www.BillHowell.ca - [Variables, notations, styles] for the review of Bill Lucas's book "The Universal Force, vol1"
initial draft 25Aug2015
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This file is :
/*$ echo " $p_augmented" >>"$p_augmented"
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>>> SUMMARY
Beyond a listing of 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"
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TABLE OF CONTENTS
For a more detailed version & easy viewing alongside this text file, see "math Howell notations, Table of Contents.txt"
EQUATIONS :
For instructions on how to update the Table of Contents and Equations, and on how to generate a summary list of formulae, see the section "Document build short description" at the end of this document.
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>>> waiver, copyright :
/*$ cat >>"$p_augmented" "$d_Lucas""context/waiver, copyright.txt"
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>>> Lucas's Dedication
(This is copied directly from his book.)
/*$ cat >>"$p_augmented" "$d_Lucas""context/Lucas dedication.txt"
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>>> Introduction
/*$ cat >>"$p_augmented" "$d_Lucas""context/introduction.txt"
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>>> Mathematical notations
see also the file "5_Math symbols.odt"
Matrix notation
|, // denotes the vertical bar for "given"
M denotes [scalar, vector, matrix]
M_T denotes transpose of M, also transpose(M
|M| , //M// denotes absolute value of matrix M (each element)
||M||, ////M//// denotes spectral norm of M ////M////2
M_bar denotes an overscore on M
M_tilde denotes a tilde over a Matrix symbol
|x|, //x// is the absolute value vector of x, |x| = (|x1|, |x2|, . . . , |xn|)T
||x||2 is the vector norm of x, ||x||2 = √Σni=1 |xi|2
I is the identity matrix
M > (≥)0 means M is a positive definite(semidefinite)matrix
M > (≥)B means M − B is a positive definite(semidefinite)matrix
M ≽ 0 means M is a positive(nonnegative) matrix, i.e, mij ≥ 0,
M ≽ B means the elements of matrices M,B satisfy the inequality mij ≥ bij
|M| is the absolute value matrix of M; |M| = (|mij |)nÃ—n
(M) is spectral radius of M
λmax(M) means the maximum eigenvalue of matrix M
λmin(M) means the minimum eigenvalue of matrix M
ρ(M) is the spectral radium of matrix M
||M||2 is the spectral norm of matrix M. ||M||2 = √λmax(MTM)
Note : best viewed in text editor without automatic line wrap, constant font width
Kronecker product - If A is an m × n matrix and B is a p × q matrix, then the Kronecker product A ⊗ B is the mp × nq block matrix:
a11*B ... a1n*B
A ⊗ B = . ... .
a1n*B ... ann*B
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>>>>>> Vector & Tensor notation
(see also table of Unicode characters...)
dot product for 2 vectors - gives scalar
v1v2 = Σ(d= 1 to 2 or 3 : x1d*x2d)
cross product for 2 vectors - gives vector
X (capital X), also cross product for 2 vectors - gives vector
v1v2 = det(d=2 or 3 : d d reshape link nj vij)
ni = unit orthogonal vector basis
∇ gradient (nabla or del) of scalar field f - gives vector
v = Σ(i=1 to D : ∂/∂xi(f)*ni )
∇′ gradient in particle/system frame
NYET ?-> I'm guessing time gradient in Lucas's notation? p69h0.0 Eqn (4-20) (4-21)
∇ divergence (grad-dot) of vector field V - gives scalar
∇V = Σ(i=1 to 2 or 3: ∂/∂xi(Vi) )
∇ curl (grad-cross) of vector field v - gives vector
∇V= det(i=1 to 3 : i i reshape link ni ∂/∂xi(V) vi ...)
ni = unit orthogonal vector basis
r_hat In Lucas's notation, an "^" or "hat" over a letter - denotes a unit vector in the direction of the vector represented by the letter, eg r_h (r in Lucas writing) means a unit vector in the direction of vector r.
⊥ perpendicular to
# [Howell 17Aug2015 - in Lucas's "Universal force, volume 1" the variables t, c, n q are scalar, the rest are vector.
|r - v*t| In Lucas's book, this denotes the vector norm (usually written ||r - v*t||) rather than the absolute value of each component of a vector or matrix, as is the usual interpretation of |r - v*t|. The single verticle bars do usually denote the norm on the one-dimensional vector spaces formed by the real or complex numbers, for example |-3| (absolute value)
(see https://en.wikipedia.org/wiki/Norm_%28mathematics%29)
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>>>>>> Variable symbols and notations
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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>>>>>>>>> Variable symbols
For illustrations of the meaning of many variables, see "Howell - Figures for Lucas Universal Force.odt", including :
Figure "General reference frames"
Figure "Chapter 4 reference frames"
The lists below include BOTH Lucas' notation (as required to understand his book) and mine (to follow my derivations).
