decraig Posted November 16, 2013 Share Posted November 16, 2013 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] define [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] and [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? Link to comment Share on other sites More sharing options...

decraig Posted November 19, 2013 Author Share Posted November 19, 2013 (edited) 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 November 19, 2013 by decraig Link to comment Share on other sites More sharing options...

decraig Posted November 25, 2013 Author Share Posted November 25, 2013 solved Link to comment Share on other sites More sharing options...

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