I couldn't finish it, so I paid $35 for Alan Macdonald's Vector and Geometric Calculus. This uses geometric algebra, where vectors may be multiplied together to form bivectors, trivectors, and so forth. They are added together with abandon.(adsbygoogle = window.adsbygoogle || []).push({});

The electric fieldEis more or less 1D so it is unchanged. In N dimensions there is no such thing as a magnetic pole, so one needs to return to basic special relativity to see what will happen. Magnetism is 2D: the remainder of dimensions are irrelevant. It's plane is defined by three points: the location of the source charge, the location of the source charge after dt, and the location of the affected particle. Magnetism can be represented by a bivectorBwhich defines the plane in which magnetism operates, the magnitude, and the sign. The magnetic force is calculated by taking the inner product of B with the velocity vector of a charged particle. The inner product of a vector with a bivector is the projection of that vector onto the plane rotated 90 degrees, which just so happens to be precisely what we want.

A scalar is grade zero, an ordinary monovector is grade 1, a bivector is grade 2, etc.

In GA gradient = div + curl. Taking the div lowers the grade while the curl increases the grade. Take the time derivative leaves the grade unchanged.

Everything is as usual with the monovectorE. FieldBis the curl of the charge current potential vector, so it is a bivector. The divergence of a bivector is a vector, while the curl is a trivector. There are no trivector terms on the other side of the equation, so it must be zero.

With no charges,

div E = 0

curl E = -dB/dt

div B = -dE/dt

curl B = 0

The grade of both sides of each of the equations is, in order,

0

2

1

3

That's all there is to it, for any N dimensions.

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# Maxwell's Equations in N Dimensions

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