Latin symbols
A amplitude of neutral dipole vibrations (A1, A2 (Figure 7-2,8-1))
A or A2 is a 2D angle (radians),
A1 1D angle (+,-)
A2 2D angle (radians)
A3 3D solid angle (steridians) -> Lucas symbol Ω, omega (5-16)
Aθ=AO 2D angle in O(theta) direction. This is the angle rooted at the [observer, particle,POI_root], that is measured from [a POI_start, or the direction of a vector (typically Vonv(particle))] to [a POI_end, or vector R_end ]. Note that the plane of intersection of [Vonv(particle),POIo or POIp (POIs FIXED in RFo or RFp)] is CONSTANT - ie there is no rotation or complication as the particle moves. The O(theta) direction is like a latitude.
AO is shorthand notation for fast typing, converted later.
Aφ=AP 2D angle in P(phi) direction. This is the angle IN A PLANE (often a plane that is perpendicular to Vonv(particle)), that is measured from [a POI_start, or the direction of a vector (typically the y-axis where the z-axis is aligned with Vonv(particle))] to [a POI_end, or vector R_end]. Apply the right hand rule with thumb in the direction of the vector to get positive and negative directions. The P(phi) direction is like a longitude (as suggested by Lucas's integration limits for φ from 0 to π p95h0.9 Equation (7-16)).
AP is shorthand notation for fast typing, converted later.
AQ is the angle between Rocv(observer to particle) the directed vector from observer to particle, and Vonv(particle), the constant velocity of the particle for Chapter 4. Note that the plane of intersection of [R,V] is CONSTANT - ie there is no rotation or complication as the particle moves. Note that this angle isn't used in Chapter 4, as it is assumed that the observer is located on the line L(Vocv(particle)) that is ANCHORED on the particle, such that AQ(Rocv(observer to particle), Vonv(particle)) = 0. Note that the plane of intersection of [r,v] is CONSTANT - ie there is no rotation or complication as the particle moves. This, along with ro(t=0) from the observer to the particle, firmly "links" the observer and particle frames of reference.
a acceleration = d2[d^2t: rv] - 2nd derivative of displacement vector
ag or as acceleration wrt center of galaxy,
au or a acceleration wrt center of universe
B magnetic field, B0 - from static charges , Bi - induced
NOTE : It would help clarity if the [electric, magnetic fields] were designated [E,B]_[0,V,S] (for [static, induced from velocity across field, induced from changing field]. In Chapter 4, ALL B are induced! (i.e. no magnetic materials)
<<< 29Mar2018 - 3 types of [E,B]!! >>>
b = β b = v/c (default scalar)
c speed of light in a vacuum (default scalar, taken as dimensionless for Chapter 4 for dimensional analysis)
D electric displacement field D = ε0*E + P = ε0*εr*E
where P = polarization density
(see https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law)
d[dt : f(t)] derivative short-hand notation eg d(dt : f(t)) = d/dt(f(t) Default meaning is "total" derivative
dt[dt : f(t)] total" derivative eg dt(dt : f(x,v,a,t)) = d/dt f(x,v,a,t)
dp[dt : f(t)] "partial" derivative eg dp(dt : f(x,v,a,t)) = ∂/∂t f(x,v,a,t)
∇' I'm guessing ∇' is Lucas's notation for the time gradient dp[dt: ...]
∇' (4-28) here it is a spatial, NOT time, derivative!?!?
d^3r =dx*dy*dz
E electric field, E0 static or primary, Ei - induced, Ei′(r′,t′) - new, different from Ei but not defined (see Lucas p68,69)
NOTE : It would help clarity if the [electric, magnetic fields] were designated [E,B]_[0,V,S] (for [static, induced from velocity across field, induced from changing field strength]. In Chapter 4, ALL B are induced! (i.e. no magnetic materials)
<<< 29Mar2018 - 3 types of [E,B]!! >>>
I assume that Ei′(r′,t′) = E0(r′,t′) + Ei(r′,t′) = total E, but this contradicts statements p68h0.3 where Ei is induced only (doesn't include static)
e charge of proton,
-e charge of electron (7-13)
F force, F_G gravity, F_I inertial, F(2+,1+) etc (7-11, 7-25)
Fu Howell's notation specific to derivations [of, from] Lucas's "Universal force"
Fu_G Lucas's "Universal force" expression for gravity
f frequency (Fiigure 7-2,8-1)
G Newton's universal gravitational constant
H is the magnetic H-field (in ampere per metre, Am−1) https://en.wikipedia.org/wiki/Amp%C3%A8re's_circuital_law
H is the magnetic H field (also called "auxiliary magnetic field", "magnetic field intensity", or just "magnetic field")
H = B/μ0 - M
h Planc's constant (Barry Setterfield : h*c = constant) (2-6)
J is the total current density (in ampere per square metre, Am−2) https://en.wikipedia.org/wiki/Amp%C3%A8re's_circuital_law
J displacement current J_D = ∂/∂t(D(r,t)) (4-9)
J includes magnetization current density[13] as well as conduction and polarization current densities.
