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Symbolic vector field

asked 2016-04-19 00:53:11 +0100

Richard_L gravatar image

Is it possible to represent a partial differentiation without specifying what the derivative is acting upon? This would be handy in manipulating vector fields symbolically. For ex., suppose we have a vector field Y = y * d/dx and we'd like to compute the pushforward of Y by some diffeomorphism F. Can one express "d/dx" in the usual mathematical partial differential notation and in a way that Sage can handle?

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There is a differential forms package now, which might have some useful stuff?

kcrisman gravatar imagekcrisman ( 2016-04-19 15:38:58 +0100 )edit

Thanks. I'll have a look at the package. For now, I have resorted to positional notation, i.e., representing the field by its vector of coefficients. So Y = y*d/dx above becomes [y,0], etc.

Richard_L gravatar imageRichard_L ( 2016-04-19 17:14:21 +0100 )edit

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answered 2016-04-19 17:05:25 +0100

eric_g gravatar image

This is implemented in SageManifolds, see for instance here for an example of vector field on the sphere S^2 (in the section "Module of vector fields" of the worksheet).
SageManifolds is not fully implemented in SageMath yet, so you have to install it according to these instructions. Note that SageManifolds is installed in the SageMathCloud (see here for some SMC example including vector vectors fields on S^2).

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Btw, the method pushforward is implemented: see here for an example: the pushforward of a vector field by the standard embedding S^2 --> R^3.

eric_g gravatar imageeric_g ( 2016-04-19 18:25:56 +0100 )edit

Thanks Eric. I have now compiled SageMath 7.1 from source on openSUSE 13.2, After testing, I have installed SageManifolds per instructions given. All tests pass!

Richard_L gravatar imageRichard_L ( 2016-04-25 04:25:55 +0100 )edit

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Asked: 2016-04-19 00:53:11 +0100

Seen: 450 times

Last updated: Apr 19 '16