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Using dz /\ dx convention in SageManifolds

asked 2021-11-19 15:55:05 +0100

Donald_Munro gravatar image

Several books on Differential Forms in Electrodynamics use the convention that the dx,dz basis is defined as $dz \wedge dx$ and not $dx \wedge dz$, for example the Faraday two-form expressed in this way

$$F = -E_x\, dt \wedge dx - E_y\, dt \wedge dy - E_z\, dt \wedge dz + B_x\, dy \wedge dz + B_y\, dz \wedge dx + B_z\, dx \wedge dy$$

I can't seem to find anything in the SageManifolds documentation on defining either a manifold or a form to use this convention. Is it possible to directly use this convention in SageManifolds ?

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I suppose you already knows that $dz\wedge dx=- dx\wedge dz$ as far $dx$ and $dz$ are 1-forms. With this in mind the only difference is a change in a sign.

Masacroso gravatar imageMasacroso ( 2021-11-19 18:58:39 +0100 )edit

Yes I'm aware that they are alternating products, however:

1 Its less error prone if you don't have to remember to swap signs and

2 the exterior derivative seems to be incorrect in $dt \wedge dz \wedge dx$ term

Eg the correct term is $\left(\frac{\partial E_x}{\partial z}-\frac{\partial E_z}{\partial x}+\frac{\partial B_y}{\partial t}\right)dt \wedge dz \wedge dx$ [1] but Sage gives

$(\frac{\partial\,By}{\partial t}-\frac{\partial\,Ex}{\partial z}+\frac{\partial\,Ez}{\partial x})dt \wedge dx \wedge dz=$ $(\frac{\partial\,Ex}{\partial z}-\frac{\partial\,Ez}{\partial x}-\frac{\partial\,By}{\partial t}) dt \wedge dz \wedge dx$

See next comment for the code as I'm running out of chars

[1] A Visual Introduction to Differential Forms and Calculus on Manifolds, Fortney, J.P, Springer 20

Donald_Munro gravatar imageDonald_Munro ( 2021-11-20 17:28:20 +0100 )edit

Sorry, I could not the editor to do a markdown code display with4 tilde's. The code is at: link texthttps://gist.github.com/donaldmunro/6...

Donald_Munro gravatar imageDonald_Munro ( 2021-11-20 17:28:47 +0100 )edit
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SageMath is correct in computing the exterior derivative, simply your initialization of $F$ is not: the F[1,3] term should be F[1,3] = - B_y(u,v,w, T) instead of F[1,3] = B_y(u,v,w, T).

eric_g gravatar imageeric_g ( 2021-11-21 15:43:49 +0100 )edit

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answered 2021-11-21 15:51:27 +0100

eric_g gravatar image

At the moment, there is no option in Sage leading to the display you are asking for. It could be implemented of course, via an optional keyword of the method display of differential forms. This simply requires some work, including tests and documentation. Would you agree to implement this feature ? If yes, please take a look at the contribute page.

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Thanks, I'll have a look at the source.

Donald_Munro gravatar imageDonald_Munro ( 2021-11-22 16:40:02 +0100 )edit

Very good!

eric_g gravatar imageeric_g ( 2021-11-24 11:39:28 +0100 )edit

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Asked: 2021-11-19 15:55:05 +0100

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Last updated: Nov 21