I would like to further develop the investigation of
high order derivatives touched on in previous posts.
Writing the expression dnL/dtn out in full we have,
1st 2nd 3rd 4th 5th 6th nth
(dL/dt).(dL/dt).(dL/dt).(dL/dt).(dL/dt).(dL/dt).......(dL/dt).
For simplicity I will just take the numerator term.
1st 2nd 3rd 4th 5th 6th nth
dnL (dL).(dL).(dL).(dL).(dL).(dL).......(dL).
Now this form of expression is not very insightful since it does
not diagrammatically show the nested nature of the derivative string.
A more meaningful representation is the string form familiar to
anyone who has written computer programs.
dnL = (dL(dL(dL(dL(dL(dL.......(dL))))))).
or even,
http://www.grimer2.freeserve.co.uk/pge24.htm 8-)
Since we are never likely to need more than 26 derivatives we may
as well use Dr.Seuss's Cat nomenclature to distinguish between the
different orders.
d*L = (AdL(BdL(CdL(EdL(FdL(GdL.......(ZdL)....))))))
With the exception of the first and last term which are special
boundary cases, each length in this string has an upper and lower
bound. For example, RdL > SdL > TdL, and so on.
This means that concept of length we are dealing with here is quite
distinct from the Euclidean concept of length.
In Euclidean geometry the concept of length is unbounded.
In Euclidean geometry one can make length as small as one likes.
There is no such thing as a smallest length.
There is not 'atom' of length, no unit of length.
There is no such thing as a largest length.
There is no Universe of length.
The concept of length in Euclid's geometry is metaphysical, not
physical.
In contrast, the concept of length implicit in GI (Gottfried-Isaac)
calculus is eminently physical - which no doubt accounts for its
success in real life situations.
Now the problem with Cartesian Geometry (CG) is though it moves some way
from the one dimensional concept of length to the extent that it
introduces three independent dimensions of length, x, y and z.
which is an improvement on a 'ball of wool' geometry for example
which, like Euclid, has only one dimension.....
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A slight digression onto a piece by Fred Hoyle in Nature which may
be regarded as "ball of wool geometry" in that he reduces everything
(including Time to one nominal dimension. However, you will notice
that he sneaks in multi-dimensionality under the guise of powers of
L. His paper piece would have been much improved if he had got to
grips with the nesting aspect of powers of L.
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It is well known that when c and h(bar) are set equal to unity.
only a single dimensionality is needed for the whole of physics.
We take this to be length and we denote the unit by L. Every
quantity has a dimensional form L^n, for example, mass �L^-1.
frequency �L^-1. charge �Lo. magnetic field �L^-2 and the
gravitational 'constant' has dimensionality L^2.
Every observation is concerned with a dimensionless number.
so that every observation is concerned with a product of quan-
tities such that the dimensional dependencies on L cancel to
zero So far as experimental physics and engineering are
concerned it is possible to convert a quantity of dimensionality
L^n into a quantity of dimensionality L^m by means of a linear
device provided n=m Non-linear devices are needed if n<>m.
This property makes it comparatively easy to see what kind of
physical device is needed to relate one quantity to another.
There is no doubt that physics and engineering are made quite
unnecessarily complicated by the current practice of using
multiunit systems.
F Hoyle and J V NARLIKAR
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....unfortunately CG is still lumbered with the Euclidean concept in
each of its three independent dimensions. This has the additional
disadvantage of disguising the hierarchical nature of x, y and
z which are not normally thought of as implicitly referring to
length area and volume.
(To be continued)
Frank Grimer
