decraig Posted November 16, 2013 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?
decraig Posted November 19, 2013 Author 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
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