J the current density contribution actually due to movement of charges, both free and bound
J = Jf+ Jd + Jm
Jd displacement current
Jf enclosed conduction current or free current density
Jm magnetization current density?
J charge density (Jackson1999 p554h0.7 Equation (11.127)
L(v) = λ(v) velocity-dependent proportionality between Ei and E0 (4-31) (default scalar)
L(POI) is a line running through the POI that is parallel to Vonv(particle)
L(particle) = L(POI) for POI = particle
L(observer) = L(POI) for POI = observer
Lx(vo) is a line that is colinear with rx(POI), but note that it is the SAME for ALL POI!!
l path or line (4-2)
M magnetization field - how strongly magnetized a region is
Mg mass of spiral galaxy hosting a neutral electric dipole (10-6)
Mu mass of the universe (10-6)
magnetic field - The term is used for two distinct but closely related fields denoted by the symbols B and H, where H is measured in units of amperes per meter (symbol: Am−1 or A/m) in the SI. B is measured in teslas (symbol:T) and newtons per meter per ampere (symbol: Nm−1A−1 or N/(mA)) in the SI.
B is most commonly defined in terms of the Lorentz force it exerts on moving electric charges.
m mass (see also M - of galaxy, Mu of universe (10-6) )
n quantum level or number (2-7)
n unit basis vector, eg Cartesian coordinates x = n1*x1 + n2*x2 + n3*x3
n1,n2,n3 (note : often i j k are used)
n_hat unit vector normal to a surface (4-2, 4-3)
PI= Shorthand notation for fast typing, converted later. see constant π
POI (POIo,POIp) = "Point Of Interest" is a specific point, and is typically where estimates of [B,E,F, etc] are required. Normally, the POI is STATIONARY in EITHER :
RFp - particle reference frame (eg POIp,POIo(t))
RFo- observer reference frame (eg POIo,POIp(t))
Note that a POI cannot be simultaneously be fixed in BOTH RFo and RFp at a specific time t, EXCEPT when BOTH vo & vp = zero.
POIo(t) = POIp fixed in particle RFp, but seen in the observer Rfo at time t (eg for R,A,E,B,etc)
POIp(t) = POIo fixed in observer RFo, but seen in the particle RFp at time t(eg for R,A,E,B,etc)
See Section ""Points of Interest" (POI)s" for more details.
ρ charge density
Q is the angle between r (directed vector from observer to particle) and v (the constant velocity of the particle for Chapter 4). Note that the plane of intersection of [r,v] is CONSTANT - ie there is no rotation or complication as the particle moves.
q charge (4-3, 4-6, 4-14)
R displacement (distance, direction) between two masses
Re radius of Earth (position on surface of the Earth (8-15)
Rg displacement (distance, direction) to center of galaxy
Ruc for a universe with spherical symmetry, Ruc is the distance to the center of the universe from the vibrating neutral electric dipole (8-13)_(10-6)
R(POIstart,POIend) is used to denote [scalar distances, vectors, unit vectors, displacement vectors, coordinate displacement vectors] depending on the qualifiers of the symbol (eg [s,v,h,c]). If both POI are present, POIstart indicates the start point, POIend the endpoint. If only one POI is present, then it denotes the endpoint, whereas POIstart is the coordinate origin - but this is also emphasized by using the qualifier "c", giving "redundancy" to reduce errors .
Examples :
using the particle frame of reference (RFp) : eg [Rpcv,Rpds,Rpnh,Rpd]
using the observer frame of reference (RFo) : eg [Rocv,Rods,Ronh,Ros,]
Special notations for r :
[R2,R1] often used by Lucas for polar ends of vibrating dipole etc, r21 = ?? (7-11), R to center of universe,
RDEpdh(Rpch(POIo,t,dt)) unit vector in direction of dp[dt : E0odv(POIo,t)] , that is, the differential change in static electric field due to particle movement relative to a POIo.
Rθ0och unit vector in the direction of θ=0 (theta=0). In Chapter 4 this is the direction of the particle velocity in both the particle and observer reference frames (RFp,RFo), ie Rθ0pch = Rθ0och.
Rθ0ocs(POIo) Rθ0pcs(POIp) etc
Rθ0ocs(POIp(t),t) Rθ0pcs(POIo(t),t) etc
scalar distance that is the Rθ0ch direction component of a coordinate displacement vector from the origin of the observer reference frame (RFo) to a point of interest (POI,t) (i.e. a distance from coordinate origin along the direction of the particle movement).
As another way to say it, Rθ0cs is the magnitude of a displacement vector that is perpendicular to L(POI) and that is anchored to the POI (Point Of Interest). See Figure ???
