Timeline for Why do I need densities in order to integrate on a non-orientable manifold?
Current License: CC BY-SA 4.0
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| Feb 11 at 9:49 | history | edited | Liviu Nicolaescu | CC BY-SA 4.0 |
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| Aug 16, 2017 at 13:59 | history | edited | Liviu Nicolaescu | CC BY-SA 3.0 |
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| Mar 1, 2013 at 22:01 | comment | added | Toby Bartels | Following up on pseudo-forms: A degree-$n$ pseudo-form on an n-dimensional manifold is the same thing as a density (and hence the same thing as an absolutely continuous Radon measure) so can be integrated directly, while a degree-$k$ pseudo-form for $k < n$ can be integrated only with the help of a pseudo-orientation (on the region of integration). Flux is a good example of the integral of a pseudo-form (of degree $n - 1$); the pseudo-orientation specifies in which direction one is passing through. | |
| Mar 13, 2012 at 10:57 | history | edited | Liviu Nicolaescu | CC BY-SA 3.0 |
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| Mar 8, 2012 at 10:17 | history | edited | Liviu Nicolaescu | CC BY-SA 3.0 |
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| Mar 7, 2012 at 14:57 | comment | added | Liviu Nicolaescu | If integration is your goal, then you cannot avoid densities. as a matter of fact it is easier to work with densities then with forms. There is a concept of pseudo-form which a bit tricky; see the book The Geometry of Physics by Theodore Frankel. | |
| Mar 7, 2012 at 14:25 | comment | added | ISH | Thank you for the link. I guess what I am asking is if I could forget about densities and, with my above argument just resort to n-forms also in the non-orientable case. It is just that forms appear all over the place and up to now, I've seen densities used only for integration on non-orientable manifolds. So, is there no way I could get around having to use densities? | |
| Mar 7, 2012 at 14:19 | history | answered | Liviu Nicolaescu | CC BY-SA 3.0 |