Tangent Vectors#

The class TangentVector implements tangent vectors to a differentiable manifold.

AUTHORS:

  • Eric Gourgoulhon, Michal Bejger (2014-2015): initial version

  • Travis Scrimshaw (2016): review tweaks

REFERENCES:

class sage.manifolds.differentiable.tangent_vector.TangentVector(parent, name=None, latex_name=None)#

Bases: FiniteRankFreeModuleElement

Tangent vector to a differentiable manifold at a given point.

INPUT:

  • parent – 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:

A tangent vector \(v\) on a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: X.<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 ∂/∂x + ∂/∂y
sage: v.parent()
Tangent space at Point p on the 2-dimensional differentiable manifold M
sage: v in Tp
True

Tangent vectors can also be constructed via the manifold method tangent_vector():

sage: v = M.tangent_vector(p, (-2, 1), name='v'); v
Tangent vector v at Point p on the 2-dimensional differentiable
 manifold M
sage: v.display()
v = -2 ∂/∂x + ∂/∂y

or via the method at() of vector fields:

sage: vf = M.vector_field(x - 4*y/3, (x-y)^2, name='v')
sage: v = vf.at(p); v
Tangent vector v at Point p on the 2-dimensional differentiable
 manifold M
sage: v.display()
v = -2 ∂/∂x + ∂/∂y

By definition, a tangent vector at \(p\in M\) is a derivation at \(p\) on the space \(C^\infty(M)\) of smooth scalar fields on \(M\). Indeed let us consider a generic scalar field \(f\):

sage: f = M.scalar_field(function('F')(x,y), name='f')
sage: f.display()
f: M → ℝ
   (x, y) ↦ F(x, y)

The tangent vector \(v\) maps \(f\) to the real number \(v^i \left. \frac{\partial F}{\partial x^i} \right|_p\):

sage: v(f)
-2*D[0](F)(2, 3) + D[1](F)(2, 3)
sage: vdf(x, y) = v[0]*diff(f.expr(), x) + v[1]*diff(f.expr(), y)
sage: X(p)
(2, 3)
sage: bool( v(f) == vdf(*X(p)) )
True

and if \(g\) is a second scalar field on \(M\):

sage: g = M.scalar_field(function('G')(x,y), name='g')

then the product \(f g\) is also a scalar field on \(M\):

sage: (f*g).display()
f*g: M → ℝ
   (x, y) ↦ F(x, y)*G(x, y)

and we have the derivation law \(v(f g) = v(f) g(p) + f(p) v(g)\):

sage: bool( v(f*g) == v(f)*g(p) + f(p)*v(g) )
True

See also

FiniteRankFreeModuleElement for more documentation.

plot(chart=None, ambient_coords=None, mapping=None, color='blue', print_label=True, label=None, label_color=None, fontsize=10, label_offset=0.1, parameters=None, scale=1, **extra_options)#

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) 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 arrow2d() and arrow3d() for details)

OUTPUT:

  • a graphic object, either an instance of Graphics for a 2D plot (i.e. based on 2 coordinates of chart) or an instance of 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()                                                              # needs sage.plot
Graphics object consisting of 2 graphics primitives

Plot atop of the chart grid:

sage: X.plot() + v.plot()                                                   # needs sage.plot
Graphics object consisting of 20 graphics primitives
../../../_images/tangent_vector-1.svg

Plots with various options:

sage: X.plot() + v.plot(color='green', scale=2, label='V')                  # needs sage.plot
Graphics object consisting of 20 graphics primitives
../../../_images/tangent_vector-2.svg
sage: X.plot() + v.plot(print_label=False)                                  # needs sage.plot
Graphics object consisting of 19 graphics primitives
../../../_images/tangent_vector-3.svg
sage: X.plot() + v.plot(color='green', label_color='black',                 # needs sage.plot
....:                   fontsize=20, label_offset=0.2)
Graphics object consisting of 20 graphics primitives
../../../_images/tangent_vector-4.svg
sage: X.plot() + v.plot(linestyle=':', width=4, arrowsize=8,                # needs sage.plot
....:                   fontsize=20)
Graphics object consisting of 20 graphics primitives
../../../_images/tangent_vector-5.svg

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) ∂/∂x - b^2 ∂/∂y
sage: X.plot() + v.plot(parameters={a: -2, b: 3})                           # needs sage.plot
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()                                                   # needs sage.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()                                                              # needs sage.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))                                          # needs sage.plot
Graphics object consisting of 2 graphics primitives
../../../_images/tangent_vector-6.svg

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))                                    # needs sage.plot
sage: print(g)                                                              # needs sage.plot
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                          # needs sage.plot
....:  + v.plot(ambient_coords=(t,x,z),
....:           label_offset=0.5, width=6))
Graphics3d Object
../../../_images/tangent_vector-7.svg

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 ∂/∂phi at p
Tangent vector ∂/∂ph at Point p on the 2-dimensional differentiable
 manifold S^2
sage: graph_v = v.plot(mapping=F)                                           # needs sage.plot
sage: graph_S2 = XS.plot(chart=X3, mapping=F, number_values=9)  # long time, needs sage.plot
sage: graph_v + graph_S2                                        # long time, needs sage.plot
Graphics3d Object
../../../_images/tangent_vector-8.svg