'left' and 'right' arguments 

is not entirely free since it must comply with Python 

prescriptions. 

INPUT: 

- ``left`` -- (default: ``D``) string to concatenate to the left 

of each coordinate functions of the chart 

- ``right`` -- (default: ``None``) string to concatenate to the 

right of each coordinate functions of the chart 

OUTPUT: 

- a list of symbolic expressions with the desired names 

EXAMPLES: 

Symbolic derivatives of the Cartesian coordinates of the 

3-dimensional Euclidean space:: 

sage: R3 = Manifold(3, 'R3', start_index=1) 

sage: cart.<X,Y,Z> = R3.chart() 

sage: D = cart.symbolic_velocities(); D 

[DX, DY, DZ] 

sage: D = cart.symbolic_velocities(left='d', right="/dt"); D 

Traceback (most recent call last): 

... 

ValueError: The name "dX/dt" is not a valid Python 

identifier. 

sage: D = cart.symbolic_velocities(left='d', right="_dt"); D 

[dX_dt, dY_dt, dZ_dt] 

sage: D = cart.symbolic_velocities(left='', right="'"); D 

Traceback (most recent call last): 

... 

ValueError: The name "X'" is not a valid Python 

identifier. 

sage: D = cart.symbolic_velocities(left='', right="_dot"); D 

[X_dot, Y_dot, Z_dot] 

sage: R.<t> = RealLine() 

sage: canon_chart = R.default_chart() 

sage: D = canon_chart.symbolic_velocities() ; D 

[Dt] 

""" 

from sage.symbolic.ring import var 

# The case len(self[:]) = 1 is treated apart due to the 

# following fact. 

# In the case of several coordinates, the argument of 'var' (as 

# implemented below after the case len(self[:]) = 1) is a list 

# of strings of the form ['Dx1', 'Dx2', ...] and not a unique 

# string of the form 'Dx1 Dx2 ...'. 

# Although 'var' is supposed to accept both syntaxes, the first 

# one causes an error when it contains only one argument, due to 

# line 784 of sage/symbolic/ring.pyx : 

# "return self.symbol(name, latex_name=formatted_latex_name, domain=domain)" 

# In this line, the first argument 'name' of 'symbol' is a list 

# and not a string if the argument of 'var' is a list of one 

# string (of the type ['Dt']), which causes error in 'symbol'. 

# This might be corrected. 

if len(self[:]) == 1: 

string_vel = left + format(self[:][0]) # will raise an error 

# in case left is not a string 

if right is not None: 

string_vel += right # will raise an error in case right 

# is not a string 

# If the argument of 'var' contains only one word, for 

# instance: 

# sage: var('Dt') 

# then 'var' does not return a tuple containing one symbolic 

# expression, but the symbolic expression itself. 

# This is taken into account below in order to return a list 

# containing one symbolic expression. 

return [var(string_vel)] 

list_strings_velocities = [left + format(coord_func) 

for coord_func in self[:]] # will 

# raise an error in case left is not a string 

if right is not None: 

list_strings_velocities = [string_vel + right for string_vel 

in list_strings_velocities] # will 

# raise an error in case right is not a string 

return list(var(list_strings_velocities)) 

#***************************************************************************** 

class RealDiffChart(DiffChart, RealChart): 

r""" 

Chart on a differentiable manifold over `\RR`. 

Given a differentiable manifold `M` of dimension `n` over `\RR`, 

a *chart* is a member `(U,\varphi)` of the manifold's 

differentiable atlas; `U` is then an open subset of `M` and 

`\varphi: U \rightarrow V \subset \RR^n` is a homeomorphism from 

`U` to an open subset `V` of `\RR^n`. 

The components `(x^1,\ldots,x^n)` of `\varphi`, defined by 

`\varphi(p) = (x^1(p),\ldots,x^n(p))\in \RR^n` for any point `p\in U`, are 

called the *coordinates* of the chart `(U,\varphi)`. 

