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r""" Tangent Vectors
The class :class:`TangentVector` implements tangent vectors to a differentiable manifold.
AUTHORS:
- Eric Gourgoulhon, Michal Bejger (2014-2015): initial version - Travis Scrimshaw (2016): review tweaks
REFERENCES:
- Chap. 3 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/ #******************************************************************************
r""" Tangent vector to a differentiable manifold at a given point.
INPUT:
- ``parent`` -- :class:`~sage.manifolds.differentiable.tangent_space.TangentSpace`; the tangent space to which the vector belongs - ``name`` -- (default: ``None``) string; symbol given to the vector - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to denote the vector; if ``None``, ``name`` will be used
EXAMPLES:
Tangent vector on a 2-dimensional manifold::
sage: M = Manifold(2, 'M') sage: c_xy.<x,y> = M.chart() sage: p = M.point((2,3), name='p') sage: Tp = M.tangent_space(p) sage: v = Tp((-2,1), name='v') ; v Tangent vector v at Point p on the 2-dimensional differentiable manifold M sage: v.display() v = -2 d/dx + d/dy sage: v.parent() Tangent space at Point p on the 2-dimensional differentiable manifold M sage: v in Tp True
.. SEEALSO::
:class:`~sage.tensor.modules.free_module_element.FiniteRankFreeModuleElement` for more documentation.
""" r""" Construct a tangent vector.
TESTS::
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: p = M.point((1,-2), name='p') sage: Tp = M.tangent_space(p) sage: v = Tp.element_class(Tp, name='v') ; v Tangent vector v at Point p on the 2-dimensional differentiable manifold M sage: v[:] = 5, -3/2 sage: TestSuite(v).run()
""" latex_name=latex_name) # Extra data (with respect to FiniteRankFreeModuleElement):
r""" String representation of ``self``.
TESTS::
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: p = M.point((1,-2), name='p') sage: Tp = M.tangent_space(p) sage: v = Tp([-3,2], name='v') sage: v._repr_() 'Tangent vector v at Point p on the 2-dimensional differentiable manifold M' sage: repr(v) # indirect doctest 'Tangent vector v at Point p on the 2-dimensional differentiable manifold M'
"""
color='blue', print_label=True, label=None, label_color=None, fontsize=10, label_offset=0.1, parameters=None, **extra_options): r""" Plot the vector in a Cartesian graph based on the coordinates of some ambient chart.
The vector is drawn in terms of two (2D graphics) or three (3D graphics) coordinates of a given chart, called hereafter the *ambient chart*. The vector's base point `p` (or its image `\Phi(p)` by some differentiable mapping `\Phi`) must lie in the ambient chart's domain. If `\Phi` is different from the identity mapping, the vector actually depicted is `\mathrm{d}\Phi_p(v)`, where `v` is the current vector (``self``) (see the example of a vector tangent to the 2-sphere below, where `\Phi: S^2 \to \RR^3`).
INPUT:
- ``chart`` -- (default: ``None``) the ambient chart (see above); if ``None``, it is set to the default chart of the open set containing the point at which the vector (or the vector image via the differential `\mathrm{d}\Phi_p` of ``mapping``) is defined
- ``ambient_coords`` -- (default: ``None``) tuple containing the 2 or 3 coordinates of the ambient chart in terms of which the plot is performed; if ``None``, all the coordinates of the ambient chart are considered
- ``mapping`` -- (default: ``None``) :class:`~sage.manifolds.differentiable.diff_map.DiffMap`; differentiable mapping `\Phi` providing the link between the point `p` at which the vector is defined and the ambient chart ``chart``: the domain of ``chart`` must contain `\Phi(p)`; if ``None``, the identity mapping is assumed
- ``scale`` -- (default: 1) value by which the length of the arrow representing the vector is multiplied
