ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Fri, 10 Jul 2020 06:13:31 +0200Defining $S^1$ with four chartshttps://ask.sagemath.org/question/52400/defining-s1-with-four-charts/Hello,
I am trying to implement the construction of the manifold $S^1$ using the four charts, as described at en.wikipedia.org/wiki/Manifold#Circle . It's not clear to me how to define the charts in terms of the projection operators within SageMath. I'm also not sure how to isolate open sets that I would like to define the charts over. The circle is parameterized using polar coordinates, and the four charts have the form:
$
\begin{align*}
\varphi_1(r, \theta) &= \theta && \theta \in (-\frac{\pi}{3}, \frac{\pi}{3}) \\
\tag{1}
\end{align*}
$
$
\begin{align*}
\varphi_2(r, \theta) &= \theta && \theta \in (\frac{\pi}{6}, \frac{5\pi}{6}) \\
\tag{2}
\end{align*}
$
$
\begin{align*}
\varphi_3(r, \theta) &= \theta && \theta \in (\frac{2\pi}{3}, \frac{4\pi}{3}) \\
\tag{3}
\end{align*}
$
$
\begin{align*}
\varphi_4(r, \theta) &= \theta && \theta \in (\frac{7\pi}{6}, \frac{11\pi}{6}) \\
\tag{4}
\end{align*}
$
Once we have these charts, how would I define the four transition functions to complete the description?
Thank you for your help!wintermuteFri, 10 Jul 2020 06:13:31 +0200https://ask.sagemath.org/question/52400/Inverse of the transition map on a Manifold doesn't holdhttps://ask.sagemath.org/question/48264/inverse-of-the-transition-map-on-a-manifold-doesnt-hold/I was trying to represent S³ as a three dimensional manifold, with coordinates (x,y,z,w) in R⁴, and make the transition map from the upper cap w > 0 to the lateral cap z<0, with the charts being the graphs of the caps as functions. I came up with the following code:
M = Manifold(3, 'S^3')
N = M.open_subset('N')
projN.<x,y,z> = N.chart()
E = M.open_subset('E')
projE.<x,y,w> = E.chart()
ProjNE = projN.transition_map(projE,
[x,y, sqrt(1-x^2-y^2-z^2)], intersection_name='D',
restrictions1= z < 0, restrictions2= w>0)
It sounds reasonable, but calling
ProjNE.inverse()
failed. No problem, i tried using
ProjNE.set_inverse(x,y, -sqrt(1-x^2-y^2-w^2))
but i got the following warning:
Check of the inverse coordinate transformation:
x == x *passed*
y == y *passed*
z == -abs(z) **failed**
x == x *passed*
y == y *passed*
w == abs(w) **failed**
NB: a failed report can reflect a mere lack of simplification.
i don't know why the test is failing. The math sounds ok, where did it go wrong?JGCThu, 10 Oct 2019 22:07:30 +0200https://ask.sagemath.org/question/48264/Transformation of derivative under a change of charthttps://ask.sagemath.org/question/48085/transformation-of-derivative-under-a-change-of-chart/Consider the following code. When it display the connection coefficient in the Y frame at the end, we obtain derivative expressions that are quite complicated (for example derivative with respect to "r sin(\theta)". I would rather expect simpler expressions containing derivatives with respect to \lambda, \theta or r alone. Like what I would obtain by applying the "textbook" transformation rules for the partial derivatives.
M = Manifold(4, 'M', latex_name=r'\mathcal{M}')
X.<t,x,y,z> = M.chart()
U = M.open_subset('U', coord_def={X: (y!=0, x<0)})
X_U = X.restrict(U)
var('l', latex_name='\lambda')
Y.<t,l,th,r> = U.chart(r't:(0,+oo) l:(0,pi) th:(0,2*pi):\theta r:(0,+oo)')
Omega = var('Omega')
transit_Y_to_X = Y.transition_map(X_U, [t, r*cos(th)*cos(l+Omega*t), r*cos(th)*sin(l+Omega*t), r*sin(th)])
transit_Y_to_X.set_inverse(t, atan2(y, x) - Omega*t, atan2(z, sqrt(x^2+y^2)), sqrt(x^2+y^2+z^2))
nabla = M.affine_connection('nabla', r'\nabla')
phi = M.scalar_field(function('Phi', latex_name='\Phi')(x, y, z), name='phi', latex_name='\phi')
e = X_U.frame()
nabla[1,0,0] = e[1](phi).expr()
nabla.display(coordinate_labels=False, only_nonredundant=True)
nabla.display(frame=Y.frame(), chart=Y, coordinate_labels=False, only_nonredundant=True)CDFri, 27 Sep 2019 21:19:19 +0200https://ask.sagemath.org/question/48085/Display connection coefficients under a change of charthttps://ask.sagemath.org/question/48078/display-connection-coefficients-under-a-change-of-chart/ I want to define the connection components as the derivative of a scalar field in one frame and calculate their values in another frame.
M = Manifold(4, 'M', latex_name=r'\mathcal{M}')
X.<t,x,y,z> = M.chart()
U = M.open_subset('U', coord_def={X: (y!=0, x<0)})
X_U = X.restrict(U)
var('l', latex_name='\lambda')
Y.<t,l,th,r> = U.chart(r't:(0,+oo) l:(0,pi) th:(0,2*pi):\theta r:(0,+oo)')
Omega = var('Omega')
transit_Y_to_X = Y.transition_map(X_U, [t, r*cos(th)*cos(l+Omega*t), r*cos(th)*sin(l+Omega*t), r*sin(th)])
nabla = M.affine_connection('nabla', r'\nabla')
phi = M.scalar_field(function('Phi', latex_name='\Phi')(x, y, z), name='phi', latex_name='\phi')
e = X_U.frame()
nabla[1,0,0] = e[1](phi).expr()
Then
nabla.display(coordinate_labels=False, only_nonredundant=True)
show that the coefficients are good in the X_U chart, but the change of coordinate fail
nabla.display(frame=Y.frame(), chart=Y, coordinate_labels=False, only_nonredundant=True)
What is wrong ?CDThu, 26 Sep 2019 21:59:54 +0200https://ask.sagemath.org/question/48078/