INPUT: 

- ``domain`` -- open subset `U` on which the chart is defined 

- ``coordinates`` -- (default: '' (empty string)) single string defining 

the coordinate symbols and ranges, with ' ' (whitespace) as a separator; 

each item has at most three fields, separated by ':': 

1. The coordinate symbol (a letter or a few letters) 

2. (optional) The interval `I` defining the coordinate range: if not 

provided, the coordinate is assumed to span all `\RR`; otherwise 

`I` must be provided in the form ``(a,b)`` (or equivalently 

``]a,b[``). The bounds ``a`` and ``b`` can be ``+/-Infinity``, 

``Inf``, ``infinity``, ``inf`` or ``oo``. 

For *singular* coordinates, non-open intervals such as ``[a,b]`` and 

``(a,b]`` (or equivalently ``]a,b]``) are allowed. 

Note that the interval declaration must not contain any whitespace. 

3. (optional) The LaTeX spelling of the coordinate; if not provided the 

coordinate symbol given in the first field will be used. 

The order of the fields 2 and 3 does not matter and each of them can be 

omitted. 

If it contains any LaTeX expression, the string ``coordinates`` must be 

declared with the prefix 'r' (for "raw") to allow for a proper treatment 

of LaTeX backslash characters (see examples below). 

If no interval range and no LaTeX spelling is to be set for any 

coordinate, the argument ``coordinates`` can be omitted when the 

shortcut operator ``<,>`` is used via Sage preparser (see examples below) 

- ``names`` -- (default: ``None``) unused argument, except if 

``coordinates`` is not provided; it must then be a tuple containing 

the coordinate symbols (this is guaranteed if the shortcut operator 

``<,>`` is used). 

- ``calc_method`` -- (default: ``None``) string defining the calculus 

method for computations involving coordinates of the chart; must be 

one of 

- ``'SR'``: Sage's default symbolic engine (Symbolic Ring) 

- ``'sympy'``: SymPy 

- ``None``: the default of 

:class:`~sage.manifolds.calculus_method.CalculusMethod` will be 

used 

EXAMPLES: 

Cartesian coordinates on `\RR^3`:: 

sage: M = Manifold(3, 'R^3', r'\RR^3', start_index=1) 

sage: c_cart = M.chart('x y z'); c_cart 

Chart (R^3, (x, y, z)) 

sage: type(c_cart) 

<class 'sage.manifolds.differentiable.chart.RealDiffChart'> 

To have the coordinates accessible as global variables, one has to set:: 

sage: (x,y,z) = c_cart[:] 

However, a shortcut is to use the declarator ``<x,y,z>`` in the left-hand 

side of the chart declaration (there is then no need to pass the string 

``'x y z'`` to ``chart()``):: 

sage: M = Manifold(3, 'R^3', r'\RR^3', start_index=1) 

sage: c_cart.<x,y,z> = M.chart(); c_cart 

Chart (R^3, (x, y, z)) 

The coordinates are then immediately accessible:: 

sage: y 

y 

sage: y is c_cart[2] 

True 

The trick is performed by Sage preparser:: 

sage: preparse("c_cart.<x,y,z> = M.chart()") 

"c_cart = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = c_cart._first_ngens(3)" 

Note that ``x, y, z`` declared in ``<x,y,z>`` are mere Python variable 

names and do not have to coincide with the coordinate symbols; for instance, 

one may write:: 

sage: M = Manifold(3, 'R^3', r'\RR^3', start_index=1) 

sage: c_cart.<x1,y1,z1> = M.chart('x y z'); c_cart 

Chart (R^3, (x, y, z)) 

Then ``y`` is not known as a global variable and the coordinate `y` 

is accessible only through the global variable ``y1``:: 

sage: y1 

y 

sage: y1 is c_cart[2] 

