4 dimensional curl as antisymmetric matrix

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The discussion centers on calculating the curl of a 4-dimensional matrix using Stokes' theorem. The curl is represented as an antisymmetric matrix with specific entries defined by the derivatives of vector components. There is confusion regarding the integral calculation in the x-y plane, particularly whether it should yield M1,2 ΔxΔy or -M12. The correct interpretation involves understanding that in 4 dimensions, the curl corresponds to elements of the 6-dimensional Lie algebra so(4), which consists of skew-symmetric real matrices. Further exploration of this Lie algebra is suggested for deeper comprehension.
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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 x-y 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.
 
Mathematics news on Phys.org
In 4 dimensions the curl of a vector field is, geometrically, at each point an element of the ##6-##dimensional Lie algebra ##\mathfrak{so}(4)##, all skew-symmetric real matrices: ##v^*\otimes w - w^*\otimes v##.

You can search for this Lie algebra or Lie group for further information.
 
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