RθPI2och(POIp(t),t) RθPI2pch(POIo(t),t) etc
unit vector, orthogonal to Rθ0ch and intersecting POI, that is at an angle of ???Pc(POI)??? to RθPI20ch. Note that this applies to either [RFo,RFp], and that in Chapter 4 RθPI2och = RθPI2pch.
RθPI2cs(POI,t) scalar distance that is the RθPI2ch direction component of a coordinate displacement vector rpc. Note that this applies to either [RFo,RFp], and that in Chapter 4 RθPI2ocs = RθPI2pcs
rx(POI) = perpendicular distance from the particle center to the L(POI). In Chapter 4, given constant vov(t) = vov, and Chapter 4's (constant vov, aligned particle & observer reference frames (RFp & RFo)), rx(POI) is NOT a function of time nor Pph=Poh.
rxh is a unit vector in the direction of rx
S Poynting vector (5-15)
s position
s symbol qualifier - means scalar (eg magnitude of a vector)
t time
tx time at which POIo = POIp(tx), or POIp = POIo(tx)
U generalized potential (usually wrt Universal force or component) (5-2)
v velocity
vs velocity of circular orbits (10-10)
vtan speed (component of velocity) tanjent to curve
x Position vector in Cartesian coordinates (comparing to more conventional notations :
x1 = x,i; x2=y,j; x3=z,k
Z charge state of [atom, particle,...]
z red shift (7-29))
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>>>>>>>>> Greek symbols, with shorthand
Shorthand notiations are often used for unicode characters, which are slow to [search,copy,paste]. They are converted automatically by
β, beta β= v/c (4-32), see small-case "b"
ε0 electric ?[permeability,permittivity]? in a vacuum,
ε0 = 8.854 187 817... × 10−12 F/m (farads per metre).
BUT The value of ε0 is currently defined by the formula[2]
ε0 = 1/μ0/c^2 AHHH HAH!
εr relative static permittivity
λ , lamda wavelength (equ?)
λ(v) velocity-dependent proportionality between Ei and E0 (4-31)
μ0 magnetic ?[permeability,permittivity]?.
ε0 = 1/μ0/c^2 or ε0*μ0 = c^2
µ0 = 4π×10−7 N / A2 ≈ 1.2566370614...×10−6 H / m or Tm / A or Wb / (Am) or Vs / (Am)
φ(r), phi For Appendix A as per p179h0.25 (A-3) (A-3) and (A-8) and p180h0.45 just below (A-10) from Ampere's Law
φ(r) = -3/2/c/|r|^2
But "A" drops out between p179h0.9 (A-9) and p180h0.3 (A-10) , assuming for CGS units that A/2 = 1/c (but Lucas typically uses Gaussian units?)
This is likely because A is "absorbed" into an arbitrary (at that point) ?
Shorthand PP(r)
χ(r), chi χ(r) = i*i/c/|r|^3 from p180h0.5 (A-11)
ψ(r), psi ψ(r) = B/|r|^2 as per p179h0.25 (A-3) from Ampere's Law,
from p179h0.8 (A-8) B = -3/2*A
Ω, omega solid angle (5-16)
θ, theta is the 2D angle between the plane perpendicular to v (unit vector for velocity of a POI - often of the particle in the observer frame), centered at the particle, and another point of interest(POI), measured from the direction of v.
In a sense, the angle θ is like the LATITUDE of the point of interest wrt the particle.
w, ω angular speed (radians/s (Fiigure 7-2,8-1) , w1,w2
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>>>>>> References :
electro-magnetic symbols - see https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law
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>>>>>> Gaussian versus SI units
Jackson 1999 p782h0.15 Table 3 provides conversions
?????? put table here!!!
I have NOT properly adjusted formulae from various sources for differences in the units used, whether Gaussian, SI, or other. This creates some confusion here and there in my review comments.
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>>>>>> Dimensional analysis (Gaussian units)
Most variables have units which are obvious, but E (electric field) and B (magnetic field) should be stated in terms of the Gaussian units used for derivations.
+--+
Limit CHECKS
Dimensional consistency :
From file "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt"
sub-sub-section "Dimensional analysis (Gaussian units)" :
As a convention here take [unit vectors, c] as dimensionless .
OK, as all terms reduce to the same unit dimensions.
All derivations below ignore permittivity & permeability for Gaussian coordinates, except that of F, which requires these factors to balance units?
