742:(mathH) Rpcv(POIp) = constant 744:(mathH) Aθpc(POIp) = constant 746:(mathH) Aφpc(POIp) = Aφoc(POIp(t),t) = constant 759:(mathH) Rpcv(POIo(t),t) = Rocv(POIo) - Vonv(PART)*t 761:(mathH) Rpcs(POIo(t),t) = Rocs(POIo) - Vons(PART)*t 790:(mathH) Rpcv(POIo(t),t) = [(Rocs(POIo)*cos(Aθoc(POIo)) - Vonv(PART)*t]*Rθ0och(POIo) + [Rocv(POIo)*sin(Aθoc(POIo))]*RθPI2och(POIo) 819:(mathH) Rpcs(POIp) = |Rocv(POIo) - Vonv(PART)*t| 854:(mathH) Rpcs(POIo(t),t) = { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) 901:(mathH) sin(Aθpc(POIo(t),t)) = RθPI2pcs(POIo)/Rpcs(POIo(t),t) 911:(mathH) Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) 922:(mathH) sin(Aθpc(POIo(t),t)) = Rocs(POIo)*sin(Aθoc(POIo)) / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) 942:(mathH) cos(Aθpc(POIo(t),t)) = [Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2}^(1/2) 969:(mathH) Rθ0pcs(POIo(t),t) = Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t)) 986:(mathH) Rθ0pcs(POIo(t),t) = Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t 1004:(mathH) RθPI2pcs(POIo(t),t) = Rpcs(POIp)*sin(Aθpc(POIp)) = constant 1016:(mathH) RθPI2pcs(POIo(t),t) = Rocs(POIo)*sin(Aθoc(POIo)) = constant 1029:(mathH)/* for use in differentiations /% K0 = 3/2*β^2*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 1033:(mathH)/* for use in differentiations /% K2 = (-1)*λ(Vons(PART))*Q(PART)*Rpcs(POIo(t),t)^(-2) 1037:(mathH)/* not used in derivations, see "???" /% K1 = (-3)*β^2*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t)) - 1) 1042:(mathH) f_sphereCapSurf(x) = β*Rocs(POIo)^2*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))*∂[∂(t): x]] 1055:(mathH) Rocs(POIo) = Rpcs(POIo(t),t=0) when : t=0, RFp=RFo @t=0, POIo is on the trajectory of the particle in the direction of Vons(PART) 1061:(mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% K0(t=0) = E0ods(POIo,t=0)*3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 1065:(mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% K2(t=0) = -E0ods(POIo,t=0)*λ(Vons(PART)) 1069:(mathH)/* not used in derivations, see "???" /% K1(t=0) = 0 1089:(mathH) EIods(POIo,t=0,1st stage) = K_1st + f_sphereCapSurf(EIods(POIo,t)) 1092:(mathH) EIods(POIo,t=0,2nd stage) = K_1st + f_sphereCapSurf(EIods(POIo,t=0,1st stage) 1096:(mathH) EIods(POIo,t=0,ith stage) = K_1st + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage)))} 1105:(mathH) K_1st = K0 + K2 1108:(mathH)/* differentiable form /% K_1st = + Q(PART) *( 3/2 *β^2*Rocs(POIo)^3 *Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 - λ(Vons(PART)) *Rpcs(POIo(t),t)^(-2) ) 1136:(mathH)/* HIGHLY restrictive conditions : t=0, RFo(t=0) = Rfp(t=0), Rocs(POIo) = Rpcs(POIo(t),t=0) This means that the [observer, particle] reference frames are exactly the same at t=0 (other than motion). drop as roundoff error : f_sphereCapSurf expression see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt" /% K_1st = E0pds(POIp)*3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 - E0pds(POIp)*λ(Vons(PART)) 1164:(mathH) K_2nd = + 21/8 *β^4*Q(PART) *Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t=0))^4 - λ(Vons(PART)) *1 *β^2*Q(PART)*Rocs(POIo) *Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t=0))^2 1179:(mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! K_1st(t=0) = E0ods(POIo,t)*3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 - E0ods(POIo,t)*λ(Vons(PART)) 1188:(mathH)/* HIGHLY restricted! [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% K_2nd(t=0) = E0ods(POIo,t)*21/8*β^4*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t=0))^4 - E0ods(POIo,t)*λ(Vons(PART))*β^2*sin(Aθpc(POIo(t),t=0))^2 1204:(mathH) E0pdv(POIp) = Q(PART)/Rpcs(POIp)^2*Rpch(POIp) 1207:(mathH) E0pds(POIp) = Q(PART)/Rpcs(POIp)^2 1220:(mathH) E0odv(POIo,t) = E0pdv(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 *Rpch(POIo(t),t) 1223:(mathH) E0ods(POIo,t) = E0pds(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2 1234:(mathH) B0pdv(POIp) = 0 1237:(mathH) B0odv(POIo) = 0 1253:(mathH) BIpdv(POIp) = 0 1264:(mathH) B0pdv(POIp) = 0 as given in Chapter 4 1270:(mathH) BTpdv(POIp) = 0 1278:(mathH) EIpdv(POIp) = 0 1290:(mathH) ETpdv(POIp) = Q(PART)/Rpcs(POIp)^2*Rpch(POIp) 1326:(mathH) ∂[∂(t): Rpcv(POIp)] = 0 1329:(mathH) ∂[∂(t): Aθpc(POIp)] = 0 1348:(mathH) ∂[∂(t): Rpcv(POIo(t),t)] = -Vonv(PART) 1369:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = -Vons(PART)*cos(Aθpc(POIo(t),t)) 1514:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = Vons(PART)*{- Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t} /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(-1/2) 1558:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-α)] = α*Vons(PART)*Rpcs(POIo(t),t)^(-α - 1)*cos(Aθpc(POIo(t),t)) 1593:(mathH) ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) 1623:(mathH) ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(1/2) 1703:(mathH) ∂[∂(t): Rpch(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*RDEpdh(POIo(t),t) 1735:(mathH)/* where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t), is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t), angle