/$ ∫[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)) ]