Example : From "Howell - Key math info & derivations for Lucas Universal Force.odt"
sub-sub-section "E0pdv(POIo,t) = E0odv(POIo,t)"
Units of measure
Gauss's Law for a single point charge, in the particle reference frame (FRp) :
1) E0pdv(POIo,t) = E0odv(POIo,t)
= Q(particle)/Rpcs(POIo,t)^2*Rpch(POIo,t)
E units = (charge/length^2)
(ignoring permittivity & permeability for Gaussian coordinates)
dp[dt : E] units = (charge/length^2/time)
B units Example : From "Howell - Key math info & derivations for Lucas Universal Force.odt"
(ignoring permittivity & permeability for Gaussian units)
sub-sub-section "BTodv(POIp,t) = BTpdv(POIp,t) "
Generalized Ampere's Law :
BIpdv(POIp,t) = Vpnv(POIp)/c X E0pdv(Rpcv(POIp))
units = (length/time)*(charge/length^2)
= (charge/length/time)
dp[dt : B] units = charge/length/time^2
F (force) ignoring permittivity & permeability for Gaussian coordinates
from my correction to (4-06) and taking "c" as dimensionless
F(r,v,t) = F(r,v,t) = q*[ E(r,v,t) + v/cB(r,v,t) ]
units = charge*[ (charge/length^2) + (length/time)*(charge/length/time) ]
= charge^2/length^2 + charge^2/time^2
OOPS!! This doesn't work at all!!!
If I insert permittivity (ε) and permeability (μ) :
from : https://simple.wikipedia.org/wiki/Maxwell's_equations#The_structure_of_the_magnetic_field
ε = permittivity = F/m (F = Farads)
μ = permeability = W/(A*m) = Power /charge/length
= mass*length/time^3/ charge/length
E -= electric field = V/m
B = magnetic field = H (H = Henrys)
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>>> 'Howell's FlatLiner Notation' (HFLN) formatting conventions
I rewrite formulas in a format that can be used in either simple ASCII or UNICODE text files, for several reasons :
1. peer review systems were the initial motivation. Especially in the past, these systems accepted only ASCII text comments, sometime UNICODE. Some of these systems do not accept non-ASCII characters such as Greek letters.
2. Reduce when possible the use of special UNICODE characters (eg Greek letters where an equivalent English letter will do!) An additional problem that arises with UNICODE characters, is that even for Courier constant-width character sets, these tend NOT to have constant width, which mis-aligns expressions, and therefore majkes proof-reading more difficult (see "constant-character-width fonts " in the point below.
3. Elimination of [superscripts, subscripts, special line formating], which are not handled by simple text editors (other than coding like HTML).
4. Use constant-character-width fonts like courier 10 pitch. This greatly aids in the virticle alignment of successive equations, making the derivations much easier to follow, and often making it obvious when mistakes occur in expressions!
5. Restructure a few conventional ways of writing formule, which :
makes for faster typing,
makes for faster and better review by readers
reduces ambiguities that often arise with conventional notations.
"brings togther" information about an operation in front of the expression being evaluation, being faster and easier to see. Most notably :
dp(t : f(x,v,a,t)) = ∂f(x,v,a,t) /∂t "partial" derivative dp(dt : f(x,v,a,t)) = ∂/∂t f(x,v,a,t)
(dt, t0 to t1 : f(t) ) definite integral of f(t) wrt dt, from t0 to t1
facilitates use in symbolic programs
6. Points [2-5] above make typing MUCH [faster, easier] plus allow more precise alignment of [characters, symbols, expressions, functions, formulae] that greatly improves error-checking! Careful alignments often make [errors, omissions, incmpleteness] stand out automatically.
7. Symbolic processing - [formulae, descriptive text, variables] are entered in a "everything in a simple character line" format that is directly used as a computer program (QNial - Queen's University Nested Interactive Array Language, www.nial.com - this is an interpreted language (no compiler)). In some peer reviews that I do, this allows simple and powerful "symbolic" processing, especially of large higher-dimensional arrays, as this is much more accurate than imagining precisely outputs of formulae.
8. Create many sub-sub-sections for each key symbol etc - so that hyperlinks from the table of contents make it very easy to get to each derivation. (could use the index for that as well, but a Table of Contents provides a more powerful [ordering,organisation].
Note that QNial is not a "high usage" language, and perhaps I should migrate to O'Camel or something else, but I usually end up frustrated with the limitations of other languages, and I've grown used to it. I've not used C for many years except for Linux language ports and limited work - never got into C++ or C# etc as I rarely need the programming framework and standard coding, which is easy to adapt to QNial (but much more work to go the other way.
HFLN nomenclature is intentionally redundant (like language) to help reinforce symbol meanings and make it easier to spot errors. For example :
dp[dt : E0ods(POIo,t)] = dp[dt : E0pds(POIo(t),t)]
E0pds(POIo(t),t) is a measure of the static electric field at POIo (Point of Interest fixed in the observer frame RFo) at time t. Note that POIo shifts position in RFp with time, and that the static electric field at that point that arises from the moving particle, also changes with time.
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>>>>>> General principles for symbol conventions used in this paper
Below are some principles used for variable symbols. These have the disadvantage of making expressions much longer, but the cost is compensated for greater [clarity, consistency, error-checking,etc].
Redundancy - easier to see that a mistake has been made, or that thinking is sloppy
This is like human language. For example, in the symbol "Ooca(POIp,t)" the "a" is redundant, as "O" only refers to an angle. Therefore, a reader who sees "Ooc(POIp,t)" should be on the alert that I may have made a mistake with the symbol.
