First try : ?date?., third attempt 06Sep2015, rev1 24Sep2015,
09Jan2016 rev4Lucas4-15, 22May2016 used my expression from file
"Howell - Background math for Lucas Universal Force, Chapter 4.odt" HFLN format :
# Start with (4-15) Howell's version
E,B for symmetry point charge @v_const
Adapted non-[derivative, integral] form of Amperes law
/$ Bi(ro - vo*t,t)
= vos/c*Pph
* [ q*sinOo*ros/|rov - vo*t|^3
+ sinOp*|Ei(ro - vo*t,t)| ]
/* taking the ∂/∂t of (4-15) Howell's
/$1) dp[dt : Bi(ro - vo*t,t)]
= vos/c*Pph
* [ dp[dt : q*sinOo*ros/|ro - vo*t|^3 ]
+ dp[dt : sinOp*|Ei(ro - vo*t,t)| ] ]
/*
yy)"Pseudo-constant variables/expressions" within partial derivatives wrt time
yy)Remove : [c,L(v),v,b,q,rp,|ro - vo*t|^(n),Op,sinOp,Pp] - Chap4 v=constant, Chap5 v=variable.
yy)Retain : [t,E,B,ro,Oo,sinOo] are the key variables to retain within derivatives
for Chapter 4 verifications
# yielding
/$2) dp[dt : Bi(ro - vo*t,t)]
= vos/c*Pph
* [ q /|ro - vo*t|^3 *dp[dt : ros*sinOo ]
+ sinOp *dp[dt : |Ei(ro - vo*t,t)|] ]
/* From "Howell - Key math info & derivations for Lucas Universal Force.odt"
Section II.4 :
/$II.4.9) dp[dt : ros*sinOo] = 0
/*OOOPPPS!!! - q term drops out as it is a constant
and the derivative is zero
/$ dp[dt : Bi(ro - vo*t,t)]
= vos/c*Pph
* [ q /|ro - vo*t|^3 *dp[dt : ros*sinOo ]
+ sinOp *dp[dt : |Ei(ro - vo*t,t)|] ]
/*10Jan2016 The problem here is that in (4-14) and (4-15) the "q" expressions are constant - leading to a zero derivative.
dp[dt : ros*sinOo] = vos*sinOo*(cosOo - 1)
/*therefore :
/$4) dp[dt : Bi(ro - vo*t,t)]
= vos/c*Pph
* [ q/|ro - vo*t|^3 *vos*sinOo*(cosOo - 1)
+ sinOp*dp[dt : |Ei(ro - vo*t,t)| ] ]
= vos/c*Pph
* [ q*vos*sinOo*(cosOo - 1)/|ro - vo*t|^3
+ sinOp*dp[dt : |Ei(ro - vo*t,t)| ] ]
/* This is very different from Lucas, at least in appearance.
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Possible_Lucas_error_or_omission
04_16rev4Lucas4-15
E,B for symmetry point charge @v_const
Derivative form of Amperes law
/$ ∂[∂(t): Bi(r - v*t,t)]
= vs/c*Rocs(POIo)*sin(θ)*φ´hat*
[ 3*q*vs/c*(Rocs(POIo)*cos(θ) - vs*t)/|r - v*t|^5
+ 1/Rocs(POIo)/c*∂[∂(t): |Ei(r - v*t,t)|] ]
/% dp[dt : Bi(ro - vo*t,t)]
= vos/c*Pph
* [ q*vos*sinOo*(cosOo - 1)/|ro - vo*t|^3
+ sinOp *dp[dt : |Ei(ro - vo*t,t)|] ]
/* Note how different my version is ??!!??
/% ∂[∂(t): BTpdv(POIo(t),t)] = ∂[∂(t): BTodv(POIo,t)]
= Vons(PART)^2/c*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo))
*{ 3*Q(particle) /Rpcs(POIo(t),t)^3
- EIpds(POIo(t),t)/Rpcs(POIo(t),t)
- ∂[∂(t): EIpds(POIo(t),t)]/Vons(PART)/cos(Aθpc(POIo(t),t))
}
/* 10Jan2016 OOOPPPSSS - q-term is zero (constant has zero derivative)
09Jan2016 WRONG! : I use Op not Oo for 2nd term RHS (as with (4-15), different terms
This could partially be due to differentiation, but not all.
??It seems to me that Lucas has dropped the "Vons(PART)*cos(AOpc(POIo,t))*EIpds(POIo,t)/Rpcs(POIo(t),t)" term.
But there should be an EIpds(POIo,t) term!
/* It seems clear to me that Lucas has :
- dropped the "Vons(PART)*cos(AOpc(POIo,t))*EIpds(POIo,t)/Rpcs(POIo(t),t)" term. But there should be an EIpds(POIo,t) term!
22Aug2019 NO!??? I factored out Vons(PART)*cos(AOpc(POIo,t)) ???
- has too many "c"s
22Aug2019 - do dimensional analysis!!
- has the term "(ros*cosOo - vos*t)/|ro - vo*t|" in the first term in parenthesis.
- I have NO real idea of why the (ros*cosOo - vos*t) pops up anyways, and why the cos term all of a sudden (spherical coords does not explain this!)
30Mar2018 do spherical coordinate derivative!?!!
NOTE!!! : use "/media/bill/SWAPPER/Qnial/MY_NDFS/matrix derivatives in [cartesian, cylindrical, spherical] coordinates.ndf"