True 

However, having the name of the Python variable coincide with the 

coordinate symbol is quite convenient; so it is recommended to declare:: 

sage: forget() # for doctests only 

sage: M = Manifold(3, 'R^3', r'\RR^3', start_index=1) 

sage: c_cart.<x,y,z> = M.chart() 

Spherical coordinates on the subset `U` of `\RR^3` that is the 

complement of the half-plane `\{y=0, x\geq 0\}`:: 

sage: U = M.open_subset('U') 

sage: c_spher.<r,th,ph> = U.chart(r'r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi') 

sage: c_spher 

Chart (U, (r, th, ph)) 

Note the prefix 'r' for the string defining the coordinates in the 

arguments of ``chart``. 

Coordinates are Sage symbolic variables (see 

:mod:`sage.symbolic.expression`):: 

sage: type(th) 

<type 'sage.symbolic.expression.Expression'> 

sage: latex(th) 

{\theta} 

sage: assumptions(th) 

[th is real, th > 0, th < pi] 

Coordinate are also accessible by their indices:: 

sage: x1 = c_spher[1]; x2 = c_spher[2]; x3 = c_spher[3] 

sage: [x1, x2, x3] 

[r, th, ph] 

sage: (x1, x2, x3) == (r, th, ph) 

True 

The full set of coordinates is obtained by means of the operator [:]:: 

sage: c_cart[:] 

(x, y, z) 

sage: c_spher[:] 

(r, th, ph) 

Let us check that the declared coordinate ranges have been taken into 

account:: 

sage: c_cart.coord_range() 

x: (-oo, +oo); y: (-oo, +oo); z: (-oo, +oo) 

sage: c_spher.coord_range() 

r: (0, +oo); th: (0, pi); ph: (0, 2*pi) 

sage: bool(th>0 and th<pi) 

True 

sage: assumptions() # list all current symbolic assumptions 

[x is real, y is real, z is real, r is real, r > 0, th is real, 

th > 0, th < pi, ph is real, ph > 0, ph < 2*pi] 

The coordinate ranges are used for simplifications:: 

sage: simplify(abs(r)) # r has been declared to lie in the interval (0,+oo) 

r 

sage: simplify(abs(x)) # no positive range has been declared for x 

abs(x) 

Each constructed chart is automatically added to the manifold's user atlas:: 

sage: M.atlas() 

[Chart (R^3, (x, y, z)), Chart (U, (r, th, ph))] 

and to the atlas of its domain:: 

sage: U.atlas() 

[Chart (U, (r, th, ph))] 

Manifold subsets have a *default chart*, which, unless changed via the 

method 

:meth:`~sage.manifolds.manifold.TopologicalManifold.set_default_chart`, 

is the first defined chart on the subset (or on a open subset of it):: 

sage: M.default_chart() 

Chart (R^3, (x, y, z)) 

sage: U.default_chart() 

Chart (U, (r, th, ph)) 

The default charts are not privileged charts on the manifold, but rather 

charts whose name can be skipped in the argument list of functions having 

an optional ``chart=`` argument. 

The action of the chart map `\varphi` on a point is obtained by means of 

the call operator, i.e. the operator ``()``:: 

sage: p = M.point((1,0,-2)); p 

Point on the 3-dimensional differentiable manifold R^3 

sage: c_cart(p) 

(1, 0, -2) 

sage: c_cart(p) == p.coord(c_cart) 

True 

sage: q = M.point((2,pi/2,pi/3), chart=c_spher) # point defined by its spherical coordinates 

sage: c_spher(q) 

(2, 1/2*pi, 1/3*pi) 

sage: c_spher(q) == q.coord(c_spher) 

True 

sage: a = U.point((1,pi/2,pi)) # the default coordinates on U are the spherical ones 

sage: c_spher(a) 

(1, 1/2*pi, pi) 

sage: c_spher(a) == a.coord(c_spher) 

