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grad on a generic metric

asked 2023-11-21 12:11:14 +0200

GPN gravatar image


First, thank you for SageMath Manifolds! I find it fascinating. I am a newbie, and I have indeed looked through the forum and the documentation, but am still stuck. So please excuse if my question is trivial in your eyes.

I am trying to write down the grad of a scalar field in a relatively generic way, but getting the error

ValueError: no common basis for the contraction

At a higher level, my goal is to prove a few basic lemmas about the grad, both as an exercise to learn Manifolds and as part of a larger project.

The following code just sets up the preliminaries and defines the metric:

# preliminaries
from sage. All import *
from sage.manifolds.operators import *
M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')
X.<t,x,y,z> = M.chart()
U = M.open_subset('U', coord_def={X: (y!=0, x<0)})
X_U = X.restrict(U)
F = U.scalar_field(function('f')(t,x,y,z), chart=X_U, name='F')

# definition of metric
g = M.metric('g',)
g[0,0] = function('g00')(t,x,y,z)
g[1,0] = g[0,1] = function('g01')(t,x,y,z)
g[2,0] = g[0,2] = function('g02')(t,x,y,z)
g[3,0] = g[0,3] = function('g03')(t,x,y,z)
g[1,1] = function('g11')(t,x,y,z)
g[1,2] = g[2,1] = function('g12')(t,x,y,z)
g[1,3] = g[3,1] = function('g13')(t,x,y,z)
g[2,2] = function('g22')(t,x,y,z)
g[2,3] = g[3,2] = function('g23')(t,x,y,z)
g[3,3] = function('g33')(t,x,y,z)

All of the above works (thank you). But now I try to do:


and I get the error above.

Obviously I am missing something basic.

Thank you GPN

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answered 2023-11-21 14:51:30 +0200

eric_g gravatar image

Your code works for me with SageMath 10.1 (the latest stable version).

Which version of SageMath are you using?

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Here is an additional find: that code is part of a bigger notebook which had the line


Based on your input (that the code cell is correct) I started removing lines from other cells and that line was the problem.

First of all, thank you! It was a newbie question and your input helped. Second, I don't know why that line conflicts with the grad(F) but maybe you have more insight?

Thanks again GPN

P.S. I am on SageMath version 9.5, Release Date: 2022-01-30. OS is Ubuntu 22.04 (which is itself on WSL2 on Windows 11, latest). I am using the binary installation, not building the code myself.

GPN gravatar imageGPN ( 2023-11-21 20:57:28 +0200 )edit

Actually, parallelism is broken for computations involving symbolic functions (i.e. defined via functionas in your case) because of this long standing bug: So this is certainly the reason why your initial code failed.

eric_g gravatar imageeric_g ( 2023-11-23 14:07:28 +0200 )edit

Thank you @eric_g

GPN gravatar imageGPN ( 2023-11-26 11:45:44 +0200 )edit

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Asked: 2023-11-21 12:11:14 +0200

Seen: 73 times

Last updated: Nov 21 '23