# 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,

$L=L_{[ij]} dx dy$

define

$S = \int L = \int L_{ij}dx dy$.

However, I would like to partially integrate L.

$S_y dy = ( \int L_{ij}dx ) dy$

and

$S_x dx = ( \int L_{ij}dy ) dx$

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 $S_x$ and $S_y$ so that they appear to behave under the rotation group as elements of a 1-form:

$S = S_x dx + S_y dy$

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

solved

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