www.BillHowell.ca - [Variables, notations, styles] for the review of Bill Lucas's book "The Universal Force, vol1"
initial draft 25Aug2015
/*$ echo "version= $date_ymdhm, cos - 1 inclusion : $cos_inclusion" >>"$p_augmented"
This file is :
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>>> SUMMARY
Beyond a listing of standard [Greek letters, set theory symbols, logic symbols], Lucas's "variable symbols and notations", which was a great reminder and reference for me during my verification process, this document also provides a description of my own non-standard format for [equations, array & vector notations, basic operations like integration & differentiation]. This will probably be essential for readers of "Howell - math of Lucas Universal Force.ndf"
/*$ cat >>"$p_augmented" "$d_Lucas""context/text editor - how to set up.txt"
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TABLE OF CONTENTS
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EQUATIONS :
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For instructions on how to update the Table of Contents and Equations, see the section "Document build short description" at the end of this document. There is currently a problem of both lists above "shifting" the line number counts.
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waiver, copyright
/*$ cat >>"$p_augmented" "$d_Lucas""context/waiver, copyright.txt"
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>>> Unicode characters - handy symbols and codes
from Howell's Lucas Universal Force" project :
/home/bill/Projects/Lucas - Universal Force/Howell - Symbols for Bill Lucas, Universal Force.odt
see also https://en.wikipedia.org/wiki/Unicode_symbols
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>>> Mathematical notations
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>>>>>> Greek Letters
/*$ cat >>"$p_augmented" "$d_augment""Greek letters.txt"
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>>>>>> Hollow capital letter symbols (U+2100–U+214F)
(page up & down if symbols appear as a box)
/*$ cat >>"$p_augmented" "$d_augment""symbols - hollow capital letter.txt"
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>>>>>> series - sum, product, limits
/*$ cat >>"$p_augmented" "$d_augment""symbols - series [sum,product,limits].txt"
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>>>>>> vector, calculus, sets, etc
/*$ cat >>"$p_augmented" "$d_augment""symbols - vector, calculus, sets, etc.txt"
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>>>>>> matrix notation
/*$ cat >>"$p_augmented" "$d_augment""matrix notation.txt"
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>>>>>> Notations for [indexing, inequalities, super, sub]-script
/*$ cat >>"$p_augmented" "$d_augment""indexing.txt"
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>>> Lucas variable symbols - definitions
It was sometimes problematic to dig around in Lucas's book to find definitions, and in some cases I've done some guessing. But at least this list will help to explain some of my review, and it may be handy to others.
I also struggled with
- the basis of units [Gaussian, SI, CGS]
- the frame of reference (in a sub-section below)
A key point is the choice of units, which I think should be SI, especially for a "Universal Force" theory. Other units are too "specialized", and are prone to inciting too many conceptual errors in developments.
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>>>>>> Lucas variable symbols - Latin
/*$ cat >>"$p_augmented" "$d_augment""symbols Lucas, Latin.txt"
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>>>>>> Lucas variable symbols - Greek
/*$ cat >>"$p_augmented" "$d_augment""symbols Lucas, Greek.txt"
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>>>>>> Gaussian versus SI units
/*$ cat >>"$p_augmented" "$d_augment""Gaussian versus SI units.txt"
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>>> 'Howell's FlatLiner Notation' (HFLN) formatting conventions
/*$ cat >>"$p_augmented" "$d_augment""Howells FlatLiner Notation (HFLN) formatting conventions.txt"
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>>>>>> General principles for HFLN symbol definitions
/*$ cat >>"$p_augmented" "$d_augment""General principles for HFLN symbol definitions.txt"
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>>>>>> Symbol [notations, qualifiers] in lieu of (subscripts, superscripts, etc)
Special Unicode characters and formatting (such as vector arrows, superscripts, subscripts, character qualifiers such as primes, "hats" etc) require much more time to type and arrange than simple ASCII text. They also make it harder to [copy, paste, modify] equations, and to align equations character by character to make error-checking much easier visually. Given that I am using a text editor as required for [simple, direct] use of the verification file "Howell - math of Lucas Universal Force.ndf" as computer code, special formatting and symbols are also out of the question.
For these reasons, I use a simple notation that appends letter-qualifiers to the end of variable symbols. I often use English letters in place of special Greek letters (eg w for ω, O for θ) for the same reasons. There are two approaches for "appending variable qualifiers" :
For projects where variables have one-letter symbols, they are simply appended (eg rp)
For projects rquiring variable symbols greater than one character, an underscore is appended first, followed by the qualifiers (eg bird_p). This allows a much richer set of variable symbols and a far greater description of the symbol meaning (eg birdDuck_v)
For this review of Bill Lucas's "Universal Force" book, I used one-letter symbols withoutther underscore before qualifiers, which helped to keep expressions a bit more compact.
