/$ ∫[dEi(r - v*t,t),0 to θ´: 1)
= Ei(r - v*t,t) |for θ´ from 0 to θ´f
= Ei(r - v*t,t) - Ei(r - v*t,t,θ´ = 0)
-1/c*∂[∂(t): Bi(r - v*t,t)]
= 3*(v/c*r)^2*q *∫[∂(θ´),0 to θ´: (r*cos(θ´) - v*t)/|r - v*t|^5*sin(θ´)
+ v*r^2/c *∫[∂(θ´),0 to θ´: 1/r/c*∂[∂(t): Ei(r - v*t,t)*sin(θ´))???brackets!!!!
/* therefore
/$ Ei(r - v*t,t) - Ei(r - v*t,t,θ´ = 0)
= 3*(v/c*r)^2*q *∫[∂(θ´),0 to θ´: (r*cos(θ´) - v*t)/|r - v*t|^5*sin(θ´)
+ v*r^2/c *∫[∂(θ´),0 to θ´: 1/r/c*∂[∂(t): Ei(r - v*t,t)*sin(θ´))???brackets!!!!
/* 1. Take Lucas's result for (4-29a), remove the minus signs
04_29arev1 Faradays_law_spherical_coordinates - reduced, integral form
/$1) -1/rs*∂[∂(θ´): Ei(r - v*t,t))*φ´
= -vs/c*rs*sin(θ´)*φ´*[ 3*q*vs/c*(r*cos(θ´) - v*t)/|r - v*t|^5
+ 1/rs/c*∂[∂(t): Eis(r - v*t,t)] ]
/* Taking (1), expand while keeping a "*1/rs/c" term with "dp[dt : Eis]", drop - signs
/$ 1/rs*∂[∂(θ´): Ei(r - v*t,t))*φ´
= + vs/c *rs*sin(θ´)*φ´* 3*q*vs/c*(rs*cos(θ´) - vs*t)/|r - v*t|^5
+ vs/c *rs*sin(θ´)*φ´* 1/rs/c*∂[∂(t): Eis(r - v*t,t)]
= + 3*(vs/c)^2*rs*sin(θ´)*φ´*q *(rs*cos(θ´) - vs*t)/|r - v*t|^5
+ vs/c *rs*sin(θ´)*φ´ *1/rs/c*∂[∂(t): Eis(r - v*t,t)]
/* Integrate this, switching again from Op to Op? (MUST check later!)
/$ ∫[∂(θ´),0 to θ´f: - 1/rs*∂[∂(θ´): Ei(r - v*t,t))*φ´]
= ∫[∂(θ´),0 to θ´f: 3*(vs/c)^2*rs*sin(θ´)*φ´*q*(rs*cos(θ´) - vs*t)/|r - v*t|^5]
∫[∂(θ´),0 to θ´f: vs/c*rs*sin(θ´)*φ´*1/rs/c*∂[∂(t): Eis(r - v*t,t)]]
/*zz)see "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" :
zz)"Pseudo-constant variables/expressions" within integrals wrt Op
/$zz)Remove : [q,v,λ(v),r´,c,β]
zz)Retain : [sin(θ´),θ´] are the key retained variables for Chapter 4 verifications
1/rs*φ´ *∫[∂(θ´),0 to θ´f: ∂[∂(θ´): Ei(r - v*t,t))]
= 3*(vs/c)^2*rs*φ´*q/|r - v*t|^5*∫[∂(θ´),0 to θ´f: sin(θ´)*(rs*cos(θ´) - vs*t)]
+ vs/c *rs*φ´ *∫[∂(θ´),0 to θ´f: sin(θ´)/rs/c*∂[∂(t): Eis(r - v*t,t)]]
/* HOWEVER, Lucas prefers to leave |ro - vo*t|^5 within the integral, so
Ill re-insert it :
/$ 1/rs*φ´ *∫[∂(θ´),0 to θ´f: ∂[∂(θ´): Ei(r - v*t,t))]
= 3*(vs/c)^2*rs*φ´*q*∫[∂(θ´),0 to θ´f: (rs*cos(θ´) - vs*t)*sin(θ´)/|r - v*t|^5]
+ vs/c *rs*φ´ *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r - v*t,t)]*sin(θ´)]
/* Cancel variables LHS vs RHS
/$ ∫[∂(θ´),0 to θ´f: ∂[∂(θ´): Ei(r - v*t,t))]
= 3*(vs/c*rs)^2*q *∫[∂(θ´),0 to θ´f: (rs*cos(θ´) - vs*t)*sin(θ´)/|r - v*t|^5]
+ vs/c*rs^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r - v*t,t)]*sin(θ´)]
/* While dp(dOp : Ei(ro - vo*t,t)) is nominally zero for the simple case of
a single, stationary point charge, it will not be in general. So taking
the integral of the derivative simply leaves me with :
/$ Ei(r - v*t,t)|θ´=θ´f - Ei(r - v*t,t,θ´ = 0)
= 3*(vs/c*rs)^2*q*∫[∂(θ´),0 to θ´f: (rs*cos(θ´) - vs*t)/|r - v*t|^5*sin(θ´)]
+ vs/c*rs^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r - v*t,t)]*sin(θ´)]
/* but
/$ Rpcs(POIo(t),t)*sin(AOpc(POIo,t)) = Rocs(POIo)*sin(AOoc(POIo))
= 3*(vs/c*rs)^2*q*∫[∂(θ´),0 to θ´f: (r*cos(θ) - vs*t)/|r - v*t|^5*sin(θ´)]
+ vs/c*rs^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r - v*t,t)]*sin(θ´)]
/* Im NOT comfortable with this!!