Sparseness - helps to reduce errors from [missing,switched] letters of symbols. Only a few of the possible combinations of [symbols,qualifiers] are actually used.
Uniqueness from symbol length Often "copy&replace" operations are used to update or change symbol conventions. Short symbols like "r" require much more attention and manual colow-level objective, not yet implemented, would be to implement a constant charater-length standard for variable symbols (eg 5 character), so that alignment of symbols in multi-line formulae, or sequences of equations, becomes "near-automatic". However, this causes other problems.
Context While it is normal to pre-specify the context of derivations, by incorporating key context into the symbols themselves, it is easier to remember specifics that are important to derivations.
Common concepts To reduce a proliferation of symbols that are hard to rememeber, a common "basic symbol" (i.e. first letter or so) is used as much as possible. For example "r" is used to denote [length, vector, displacement vector, etc].
Capital letters for the basic variable symbol - To clearly distinguish the basic symbol of a variable from qualifiers, the basic symbols are only in CAPITAL letters.
Also, there aren't enough letters in the alphabet to cover all basic sysmbols, so if a new symbol is needed, sometimes two or more capital letters are used to denote it. For example (POI) dentotes "Point Of Interest".
Small letters for qualifiers - To clearly distinguish the basic symbol of a variable from qualifiers, the qualifiers are only in small letters.
"Find-and-Replace-ability" - When changing symbols or correcting errors in formulae, having [long, distinct] symbols can greatly aid with cut&paste operations. For example, if [r, roc, rocs] are used, then a mass " Find-and-Replace" operation on an entire file, or section of a file, for "r" becomes an intensely manual "r-by-r" operation. The use of [rocv,rocs] is much easier.
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>>>>>> Notations for [indexing, inequalities, super, sub]-script, calculus, series [sums, products]]
Indexing :
yi y(i), simple concatenation (might be confused with y*i)
y(i+1) array type notation
Note the more clear & easy to read indexing used in this review for mixing indexes and t or other variables
... example ai(t) -> a(i,t)
Inequalities (for faster typing, symbolic processing)
≤ <=
≥ >=
≠ !=
Special notations
~= "not equal to", a Q'Nial programming language convention, which is often denoted in the literature as "<>", "!=" etc
This is NOT used in this work.
[super, sub]-script symbols
"_cup" denotes variable values during ??? eg p4L23 Eq (9d)
"_ovr" denotes variable values during times s ∈ [t_til(0),0]??? eg ξ_ovr(t)
"_til" denotes variable values during ??? eg p4L38
/**********************************************
>>>>>> Calculus notations :
d[dt: x] total derivative of x with respect to t
dp[dt: x] partial derivative of x with respect to t
d^n[dt^n: x] nth total derivative of x with respect to t
dp^n[dt^n: x] nth partial derivative of x with respect to t
∫[ds: f] indefinite integral of f with respect to s
∫[ds, a to b: f] definite integral of f with respect to s, from a to b
∮[ds: f(t)] integral of f(t) over the closed [2D curve, 3D surface] S
∮[dl: f(l)] closed-loop integral f(l) wrt dl (pathway)
∮[dl: f(l)] closed-loop integral with dot product of integranddl (eg in Green's theorem etc)
Other :
sum[i=1 to n: f(n,t)] sum of a series
Π[tt_plus : (r(t,t) - r(s,t))/(t - s)) }
D_minus D_minus(t) = lim{s->t_minus: (r(t,t) - r(s,t))/(t - s)) }
/**********************************************
>>>>>> 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 : :
/**********************************************
>>>>>>>>> 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
/**********************************************
>>>>>>>>> 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.)
/**********************************************
>>>>>>>>> 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
/**********************************************
>>>>>>>>> Fourth qualifiers
0 angle of 0 (zero) wrt angle coordinate origin
/**********************************************
>>>>>>>>> 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.
/**********************************************
>>>>>> 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 ∇′ )
/**********************************************
>>>>>> Points of Interest (POI)s
"Points of Interest" (POI) are points in a coordinate space of a frame of reference. Note that POI is a point of interest, which may refer to a particle, but more generally simply refers to a point in space.
The notation used here is :
POI (POIo,POIp) = "Point Of Interest" is a specific point, and is typically where estimates of [B,E,F, etc] are required. Normally, the POI is STATIONARY in EITHER :
RFp - particle reference frame (eg POIp,POIo(t))
RFo- observer reference frame (eg POIo,POIp(t))
Note that a POI cannot be simultaneously be fixed in BOTH RFo and RFp at a specific time t, EXCEPT when BOTH vo & vp = zero.
POIo(t) = POI fixed in particle RFp, but seen in the observer Rfo (eg for EIocv(POIp(t),t),B,etc)
POIp(t) = particle location, as a POI, usually expressed in the RFo.
