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A problem in integrating differential forms

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Is this the correct folder in which to pose a question on differential topology?



To help keep things simple, suppose you have a 2-form on a manifold of dimension 2, having metric type (+, +), rather than an arbitrary number of dimensions and other metrics.


As I understand it, we can integrate a k-form over an n dimensional manifold where k=n.


But I want to do an incomplete integration:


Given, L an antisemmetric tensor,

[math]L=L_{[ij]} dx dy [/math]


[math]S = \int L = \int L_{ij}dx dy[/math].


However, I would like to partially integrate L.


[math]S_y dy = ( \int L_{ij}dx ) dy[/math]


[math]S_x dx = ( \int L_{ij}dy ) dx[/math]


Is this even legal? I could be missing the obvious.


I’ve left out the limits of integration because I don’t know to use LaTex well. Assume integration over some given intervals.


I would like to obtain [math]S_x[/math] and [math]S_y[/math] so that they appear to behave under the rotation group as elements of a 1-form:


[math]S = S_x dx + S_y dy[/math]


Is this possible?

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This is really a problem in relativity.


To simplify, I reduced the problem to 2 dimensions.


L is the action density, represented not as a pseudo scalar, but an oriented 3-form.

The integral is a Lagrangian.

S is the scalar action of a space time 4-volume. It is an invariant over a curved manifold.


S_x would be energy density of a system.

y is the temporal coordinate.

S_y is a momentum flux density term.


Does that help?

Edited by decraig
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