True 

Cartesian coordinates on `U` as an example of chart construction with 

coordinate restrictions: since `U` is the complement of the half-plane 

`\{y=0, x\geq 0\}`, we must have `y\not=0` or `x<0` on U. Accordingly, 

we set:: 

sage: c_cartU.<x,y,z> = U.chart() 

sage: c_cartU.add_restrictions((y!=0, x<0)) # the tuple (y!=0, x<0) means y!=0 or x<0 

sage: # c_cartU.add_restrictions([y!=0, x<0]) would have meant y!=0 AND x<0 

sage: U.atlas() 

[Chart (U, (r, th, ph)), Chart (U, (x, y, z))] 

sage: M.atlas() 

[Chart (R^3, (x, y, z)), Chart (U, (r, th, ph)), Chart (U, (x, y, z))] 

sage: c_cartU.valid_coordinates(-1,0,2) 

True 

sage: c_cartU.valid_coordinates(1,0,2) 

False 

sage: c_cart.valid_coordinates(1,0,2) 

True 

A vector frame is naturally associated to each chart:: 

sage: c_cart.frame() 

Coordinate frame (R^3, (d/dx,d/dy,d/dz)) 

sage: c_spher.frame() 

Coordinate frame (U, (d/dr,d/dth,d/dph)) 

as well as a dual frame (basis of 1-forms):: 

sage: c_cart.coframe() 

Coordinate coframe (R^3, (dx,dy,dz)) 

sage: c_spher.coframe() 

Coordinate coframe (U, (dr,dth,dph)) 

Chart grids can be drawn in 2D or 3D graphics thanks to the method 

:meth:`~sage.manifolds.chart.RealChart.plot`. 

""" 

def __init__(self, domain, coordinates='', names=None, calc_method=None): 

r""" 

Construct a chart on a real differentiable manifold. 

TESTS:: 

sage: forget() # for doctests only 

sage: M = Manifold(2, 'M') 

sage: X.<x,y> = M.chart() 

sage: X 

Chart (M, (x, y)) 

sage: type(X) 

<class 'sage.manifolds.differentiable.chart.RealDiffChart'> 

sage: assumptions() # assumptions set in X._init_coordinates 

[x is real, y is real] 

sage: TestSuite(X).run() 

""" 

RealChart.__init__(self, domain, coordinates=coordinates, names=names, 

calc_method = calc_method) 

# Construction of the coordinate frame associated to the chart: 

self._frame = CoordFrame(self) 

self._coframe = self._frame._coframe 

def restrict(self, subset, restrictions=None): 

r""" 

Return the restriction of the chart to some subset. 

If the current chart is `(U, \varphi)`, a *restriction* (or 

*subchart*) is a chart `(V, \psi)` such that `V \subset U` 

and `\psi = \varphi |_V`. 

If such subchart has not been defined yet, it is constructed here. 

The coordinates of the subchart bare the same names as the 

coordinates of the original chart. 

INPUT: 

- ``subset`` -- open subset `V` of the chart domain `U` 

- ``restrictions`` -- (default: ``None``) list of coordinate 

restrictions defining the subset `V` 

A restriction can be any symbolic equality or inequality involving 

the coordinates, such as ``x > y`` or ``x^2 + y^2 != 0``. The items 

of the list ``restrictions`` are combined with the ``and`` operator; 

if some restrictions are to be combined with the ``or`` operator 

instead, they have to be passed as a tuple in some single item 

of the list ``restrictions``. For example:: 

restrictions = [x > y, (x != 0, y != 0), z^2 < x] 

means ``(x > y) and ((x != 0) or (y != 0)) and (z^2 < x)``. 

If the list ``restrictions`` contains only one item, this 

item can be passed as such, i.e. writing ``x > y`` instead 

of the single element list ``[x > y]``. 