Aφpc(POIo(t),t) doesn't change /% ∂[∂(t): Rpch(POIo(t),t)] = Vons(PART)*RDEpdh(POIo(t),t)*Rocs(POIo)*sin(Aθoc(POIo)) / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} 1786:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) 1839:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = -Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART)*{-Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t} / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2}^(3/2) 1905:(mathH) ∂[∂(t): cos(Aθpc(POIo(t),t))] = -Vons(PART)*sin(Aθpc(POIo(t),t))^2/Rpcs(POIo(t),t) 1999:(mathH) ∂[∂(t): Rpcs(POIo(t),t)*sin(Aθpc(POIo(t),t))] = 0 2002:(mathH) ∂[∂(t): ROPI2pcs(POIo(t),t)] = 0 2122:(mathH) ∂[∂(t): Rpcs(POIo(t),t)*cos(Aθpc(POIo(t),t))] = -Vons(PART) 2125:(mathH) ∂[∂(t): Rθ0pcs(POIo(t),t)] = -Vons(PART) 2256:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^a*Rpcs(POIo(t),t)^(-β)] = (a+β)*Vons(PART)*sin(Aθpc(POIo(t),t))^a*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-β - 1) 2262:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^1] = 1*Vons(PART)*sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-1) 2265:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2] = 2*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-1) 2268:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^3] = 3*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-1) 2271:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4] = 4*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-1) 2275:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^( 1)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 2278:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-1)] = Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-2) 2281:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-2)] = 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) 2284:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-3)] = 3*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) 2288:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)] = 0 2292:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-3)] = 4*Vons(PART)*sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) 2294:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-4)] = 5*Vons(PART)*sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) 2296:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-5)] = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) 2298:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-2)] = 4*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) 2300:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-3)] = 5*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) 2302:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-4)] = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) 2304:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-5)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) 2306:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-3)] = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) 2308:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-4)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) 2310:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-6)] = 9*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) 2312:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-2)] = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) 2314:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-4)] = 8*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) 2316:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-5)] = 9*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) 2318:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-6)] = 10*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) 2320:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-7)] = 11*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) 2322:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-3)] = 8*Vons(PART)*sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) 2324:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-5)] = 10*Vons(PART)*sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) 2326:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-7)] = 12*Vons(PART)*sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) 2328:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-8)] = 13*Vons(PART)*sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-9) 2330:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-1)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-2) 2332:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-4)] = 10*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) 2334:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-6)] = 12*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) 2336:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-7)] = 13*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) 2338:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-8)] = 14*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-9) 