For the overall solution approach, the Eis integral is left untouched.
Eventually, it will be argued that successive iterations result in smaller &
smaller terms that can be dropped from a series expansion.
/* 30Aug2019 see "Howell - Background math for Lucas Universal Force, Chapter 4.txt"
section '"Rpcs(POIo(t),t)" is a constant for integrals like "∫{∂(Aθtc),0 to Aθoc(POIp(t),t=0):" ?'
/% EIods(POIp(t),t) - EIods(POIp(t),t=0)
= 3*Q*Vons(PART)^2/c^2*Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*Rpcs(POIo(t),t=0)^( - 5)
*{ Rocs(POIo)*cos(Aθpc(POIo(t),t)) - Vons(PART)*t }
]
+ Vons(PART) /c *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* Clearly, the integral with respect to ∂(Aθpc),0 to Aθpc(POIo(t),t=0) is taken at a "snapshot of time", so t is a constant, independent, for the purposes of this integral, from ∂(Aθpc).
The expression above becomes :
/% EIods(POIp(t),t) - EIods(POIp(t),t=0)
= 3*Q*Vons(PART)^2/c^2*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))
*{ Rocs(POIo)*cos(Aθpc(POIo(t),t)) - Vons(PART)*t }
]
+ Vons(PART) /c *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]
/* |--???-->
/* 30Aug2019 (4-30) Shouldn't this be a full integral [2*PI = Aθpc(POIo(t),t=0) for 2D, or spherical surface]? :
∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t)) ]
eg. ∂(Aθpc)/∂(2*PI) -> fraction of a uniform
No? This is not an integral in time, it is an integral over a spherical surface in the direction of θ at an instant in time. Aθpc(POIo(t),t) = 0 is point either [towards, backwards] the direction of motion of the particle in RFo.
Yes? These equations are really only set up for constant normal [fields, forces] at a constant radius Rpcs(POIo(t),t=0)
/* <--???--|
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn OK - just take Rpcs(POIo(t),t=0) out of the integral as a constant wrt θ
04_30rev4
Lorentz force - Faradays_law_integrated
/$L Ei(r - v*t,t)|θ´=θ´f - Ei(r - v*t,t,θ´ = 0)
= 3*(vs/c*rs)^2*q*∫[∂(θ´),0 to θ´f: (rs*cos(θ´) - vs*t)/|r - v*t|^5*sin(θ´)]
+ vs/c*rs ^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r - v*t,t)]*sin(θ´)]
/$H Ei(r - v*t,t)|θ´=θ´f - Ei(r - v*t,t,θ´ = 0)
= 3*(vs/c*rs)^2*q*∫[∂(θ´),0 to θ´f: (rs*cos(θ´) - vs*t)/|r - v*t|^5*sin(θ´)]
+ vs/c*rs ^2 *∫[∂(θ´),0 to θ´f: 1/rs/c*∂[∂(t): Eis(r - v*t,t)]*sin(θ´)]
/* OK - simple, but I must redo with "current" derivations I have.
Note that the derivative in the second line below has RFo notation (important to keep me on track!).
The expression below may un-necessarily restrict the integrand to t=0 even though that is the ultimate context?
/* |--???--> 27Aug2019 Where Lucas uses r*cos(θ'), I am using Rocs(POIo)*cos(Aθpc(POIo(t),t))
as that is the only thing that makes sense to me at this stage! but it doesn't make sense!!
WRONG? -> Lucas seems to be using θ' (prime) as an arbitrary integration variable, which is a VERY BAD idea!!??!!
<--???--|
/%
(mathL)/* (4-30) /%
EIods(POIp(t),t) - EIods(POIp(t),t=0)
= 3*Q*Vons(PART)^2/c^2*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))
*{ Rocs(POIo)*cos(Aθpc(POIo(t),t)) - Vons(PART)*t }
]
+ Vons(PART) /c *Rocs(POIo)^2
*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ]