4 dimensional curl as antisymmetric matrixby purplegreen Tags: antisymmetric, curl, dimensional, matrix 

#1
May512, 01:27 PM

P: 4

I'm a bit confused. I'm trying to calculate the curl of a 4 dimensional matrix. It's an attempt to use stokes theorem for 4 dimensions.
The curl can be written as a antisymmetric matrix from what I understand with entries, Mi,j = d Ai/d j  dAj/di where i and j would be the different coordinates like x, y, z etc... However, from what I understood if you looked the integral about an infinitesimal square in the xy plane you could work out the integral as: (dAy/d x  dAx/dy)ΔxΔy I was informed that this would gives M1,2 ΔxΔy which would be wrong, you would get M12 surely? So how does the curl look in matrix/ tensor form for 4 dimensions? Hopefully this makes some sense, sorry if it's a slightly confused question. 


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