- ``color`` -- (default: 'blue') color of the arrow representing the vector
- ``print_label`` -- (boolean; default: ``True``) determines whether a label is printed next to the arrow representing the vector
- ``label`` -- (string; default: ``None``) label printed next to the arrow representing the vector; if ``None``, the vector's symbol is used, if any
- ``label_color`` -- (default: ``None``) color to print the label; if ``None``, the value of ``color`` is used
- ``fontsize`` -- (default: 10) size of the font used to print the label
- ``label_offset`` -- (default: 0.1) determines the separation between the vector arrow and the label
- ``parameters`` -- (default: ``None``) dictionary giving the numerical values of the parameters that may appear in the coordinate expression of ``self`` (see example below)
- ``**extra_options`` -- extra options for the arrow plot, like ``linestyle``, ``width`` or ``arrowsize`` (see :func:`~sage.plot.arrow.arrow2d` and :func:`~sage.plot.plot3d.shapes.arrow3d` for details)
OUTPUT:
- a graphic object, either an instance of :class:`~sage.plot.graphics.Graphics` for a 2D plot (i.e. based on 2 coordinates of ``chart``) or an instance of :class:`~sage.plot.plot3d.base.Graphics3d` for a 3D plot (i.e. based on 3 coordinates of ``chart``)
EXAMPLES:
Vector tangent to a 2-dimensional manifold::
sage: M = Manifold(2, 'M') sage: X.<x,y> = M.chart() sage: p = M((2,2), name='p') sage: Tp = M.tangent_space(p) sage: v = Tp((2, 1), name='v') ; v Tangent vector v at Point p on the 2-dimensional differentiable manifold M
Plot of the vector alone (arrow + label)::
sage: v.plot() Graphics object consisting of 2 graphics primitives
Plot atop of the chart grid::
sage: X.plot() + v.plot() Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M') X = M.chart('x y'); x, y = X[:] p = M((2,2), name='p'); Tp = M.tangent_space(p) v = Tp((2, 1), name='v') g = X.plot() + v.plot() sphinx_plot(g)
Plots with various options::
sage: X.plot() + v.plot(color='green', scale=2, label='V') Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M') X = M.chart('x y'); x, y = X[:] p = M((2,2), name='p'); Tp = M.tangent_space(p) v = Tp((2, 1), name='v') g = X.plot() + v.plot(color='green', scale=2, label='V') sphinx_plot(g)
::
sage: X.plot() + v.plot(print_label=False) Graphics object consisting of 19 graphics primitives
.. PLOT::
M = Manifold(2, 'M') X = M.chart('x y'); x, y = X[:] p = M((2,2), name='p'); Tp = M.tangent_space(p) v = Tp((2, 1), name='v') g = X.plot() + v.plot(print_label=False) sphinx_plot(g)
::
sage: X.plot() + v.plot(color='green', label_color='black', ....: fontsize=20, label_offset=0.2) Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M') X = M.chart('x y'); x, y = X[:] p = M((2,2), name='p'); Tp = M.tangent_space(p) v = Tp((2, 1), name='v') g = X.plot() + v.plot(color='green', label_color='black', fontsize=20, label_offset=0.2) sphinx_plot(g)
::
sage: X.plot() + v.plot(linestyle=':', width=4, arrowsize=8, ....: fontsize=20) Graphics object consisting of 20 graphics primitives
.. PLOT::
M = Manifold(2, 'M') X = M.chart('x y'); x, y = X[:] p = M((2,2), name='p'); Tp = M.tangent_space(p) v = Tp((2, 1), name='v') g = X.plot() + v.plot(linestyle=':', width=4, arrowsize=8, fontsize=20) sphinx_plot(g)
Plot with specific values of some free parameters::
sage: var('a b') (a, b) sage: v = Tp((1+a, -b^2), name='v') ; v.display() v = (a + 1) d/dx - b^2 d/dy sage: X.plot() + v.plot(parameters={a: -2, b: 3}) Graphics object consisting of 20 graphics primitives
Special case of the zero vector::
sage: v = Tp.zero() ; v Tangent vector zero at Point p on the 2-dimensional differentiable manifold M sage: X.plot() + v.plot() Graphics object consisting of 19 graphics primitives
Vector tangent to a 4-dimensional manifold::
