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:
- Chap. 3 of [Lee2013] 
- class sage.manifolds.differentiable.tangent_vector.TangentVector(parent, name=None, latex_name=None)[source]¶
- 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,- namewill 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 - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2) >>> p = M.point((Integer(2),Integer(3)), name='p') >>> Tp = M.tangent_space(p) >>> v = Tp((-Integer(2),Integer(1)), name='v') ; v Tangent vector v at Point p on the 2-dimensional differentiable manifold M >>> v.display() v = -2 ∂/∂x + ∂/∂y >>> v.parent() Tangent space at Point p on the 2-dimensional differentiable manifold M >>> 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 - >>> from sage.all import * >>> v = M.tangent_vector(p, (-Integer(2), Integer(1)), name='v'); v Tangent vector v at Point p on the 2-dimensional differentiable manifold M >>> 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 - >>> from sage.all import * >>> vf = M.vector_field(x - Integer(4)*y/Integer(3), (x-y)**Integer(2), name='v') >>> v = vf.at(p); v Tangent vector v at Point p on the 2-dimensional differentiable manifold M >>> 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) - >>> from sage.all import * >>> f = M.scalar_field(function('F')(x,y), name='f') >>> 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 - >>> from sage.all import * >>> v(f) -2*D[0](F)(2, 3) + D[1](F)(2, 3) >>> __tmp__=var("x,y"); vdf = symbolic_expression(v[Integer(0)]*diff(f.expr(), x) + v[Integer(1)]*diff(f.expr(), y)).function(x,y) >>> X(p) (2, 3) >>> 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') - >>> from sage.all import * >>> 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) - >>> from sage.all import * >>> (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 - >>> from sage.all import * >>> bool( v(f*g) == v(f)*g(p) + f(p)*v(g) ) True - See also - FiniteRankFreeModuleElementfor 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)[source]¶
- 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- chartmust 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- coloris 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,- widthor- arrowsize(see- arrow2d()and- arrow3d()for details)
 - OUTPUT: - a graphic object, either an instance of - Graphicsfor a 2D plot (i.e. based on 2 coordinates of- chart) or an instance of- Graphics3dfor 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 - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2) >>> p = M((Integer(2),Integer(2)), name='p') >>> Tp = M.tangent_space(p) >>> v = Tp((Integer(2), Integer(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 - >>> from sage.all import * >>> 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 - >>> from sage.all import * >>> X.plot() + v.plot() # needs sage.plot Graphics object consisting of 20 graphics primitives - 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 - >>> from sage.all import * >>> X.plot() + v.plot(color='green', scale=Integer(2), label='V') # needs sage.plot Graphics object consisting of 20 graphics primitives - sage: X.plot() + v.plot(print_label=False) # needs sage.plot Graphics object consisting of 19 graphics primitives - >>> from sage.all import * >>> X.plot() + v.plot(print_label=False) # needs sage.plot Graphics object consisting of 19 graphics primitives - 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 - >>> from sage.all import * >>> X.plot() + v.plot(color='green', label_color='black', # needs sage.plot ... fontsize=Integer(20), label_offset=RealNumber('0.2')) Graphics object consisting of 20 graphics primitives - sage: X.plot() + v.plot(linestyle=':', width=4, arrowsize=8, # needs sage.plot ....: fontsize=20) Graphics object consisting of 20 graphics primitives - >>> from sage.all import * >>> X.plot() + v.plot(linestyle=':', width=Integer(4), arrowsize=Integer(8), # needs sage.plot ... fontsize=Integer(20)) Graphics object consisting of 20 graphics primitives - 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 - >>> from sage.all import * >>> var('a b') (a, b) >>> v = Tp((Integer(1)+a, -b**Integer(2)), name='v') ; v.display() v = (a + 1) ∂/∂x - b^2 ∂/∂y >>> X.plot() + v.plot(parameters={a: -Integer(2), b: Integer(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 - >>> from sage.all import * >>> v = Tp.zero() ; v Tangent vector zero at Point p on the 2-dimensional differentiable manifold M >>> 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 - >>> from sage.all import * >>> M = Manifold(Integer(4), 'M') >>> X = M.chart(names=('t', 'x', 'y', 'z',)); (t, x, y, z,) = X._first_ngens(4) >>> p = M((Integer(0),Integer(1),Integer(2),Integer(3)), name='p') >>> Tp = M.tangent_space(p) >>> v = Tp((Integer(5),Integer(4),Integer(3),Integer(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 - >>> from sage.all import * >>> 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 - >>> from sage.all import * >>> v.plot(ambient_coords=(x,y)) # needs sage.plot Graphics object consisting of 2 graphics primitives - 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 - >>> from sage.all import * >>> g = v.plot(ambient_coords=(t,x,z)) # needs sage.plot >>> 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 - >>> from sage.all import * >>> (X.plot(ambient_coords=(t,x,z)) # long time # needs sage.plot ... + v.plot(ambient_coords=(t,x,z), ... label_offset=RealNumber('0.5'), width=Integer(6))) Graphics3d Object - 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 - >>> from sage.all import * >>> S2 = Manifold(Integer(2), 'S^2') >>> U = S2.open_subset('U') # the open set covered by spherical coord. >>> XS = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi', names=('th', 'ph',)); (th, ph,) = XS._first_ngens(2) >>> R3 = Manifold(Integer(3), 'R^3') >>> X3 = R3.chart(names=('x', 'y', 'z',)); (x, y, z,) = X3._first_ngens(3) >>> F = S2.diff_map(R3, {(XS, X3): [sin(th)*cos(ph), ... sin(th)*sin(ph), ... cos(th)]}, name='F') >>> 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)) >>> p = U.point((pi/Integer(4), Integer(7)*pi/Integer(4)), name='p') >>> v = XS.frame()[Integer(1)].at(p) ; v # the coordinate vector ∂/∂phi at p Tangent vector ∂/∂ph at Point p on the 2-dimensional differentiable manifold S^2 >>> graph_v = v.plot(mapping=F) # needs sage.plot >>> graph_S2 = XS.plot(chart=X3, mapping=F, number_values=Integer(9)) # long time, needs sage.plot >>> graph_v + graph_S2 # long time, needs sage.plot Graphics3d Object