For a (POI) fixed in the particle reference frame (RFp) => (POIp) :
(POIp) applies to to basic measures and their derivatives for (POI)s fixed with respect to (RFp). This is often used for coordinate based measures such as :[rpcs,Rθ0pcs,RθPI2pcs,sin(Opc),cos(Opc)], etc)
CAUTION : The following two symbols are a "flip of the reference frame" of POIp, in the sense that they refer to points in the OBSERVER frame of reference RFo that describe the trajectory of POIp through RFo.
(POIp(t)) refers to the position over time, in the OBSERVER frame of reference RFo, of a point of interest that is fixed in the particle reference frame RFp. Effectively, this is a trajectory of POIp in RFo.
(POIp(tx)) refers to a point of interst in the OBSERVER reference frame (RFo) of a fixed point in the particle reference frame RFp (POIp), the trajectory of which, at the specific time tx, coincides with a fixed POIo, defining a common point FIXED in RFo.
Note that :
dp[dt : POIp(tx)] = dp[dt : POIo] = 0,
whereas
dp[dt : POIp(t)] = Vonv(particle)
applies to to basic measures and their derivatives for (POI)s that move with respect to (RFp). This is often used for (RFo) coordinate based measures with respect to the observer origin such as [rocs,Rθ0ocs,RθPI2ocs,sin(Ooc),cos(Ooc)], etc).
It is also used for derived measures and their derivatives for (POI)s fixed with respect to (RFp), but which are time dependent (no examples yet 03Feb2016).
It is also used for derived measures and their derivatives for (POI)s that move with respect to (RFp), and which are obviously time dependent (no examples yet 03Feb2016).
For a (POI) fixed in the OBSERVER reference frame (RFo) => (POIo) :
(POIo) applies to to basic measures and their derivatives for (POI)s fixed with respect to (RFo). This is often used for coordinate based measures such as [rocs,Rθ0ocs,RθPI2ocs,sin(Ooc),cos(Ooc)], etc)
CAUTION : The following two symbols are a "flip" of the meaning of POIo, in the sense that they refer to points in the PARTICLE frame of reference RFp that describe the trajectory of POIo through RFp.
(POIo(t)) refers to the position over time, in the PARTICLE frame of reference RFp, of a point of interest that is fixed in the particle reference frame RFo. Effectively, this is a trajectory of POIo in RFp.
(POIo(tx)) refers to a point of interst in the PARTICLE reference frame (RFp) of a fixed point in the observer reference frame RFo (POIo), the trajectory of which, at the specific time tx, coincides with a fixed POIp, defining a common point FIXED in RFp.
Note that :
dp[dt : POIo(tx)] = dp[dt : POIp] = 0,
whereas
dp[dt : POIo(t)] = -Vonv(particle)
applies to basic measures and their derivatives for (POI)s that move with respect to (RFo). This is often used for coordinate based measures with respect to the particle such as [rpcs,Rθ0pcs,RθPI2pcs,sin(Opc),cos(Opc)] etc).
It is also used for and their derivatives for (POI)s fixed with respect to (RFo), but which are time dependent such as [E0odv,BTodv,etc] and their derivatives..
It is also used for derived measures and their derivatives for (POI)s that move with respect to (RFo), and which are obviously time dependent([ E0odv(POIo), BTodv(POIo,t), etc).
Examples :
rocs(POIo) = rpcs(POIo,t) but which also = rocs(POIo,t) = rpcs(POIo)
For the electrostatic field :
E0odv(POIo,t) = E0pdv(POIo(tx),t) != E0pdv((POIo,t) = Eodv(POIo)
For an induced electroic field :
EIpdv((POIo) = EIodv(POIo,t) but which also != EIpdv((POIo,t) = EIodv(POIo)
??? need more explanation.
??nyet : My interpretation is that this relates to Bill Lucas' comments :
p71h0.15 "... In order to obtain the results observed in the laboratory rame, it is necessary to evaluate Ei(rvt,t) at t=0 when the moving frame coincides with the laboratory frame after all terms for a given iteration have been evaluated. In order to simplify the iteration f successive terms the r'= r -vt terms are left in place in order to keep track of the correct power of r' for the derivative in the iterative term. The vt(cos(theta')) terms are explicitly dropped. ..."
CAUTIONS :
???You should ?ALWAYS? see the same "o" or "p" qualifier (meaning observer or particle reference frame) for the variable symbol and the point of interest symbol (for example, both "o"s in Rocs(POIo)). ???
This redundancy provides a good check for some errors that may srise in formulae (mostly when not paying enough attention to the frames of reference).
Note that time lags SHOULD affect field formulae unless field movement across infinite distance is instantaneous, but this issue is not addressed in this document..
Vibrating neutral diploes - are also looked at in a previous Chapter of Lucas's book. My nomenclature specific to that situation have yet to be specified.