sage: M = Manifold(4, 'M') sage: X.<t,x,y,z> = M.chart() sage: p = M((0,1,2,3), name='p') sage: Tp = M.tangent_space(p) sage: v = Tp((5,4,3,2), name='v') ; v Tangent vector v at Point p on the 4-dimensional differentiable manifold M
We cannot make a 4D plot directly::
sage: v.plot() Traceback (most recent call last): ... ValueError: the number of coordinates involved in the plot must be either 2 or 3, not 4
Rather, we have to select some chart coordinates for the plot, via the argument ``ambient_coords``. For instance, for a 2-dimensional plot in terms of the coordinates `(x, y)`::
sage: v.plot(ambient_coords=(x,y)) Graphics object consisting of 2 graphics primitives
.. PLOT::
M = Manifold(4, 'M') X = M.chart('t x y z'); t,x,y,z = X[:] p = M((0,1,2,3), name='p'); Tp = M.tangent_space(p) v = Tp((5,4,3,2), name='v') g = X.plot(ambient_coords=(x,y)) + v.plot(ambient_coords=(x,y)) sphinx_plot(g)
This plot involves only the components `v^x` and `v^y` of `v`. Similarly, for a 3-dimensional plot in terms of the coordinates `(t, x, y)`::
sage: g = v.plot(ambient_coords=(t,x,z)) sage: print(g) Graphics3d Object
This plot involves only the components `v^t`, `v^x` and `v^z` of `v`. A nice 3D view atop the coordinate grid is obtained via::
sage: (X.plot(ambient_coords=(t,x,z)) # long time ....: + v.plot(ambient_coords=(t,x,z), ....: label_offset=0.5, width=6)) Graphics3d Object
.. PLOT::
M = Manifold(4, 'M') X = M.chart('t x y z'); t,x,y,z = X[:] p = M((0,1,2,3), name='p'); Tp = M.tangent_space(p) v = Tp((5,4,3,2), name='v') g = X.plot(ambient_coords=(t,x,z)) + v.plot(ambient_coords=(t,x,z), label_offset=0.5, width=6) sphinx_plot(g)
An example of plot via a differential mapping: plot of a vector tangent to a 2-sphere viewed in `\RR^3`::
sage: S2 = Manifold(2, 'S^2') sage: U = S2.open_subset('U') # the open set covered by spherical coord. sage: XS.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi') sage: R3 = Manifold(3, 'R^3') sage: X3.<x,y,z> = R3.chart() sage: F = S2.diff_map(R3, {(XS, X3): [sin(th)*cos(ph), ....: sin(th)*sin(ph), ....: cos(th)]}, name='F') sage: F.display() # the standard embedding of S^2 into R^3 F: S^2 --> R^3 on U: (th, ph) |--> (x, y, z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th)) sage: p = U.point((pi/4, 7*pi/4), name='p') sage: v = XS.frame()[1].at(p) ; v # the coordinate vector d/dphi at p Tangent vector d/dph at Point p on the 2-dimensional differentiable manifold S^2 sage: graph_v = v.plot(mapping=F) sage: graph_S2 = XS.plot(chart=X3, mapping=F, number_values=9) # long time sage: graph_v + graph_S2 # long time Graphics3d Object
.. PLOT::
S2 = Manifold(2, 'S^2') U = S2.open_subset('U') XS = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi') th, ph = XS[:] R3 = Manifold(3, 'R^3') X3 = R3.chart('x y z') F = S2.diff_map(R3, {(XS, X3): [sin(th)*cos(ph), sin(th)*sin(ph), cos(th)]}, name='F') p = U.point((pi/4, 7*pi/4), name='p') v = XS.frame()[1].at(p) graph_v = v.plot(mapping=F) graph_S2 = XS.plot(chart=X3, mapping=F, number_values=9) sphinx_plot(graph_v + graph_S2)
"""
# # The "effective" vector to be plotted # else: #!# check # For efficiency, the method FiniteRankFreeModuleMorphism._call_() # is called instead of FiniteRankFreeModuleMorphism.__call__() # # The chart w.r.t. which the vector is plotted # raise TypeError("{} is not a chart".format(chart)) # # Coordinates of the above chart w.r.t. which the vector is plotted # "plot must be either 2 or 3, not {}".format(n_pc)) # indices coordinates involved in the plot: # # Components of the vector w.r.t. the chart frame # # # The arrow # for i in ind_pc] else: for i in ind_pc] (xp[i] + scale*vcomp[i]).substitute(parameters)) for i in ind_pc] color=color, **extra_options) else: **extra_options) # # The label # color=label_color) else: color=label_color)
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