OUTPUT: 

- a :class:`RealDiffChart` `(V, \psi)` 

EXAMPLES: 

Cartesian coordinates on the unit open disc in `\RR^2` as a subchart 

of the global Cartesian coordinates:: 

sage: M = Manifold(2, 'R^2') 

sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2 

sage: D = M.open_subset('D') # the unit open disc 

sage: c_cart_D = c_cart.restrict(D, x^2+y^2<1) 

sage: p = M.point((1/2, 0)) 

sage: p in D 

True 

sage: q = M.point((1, 2)) 

sage: q in D 

False 

Cartesian coordinates on the annulus `1 < \sqrt{x^2+y^2} < 2`:: 

sage: A = M.open_subset('A') 

sage: c_cart_A = c_cart.restrict(A, [x^2+y^2>1, x^2+y^2<4]) 

sage: p in A, q in A 

(False, False) 

sage: a = M.point((3/2,0)) 

sage: a in A 

True 

""" 

if subset == self._domain: 

return self 

if subset not in self._dom_restrict: 

resu = RealChart.restrict(self, subset, restrictions=restrictions) 

# Update of superframes and subframes: 

resu._frame._superframes.update(self._frame._superframes) 

for sframe in self._frame._superframes: 

sframe._subframes.add(resu._frame) 

sframe._restrictions[subset] = resu._frame 

# The subchart frame is not a "top frame" in the supersets 

# (including self._domain): 

for dom in self._domain._supersets: 

if resu._frame in dom._top_frames: 

# it was added by the Chart constructor invoked in 

# Chart.restrict above 

dom._top_frames.remove(resu._frame) 

return self._dom_restrict[subset] 

#****************************************************************************** 

class DiffCoordChange(CoordChange): 

r""" 

Transition map between two charts of a differentiable manifold. 

Giving two coordinate charts `(U,\varphi)` and `(V,\psi)` on a 

differentiable manifold `M` of dimension `n` over a topological field `K`, 

the *transition map from* `(U,\varphi)` *to* `(V,\psi)` is the map 

.. MATH:: 

\psi\circ\varphi^{-1}: \varphi(U\cap V) \subset K^n 

\rightarrow \psi(U\cap V) \subset K^n, 

In other words, the transition map `\psi\circ\varphi^{-1}` expresses the 

coordinates `(y^1,\ldots,y^n)` of `(V,\psi)` in terms of the coordinates 

`(x^1,\ldots,x^n)` of `(U,\varphi)` on the open subset where the two 

charts intersect, i.e. on `U\cap V`. 

By definition, the transition map `\psi\circ\varphi^{-1}` must be 

of classe `C^k`, where `k` is the degree of differentiability of the 

manifold (cf. 

:meth:`~sage.manifolds.differentiable.manifold.DifferentiableManifold.diff_degree`). 

INPUT: 

- ``chart1`` -- chart `(U,\varphi)` 

- ``chart2`` -- chart `(V,\psi)` 

- ``transformations`` -- tuple (or list) `(Y_1,\ldots,Y_2)`, where 

`Y_i` is the symbolic expression of the coordinate `y^i` in terms 

of the coordinates `(x^1,\ldots,x^n)` 

EXAMPLES: 

Transition map on a 2-dimensional differentiable manifold:: 

sage: M = Manifold(2, 'M') 

sage: X.<x,y> = M.chart() 

sage: Y.<u,v> = M.chart() 

sage: X_to_Y = X.transition_map(Y, [x+y, x-y]) 

sage: X_to_Y 

Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v)) 

sage: type(X_to_Y) 

<class 'sage.manifolds.differentiable.chart.DiffCoordChange'> 

sage: X_to_Y.display() 

u = x + y 

v = x - y 

""" 

def __init__(self, chart1, chart2, *transformations): 

r""" 

Construct a transition map. 

