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Re: D=2 versus cylinders versus dots on paper



"RAUBER, JOEL" wrote:
...
One could argue that more generally the d/dz terms are zero *anywhere* due
to the translational symmetry,
...
IMO, its a bit like the chicken and the egg; I don't think
[either reflection or translation]
is more fundamental then the other.

Exactly so, assuming infinite cylinders.

================

Related thought: In the real world we never have truly
infinite cylinders. So a full industrial-strength
analysis would consider the case of exceedingly long
but finite cylinders. For points on the midplane, the
reflection argument works exactly. For other points (not
too far from the middle) you can use either the reflection
argument or the translation argument (plus some simple
scaling arguments) to prove that the (d/dz) stuff vanishes
to an excellent approximation.

Hint: Translation is equivalent to removing from stuff
from one end of the cylinder and adding it to the other
end. Estimate the contribution of these modifications.