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.Thu, 12 Aug 2021 20:32:20 +0200Vector/Algebra valued differential formshttps://ask.sagemath.org/question/58429/vectoralgebra-valued-differential-forms/ Hi, I'm new to Sage and curious about one, potentially very useful aspect regarding SageManifolds module. I've seen that part regarding vector bundles is already developed, so I'd like to ask whether it is possible to form objects such as differential forms on regular manifold M, valued in vector bundle E.
More precisely, structures such as C^{\infty} ( E \otimes \bigwedge^k (T^{\ast} M ) ) (sorry, I can't insert pictures yet). Such entities arise naturally, at least in physics (starting from general relativity) where one needs to relate structure of vector bundle over M with tangent bundle of M.
In index notation, which also works in Sage, such tensors would be naturally manipulated by structure on M and E. Is there a possibility to form such differential forms, any other way than just forming a matrix of regular differential forms? Making such a matrix would be an option in the most trivial cases, but in general one would rather like to treat E and M equally.VonbatenBachThu, 12 Aug 2021 20:32:20 +0200https://ask.sagemath.org/question/58429/Multiplying a vector to a listhttps://ask.sagemath.org/question/52213/multiplying-a-vector-to-a-list/Let's say I have a list of forms given by
U = Manifold(4, 'U')
X.<x, y, z, w> = U.chart()
f = U.diff_form(3, 'f')
f[0, 1, 2] = x^2
g = U.diff_form(3, 'g')
g[0, 1, 2] = y^2
h = U.diff_form(3, 'g')
h[0, 1, 2] = z^3
List = [f, g, h]
So I have a list of 3 elements, each a 3-form.
Let's assume I have a list of vectors given by
ListofVectors = [(0, 1, 1), (0, 2, 1), (1, 2, 3)]
I want to do a dot product but the problem is Sage won't
let me make `List` into a vector so I can do like
List.dot_product(ListofVectors[i]) for i < len(ListforVectors)
Basically, I want a new list with `[g + h, 2*h + g, f + 2*g + 3*h]`. Is there a way to dot product a list of forms with a list of vectors? Again, I think the main issue is I can't turn List into a vector for the dot product function to make sense.whatupmattThu, 25 Jun 2020 02:16:50 +0200https://ask.sagemath.org/question/52213/Real analytic Eisenstein serieshttps://ask.sagemath.org/question/51911/real-analytic-eisenstein-series/I'd to compute values of a certain function in Sage, a kind of a modular form, the so-called [real analytic Eisenstein series](https://en.wikipedia.org/wiki/Real_analytic_Eisenstein_series). Does anybody know how to do it? I could not have found the Sage name for it.
More precisely, I would like to plot the the graph of the real analytic Eisenstein series, their real and imaginary values in a square of the complex plane (variable z) each for a certain value of the parameter s. Thus s is fixed in each of the graphs.
Real analytic Eisenstein series (also this is this their name at wikipedia) are defined for a complex z and a complex s. They are not the [Eisenstein series](https://en.wikipedia.org/wiki/Eisenstein_series) defined for complex z and an integer k. If s is k and an integer, they are connected by the multiple of Im(z)^s. Thus an easy connection, but I'd like to know the value of the series for a complex s. They are modular functions, not holomorphic and connected to theta series.svarotThu, 11 Jun 2020 20:40:09 +0200https://ask.sagemath.org/question/51911/Pullback computation hanginghttps://ask.sagemath.org/question/40852/pullback-computation-hanging/ I have the following code:
M = Manifold(3, 'M')
X.<x,y,z> = M.chart()
N = Manifold(3, 'N')
XN.<a,b1,b2> = N.chart()
omega = N.diff_form(2)
omega[0,1] = 2*b2/a^3
omega[0,2] = -2*b1/a^3
omega[1,2] = -2/a^2
Then I define a map M to N.
r = sqrt(x^2+y^2+z^2)
t = var('t', domain='real')
STSa = r^(1/2)*(r*cosh(2*r*t) - z*sinh(2*r*t))^(-1/2)
STSb1 = (x*sinh(2*r*t)/r)*STSa
STSb2 = (y*sinh(2*r*t)/r)*STSa
STS = M.diffeomorphism(N, [STSa, STSb1, STSb2])
Finally, I attempt to compute the pullback of omega to M by the map STS:
s = STS.pullback(omega)
Unfortunately, the program runs and runs and nothing ever comes out. Can anyone identify the issue? Of course, the Jacobian of the map STS will not be very nice, but this pullback should be perfectly computable.
Jeremy LaneMon, 29 Jan 2018 17:30:09 +0100https://ask.sagemath.org/question/40852/