The way that I simply append "qualifier letters" to a variable [symbol, name] would normally cause confusion in formulae, as it wouldn't be clear in "multiplication sequence with implied "*" operator" which letters would be variable symbols, and which would be qualifiers. However, because of my use of the text as computer code, the nature of the QNial programming language, and the need to build formulae in a manner consistent for use in symbolic processing, all OPERATORS (such as [+,-,*,/, etc] must be explicit, including multiplication. so each sequence of letters is a single "name" for a [variable, function, transformer, etc].
Qualifier letters are in "small case", whereas base variable name are in "CAPITAL LETTERS" (the latter has not yet been enforced, as it requres fixing ambiguities that would result from different variables with the same symbol).
While qualifiers don't have to be in any special order (as they are each unique), by convention I have listed them in the order :
1st [o,p]
2nd [c,d,h,n]
3rd [a.h,o,p,s,v] - unfortunately, there is "redundancy" with the use of [o,p] (see 1st qualifiers above)
4th 0 (zero)
Note that additional standards have yet to be implemented :
capital letters and extensions for the "base symbol" for a variable name (eg [BT,B0,BI] for [total,static,induced] magnetic fields)
constant symbol character length
Here is a listing of some of my qualifiers : :
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>>>>>>>>> First qualifiers :
o observer frame of reference (RFo) qualifier as used by Lucas
p p is the "prime" (symbol ’), which as per Lucas's use, this indicates the frame of reference moving with the particle/system (RFp) (eg fields accompany particle)
?? Notice that other reference frames (such as for 2nd,3rd observerse or particles) are not yet accounted for, although a "floating basis" (n) s provided in "second qualifiers" below
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>>>>>>>>> Second qualifiers :
c measures in a coordinate system basis, i.e. measured from a coordinate origin (often particle or observer centric). For example :
angle (eg [Ooca,Ppca])
vector (eg [Rocv,Oocv,Pocv] [xocv,yocv,zocv] (note - rocv is a displacement vector)
distance (eg [Rocs,Oocs,Pocs] [xocs,yocs,zocs]
d DISPLACEMENT vector - for which the OR[AND,OR] (startPoint, endPoint) are important and are "specific points" in a reference frame. This symbol especially applies to situations that are NOT coordinate system displacement vectors - eg the coordinates in a reference frames are displacement vectors, although given their special status, they use the special qualifier "c". (in the context of "a" for an angle, "d" for a displacement vector, "h" for a unit vector, "s" for a scalar distance, "v" for a vector)
n "floating" basis - does not have to be "anchored to origin or specific points. This is appropriate to a field, for example (eg [velocity, E, B] where the latter are uniform.)
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>>>>>>>>> Third qualifiers
a (old notation?) denotes an angle (in the context of "a" for an angle,
h h (hat or carat ^) indicates that the symbol is a unit vector (magnitude = 1), often pointing in the direction of the displacement vector r (eg roh, rph,RθPI2h)
- Sometimes a variable normally used as a scalar, is used as a vector. In this case, the direction might be obvious (for example
[rh,vh,nh] are commonly used examples, although I normally don't use h with n.
(old notations are crossed out - now are AO and AP :
o (theta) angle is in O direction
p (phi) angle is in P direction
s s indicates that the symbol is a scalar
v v indicates that the symbol is a vector
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>>>>>>>>> Fourth qualifiers
0 angle of 0 (zero) wrt angle coordinate origin
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>>>>>>>>> Additional comments about the symbol qualifiers
Multiple qualifiers are typically used, for example :
Roch means a unit vector, anchored to the observer coordinate origin in RFo, and pointing in the direction of vonv(particle)
Rθ0pcs means a scalar distance, anchored to the particle coordinate origin in RFp, and measured in the direction of O (theta) = 0.
Vonv(particle) is the velocity of the particle in the observer frame of reference RFo, but not tied to the coordinate origin or any specific point. i.e. it is a general vector as conventionally used.
Note that I have not implemented this yet, as I generally use vov(particle)...
[origin,rh,Ph], [observerCenter,roh,Poh], [particleCenter,rph,Pph]
are a basis for spherical coordinates. The magnitude of r, and the angles [O,P] are as described above. rh MUST be orthogonal to Ph!! This defines a plane. Angle O is measured back from the direction of rh (sort of pivoting about the origin).
[xh,yh,zh], [xoh,yoh,zoh], [xph,yph,zph]
are unit vectors that serve as a basis for a Cartesian coordinate system in a reference frame. Lucas assumes the particle is moving along the z-axis in RFo (vov direction), so zh = voh. For simple convenience, I will usually take xh as the direction from the particle center to L(POI), and ych as the direction perpendicular to [xh,zh] (right hand rule? - actually, I don't use it so much for Chapter 4 with only one parrticle and observer).
