Single-Particle Electrodynamics - Assassination Science

“godlike” parameter, i.e., one whose value is set by the author, but, once
set, cannot be changed by any of the operators in the equations being considered.
Thus, we immediately rule out the operator ∂ w , the generator of
w-translations, from being able to take any part in our considerations. The
gradient operator is therefore still only a three-vector, namely, ∇; there is no
operator ˜∇ in ˜r-space.
Quantities are computed in ˜r-space as they are in r-space, except that,
where necessary, the appended dimension w is discarded, when a purely
three-vector quantity is required; this is deemed to occur implicitly if an
operation being performed is an explicit three-vector operation, e.g., dotproducts
and cross-products. We then consider the resulting expressions in
the limit w → 0, after all necessary mathematical manipulations have been
performed.
It should be noted that the process employed here is essentially a rigorous
generalisation of the “a-potential” trick used by Jackson [113, Sec. 1.7].
6.6.2 Radial magnitude
The radial magnitude ˜r of ˜r is given by the standard Euclidean result:
˜r ≡ (˜r 2 ) 1/2 ≡ (w 2 + x 2 + y 2 + z 2 ) 1/2 . (6.68)
We deem that, in extending divergent three-space expressions into appended
space, the radial magnitude r is to be replaced by ˜r wherever it appears.
6.6.3 Directional normals
From the definitions (6.67) and (6.68) for the appended position vector and
radial magnitude in ˜r-space, we can construct a unit vector ñ:
ñ ≡
˜r˜r
. (6.69)
Clearly, ñ 2 = 1.
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