TESTS:: 

sage: M = Manifold(2, 'M') 

sage: X.<x,y> = M.chart() 

sage: Y.<u,v> = M.chart() 

sage: X_to_Y = X.transition_map(Y, [x+y, x-y]) 

sage: X_to_Y 

Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v)) 

sage: type(X_to_Y) 

<class 'sage.manifolds.differentiable.chart.DiffCoordChange'> 

sage: TestSuite(X_to_Y).run(skip='_test_pickling') 

.. TODO:: 

fix _test_pickling 

""" 

CoordChange.__init__(self, chart1, chart2, *transformations) 

# Jacobian matrix: 

self._jacobian = self._transf.jacobian() 

# If the two charts are on the same open subset, the Jacobian matrix is 

# added to the dictionary of changes of frame: 

if chart1._domain == chart2._domain: 

domain = chart1._domain 

frame1 = chart1._frame 

frame2 = chart2._frame 

vf_module = domain.vector_field_module() 

ch_basis = vf_module.automorphism() 

ch_basis.add_comp(frame1)[:, chart1] = self._jacobian 

ch_basis.add_comp(frame2)[:, chart1] = self._jacobian 

vf_module._basis_changes[(frame2, frame1)] = ch_basis 

for sdom in domain._supersets: 

sdom._frame_changes[(frame2, frame1)] = ch_basis 

# The inverse is computed only if it does not exist already 

# (because if it exists it may have a simpler expression than that 

# obtained from the matrix inverse) 

if (frame1, frame2) not in vf_module._basis_changes: 

ch_basis_inv = ch_basis.inverse() 

vf_module._basis_changes[(frame1, frame2)] = ch_basis_inv 

for sdom in domain._supersets: 

sdom._frame_changes[(frame1, frame2)] = ch_basis_inv 

def jacobian(self): 

r""" 

Return the Jacobian matrix of ``self``. 

If ``self`` corresponds to the change of coordinates 

.. MATH:: 

y^i = Y^i(x^1,\ldots,x^n)\qquad 1\leq i \leq n 

the Jacobian matrix `J` is given by 

.. MATH:: 

J_{ij} = \frac{\partial Y^i}{\partial x^j} 

where `i` is the row index and `j` the column one. 

OUTPUT: 

- Jacobian matrix `J`, the elements `J_{ij}` of which being 

coordinate functions 

(cf. :class:`~sage.manifolds.chart_func.ChartFunction`) 

EXAMPLES: 

Jacobian matrix of a 2-dimensional transition map:: 

sage: M = Manifold(2, 'M') 

sage: X.<x,y> = M.chart() 

sage: Y.<u,v> = M.chart() 

sage: X_to_Y = X.transition_map(Y, [x+y^2, 3*x-y]) 

sage: X_to_Y.jacobian() 

[ 1 2*y] 

[ 3 -1] 

Each element of the Jacobian matrix is a coordinate function:: 

sage: parent(X_to_Y.jacobian()[0,0]) 

Ring of chart functions on Chart (M, (x, y)) 

""" 

return self._jacobian # has been computed in __init__ 

@cached_method 

def jacobian_det(self): 

r""" 

Return the Jacobian determinant of ``self``. 

The Jacobian determinant is the determinant of the Jacobian 

matrix (see :meth:`jacobian`). 

OUTPUT: 

- determinant of the Jacobian matrix `J` as a coordinate 

function 

(cf. :class:`~sage.manifolds.chart_func.ChartFunction`) 

EXAMPLES: 

Jacobian determinant of a 2-dimensional transition map:: 

sage: M = Manifold(2, 'M') 

sage: X.<x,y> = M.chart() 

sage: Y.<u,v> = M.chart() 

sage: X_to_Y = X.transition_map(Y, [x+y^2, 3*x-y]) 

sage: X_to_Y.jacobian_det() 

-6*y - 1 

sage: X_to_Y.jacobian_det() == det(X_to_Y.jacobian()) 

True 

The Jacobian determinant is a coordinate function:: 

sage: parent(X_to_Y.jacobian_det()) 

Ring of chart functions on Chart (M, (x, y)) 

""" 

return self._transf.jacobian_det()