[rs,O,P] or [x,y,z] are coordinates in space with respect to the [OBSERVER,PARTICLE] frame of reference
When referring to a MOVING system, such as the moving distributed charge, other moving particles or system, or a "roving camera" situation, these coordinates are functions of time.
When referring to a FIXED point in space (either reference frame), [x,y,z] are NOT functions of time.
All variables above EXCEPT [v,Q] are applicable to both the [observer, particle] frames of reference.
In Chapter 4, almost all analysis of [E,B,F,etc] using [r,Op,Pp,x,y,z] coordinates in BOTH frames of reference (observer,particle) refer to fixed points in the observer Frame of Reference (RFo) space, the particle itself being the main exception.
Additionally for Chapter 4, the velocity of the particle vo(t) = v in the OBSERVER reference frame is a constant, but as per the last paragraph, almost all analysis of [E,B,F,etc] refer to fixed points in the OBSERVER reference frame that DON'T move with the particle! In the PARTICLE frame of reference, there is no allusion to any moving system, and the particle itself of course does not move within its reference frame.
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>>>>>> Reference frames (observer, objects, ether) :
Observer - In this paper, the "observer" is the origin of a fixed coordinate system in the observer frame of reference.
Particle - In this paper, the "observer" is the origin of a fixed coordinate system in the "particle" frame of reference. The word "particle" is used to represent a "diffuse" (has size) single particle, collection of particles, or some kind of system. Typically, it's position is represented by some concept of "centroid", which I don't get into here.
Ether Although General Relativity "abolished" ether, only to re-introduce it in another form, there are many non-standard basiss in physics that still apply the concept. I do not elaborate on any for Chapter 4.
Other Obviously, there may be many [observers, particles], and for that I would use either [indexing, lables], but this is not an issue for Chapter 4.
NOTE!! : Lucas's form of a Galilean transformation :
ro - vo*t = rp
is ONLY correct for :
constant velocity of particle in observer frame of reference
[observer, particle] reference frames are the SAME [rotation, scale], and are exactly coincident at time t=0 !!!!
In Lucas's book, primes (´) indicate the particle's (moving) frame of reference, and unprimed is the observer's frame. (also denotes time derivative when used as ∇′ )
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>>>>>> Points of Interest (POI)s
/*$ cat >>"$p_augmented" "$d_augment""Points of Interest (POI)s.txt"
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>>>>>> References to equations in derivations
Derivations in "Howell - Background math for Lucas Universal Force, Chapter 4.odt" and "Howell - math of Lucas Universal Force.ndf" typically refer to preceding equations derived :
1. in the same [sub-sub] section
2. in different [sub-sub] sections, taken from the same file
3. in different [sub-sub] sections, taken from different files
4. in different [sub-sub] sections, taken from a book
The notations below remove ambiguities, and serve as a reminder to the reader of the sourcing of equations.
Style for numbering equations
??? BS!! ??? In the labels for the equations, I've used the notation that (c*), for example, means (c) with substitutions for component terms that have been processed. This is used, for example, for results that will combined with other similar expressions in a parent term.
Althpough the numbering sequence changes, towards mid-Sep2015 I favoured letter-number-letter etc (eg (1a1)).
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>>>>>>>>> Notation for equations from the same [sub-sub] section
Equation references simply give the line number equation number and text. For example
Subbing (2) into (1) :
Therefore :
1) |dp[dt : E0pdv(POIo,t)]|
= Q(particle)*Vons(particle)/Rpcs(POIo,t)^3
* | [ sin(Opca(POIo,t))*RDEpdh(POIo,t,dt) + 2*cos(Opca(POIo,t))*Rpch(POIo,t) ] |
Note that equations (2) and (1) appear in the same sub-sub-section.
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>>>>>> Dimensional analysis (Gaussian units)
/*$ cat >>"$p_augmented" "$d_augment""dimensional analysis, Gaussian units.txt"
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>>> APPENDICES
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>>>>>> Symbol checking and translation - short description
/*$ cat >>"$p_augmented" "$d_Lucas""context/symbols [check, translate].txt"
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>>>>>>>>> HFLN = Howells FlatLiner Notation !!!!!!!!!!!!!! 31May2016
/*$ cat >>"$p_augmented" "$d_Lucas""context/Howells flat-line notation short description.txt"
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>>>>>> Document build short description
/*$ cat >>"$p_augmented" "$d_Lucas""context/document build short description.txt"
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>>>>>> References :
electro-magnetic symbols - see https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law
enddoc