General comment – where a "measure" (eg [R,v,E,B, etc]) is sought in a reference frame, then a POI fixed (stationary) in the same reference frame is not a function of time (i.e. doesn't move), whereas a POI fixed (stationary) in another reference frame IS a function of time.
It is recommended that the POI symbol ALWAYS be written in terms of the frame of reference in which the POI is fixed (stationary). For example :
EIocv(POIp(t),t) → the POI is stationary in the particle reference frame (RFp), and therefore is a function of time in the observer reference frame (Rfo) as is EIocv for that POI. POIp(t) makes both aspects clear.
EIpcv(POIp,t) → the POI is stationary in the particle reference frame (RFp), and therefore is NOT a function of time in the particle reference frame (RFp). POIp makes both aspects clear. This is an intermediary expression, as EIpcv in this case is not a function of time! This expression becomes EIpcv(POIp) = 0 as there is NO induced [B,E] in this case!!! But "EIpcv(POIp,t)" is a proper starting expression, as the simplifications easily follow in subse2quent steps.
EIpcv(POIo(t),t) → the POI is stationary in the observer reference frame (RFo), and therefore is a function of time in the particle reference frame (Rfp) as is the case for EIpcv. POIp(t) makes both aspects clear.
While one could write alternate forms
EIocv(POIo(t),t)
EIpcv(POIo(t),t)
these are NOT recommended, as they tend to "confuse" the issue (for me anyways, if I am not consistent), as the expression "EIocv(POIo(t),t)" loses the general sense that the POI referred to is fixed in the "o" frame (observer), which is NOT the case.
/**********************************************
>>>>>> 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)).
/**********************************************
>>>>>>>>> Notation for equations from the same [sub-sub] section
Equation references simply give the equation number and text. For example, in the file "Howell - Background math for Lucas Universal Force, Chapter 4.odt", sub-sub-section "dp[dt : E0ods(POIo,t)] = dp[dt : E0pds(POIo,t)] - proper E0odv(POIo,t) vector approach" :
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.
/**********************************************
>>>>>>>>> Notation for equations from a different [sub-sub] section in the same file
Equation references give the title of the [sub-sub] section in addition to the equation number and text. For example, in the file "Howell - Background math for Lucas Universal Force, Chapter 4.odt", sub-sub-section "dp[dt : E0ods(POIo,t)] = dp[dt : E0pds(POIo,t)] - proper E0odv(POIo,t) vector approach" :
From "Relating [Rpcs,Rθ0pcs,RθPI2pcs,sin(Opc),cos(Opc)]@t to [Roc,Oo,Po] for (POIo)" :
2) Rpcs(POIo,t))
= { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Ooca(POIo))*Vons(particle)*t + [Vons(particle)*t]^2 }^(1/2)
6) cos(Opca(POIo,t))
= [ Rocs(POIo)*cos(Ooca(POIo)) - Vons(particle)*t]
/ { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Ooca(POIo))*Vons(particle)*t + [Vons(particle)*t]^2 }^(1/2)
Subbing (2)* and (6)* into (3) :
4) 3) dp[dt : E0pds(POIo,t)]
= Q(particle)*Vons(particle)/Rpcs(POIo,t)^3*[1 + 3*cos(Opca(POIo,t))^2]
Note that when proceeding with the substitution of equations, the asterix "*" follows (2)* and (6)* which are in a different sub-sub-section, whereas equation (3), which is in the same sub-sub-section, does not have an asterix.
Notation for equations from a different [sub-sub] section in a different file
Equation references simply give the file name and the title of the [sub-sub] section in that file, in addition to the equation number and text. For example :
???? get example later ....
Notation for equations from a book
Equation references simply give the file name and the title of the [sub-sub] section in that file, in addition to the equation number and text. For example :
???? get example later ....
/*************************************************
>>> APPENDICES
/*************************************************
>>>>>> Future extensions of the Universal Force
/*$ cat >>"$p_augmented" "$d_Lucas""context/future extensions.txt"
/********************************************** ;
>>>>>>>>> Gaussian versus SI units ;
Gaussian versus SI units :
I have NOT properly adjusted formulae from various sources for differences in the units used, whether Gaussian, SI, or other. This creates some confusion here and there in my review comments.Jackson 1999 p782h0.15 Table 3 provides conversions
/**********************
>>>>>> Text type [comment, Lucas, Howell] for automated symbol checking and translation
/*$ cat >>"$p_augmented" "$d_Lucas""context/formatting comments.txt"
/********************************************
>>>>>>>>> HFLN = Howells FlatLiner Notation !!!!!!!!!!!!!! 31May2016
/*$ cat >>"$p_augmented" "$d_Lucas""context/Howells flat-line notation short description.txt"
/**********************
>>>>>> Document build short description
/*$ cat >>"$p_augmented" "$d_Lucas""context/document build short description.txt"
/*************************************************
>>>>>> REFERENCES
/*$ cat >>"$p_augmented" "$d_Lucas""context/references.txt"
enddoc