2590:(mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] = 0 2608:(mathH)/* for use in differentiations /% ∂[∂(t): K0] = 21/2*β^2*Rocs(POIo)^3*Q(PART)*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) 2620:(mathH)/* for use in differentiations /% ∂[∂(t): K2] = (-2)*λ(Vons(PART))*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) 2640:(mathH)/* ???? where is the section on this? /% ∂[∂(t): K1] = -3*β^2*Vons(PART)*Rocs(POIo)^2*Q(PART) * 0 2653:(mathH)/* when [t=0, RFp=RFo @t=0], maybe use only AFTER differentiations??? /% ∂[∂(t): K1(t=0)] = 0 2671:(mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% ∂[∂(t): K0(t=0)] = E0ods(POIo,t)*21/2*β^2*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-1) 2682:(mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% ∂[∂(t): K2(t=0)] = (-2)*λ(Vons(PART))*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIo(t),t=0)) 2704:(mathH)/* for use in differentiations /% ∂[∂(t): K_1st] = 21/2*β^2*Q(PART)*Rocs(POIo)^3*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) - 2*λ(Vons(PART))*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) 2736:(mathH)/* for use in differentiations /% ∂[∂(t): K_2nd] = + 210/8*β^4*Q(PART) *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) - λ(Vons(PART))*5*β^2*Q(PART)*Rocs(POIo) *Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) 2759:(mathH)/* when [t=0, RFp=RFo @t=0], use only AFTER differentiation!!! /% ∂[∂(t): K_1st(t=0)] = E0ods(POIo,t) *21/2*β^2*Vons(PART)*sin(Aθpc(POIo(t),t=0))^2*cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-1) - E0ods(POIo,t)*λ(Vons(PART)) *2 *Vons(PART) *cos(Aθpc(POIo(t),t=0))*Rpcs(POIo(t),t=0)^(-1) 2787:(mathH) ∂[∂(t): E0pds(POIp)] = 0 2807:(mathH) ∂[∂(t): E0ods(POIo,t)] = 2*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t)) 2814:(mathH) ∂[∂(t): E0ods(POIo,t)] = E0ods(POIo,t) *2*Vons(PART)*Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t)) 2927:(mathH)/* theoretical target - binomial series /% EIods(POIo,t,4th stage) = + E0pds(POIp) *{ 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 3/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + -1/16*β^6*sin(Aθpc(POIo(t),t=0))^6 + 3/128*β^8*sin(Aθpc(POIo(t),t=0))^8} - E0pds(POIp)*λ(Vons(PART)) *{1 + 3/2*β^2*sin(Aθpc(POIo(t),t=0))^2 + 3/8*β^4*sin(Aθpc(POIo(t),t=0))^4 + -1/16*β^6*sin(Aθpc(POIo(t),t=0))^6} 2933:(mathH) binomial series = 1 3/2 3/8 -1/16 3/128 -3/256 7/1024 12Oct2019 ETpds(POIp) = 1 3/2 21/8 35/8 455/64 ETpds(POIp)*λ(Vons(PART)) = 1 1 5/4 5/3 3004:(mathH) ∂[∂(t): E0ods(POIo,t)*sin(Aθpc(POIo(t),t))^a] = E0ods(POIo,t) *2*Vons(PART)*Rpcs(POIo(t),t)^(-1) *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^a + E0ods(POIo,t) *a *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a-1) 3016:(mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^1] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^1 + E0ods(POIo,t=0)*1*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^0 3018:(mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^2] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^2 + E0ods(POIo,t=0)*2*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^1 3020:(mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^3] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^3 + E0ods(POIo,t=0)*3*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^2 3022:(mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^4] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^4 + E0ods(POIo,t=0)*4*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^3 3024:(mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^5] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^5 + E0ods(POIo,t=0)*5*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^4 3026:(mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^6] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^6 + E0ods(POIo,t=0)*6*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^5 3028:(mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^8] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^8 + E0ods(POIo,t=0)*8*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^7 3096:(mathH)/* old-wrong!! /% ∂[∂(t): E0ods(POIo,t=0)*Rpcs(POIo(t),t)^(-β)*sin(Aθoc(POIo))^a] = + E0ods(POIo(t)=0)*(2+b)*Vons(PART) *Rpcs(POIo(t)=0)^(-1-b)*cos(AOtc(RFt))*sin(AOtc(RFt))^a + E0ods(POIo(t)=0)*a *Rpcs(POIo(t)=0)^(-b )*cos(AOtc(RFt))*sin(AOtc(RFt))^(a-1) 3209:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))] = sin(Aθpc(POIo(t),t=0))^1/1 3211:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^1] = sin(Aθpc(POIo(t),t=0))^2/2 3213:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^2] = sin(Aθpc(POIo(t),t=0))^3/3 3215:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^3] = sin(Aθpc(POIo(t),t=0))^4/4 3217:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^4] = sin(Aθpc(POIo(t),t=0))^5/5 3219:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^5] = sin(Aθpc(POIo(t),t=0))^6/6 3221:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^6] = sin(Aθpc(POIo(t),t=0))^7/7 3223:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^7] = sin(Aθpc(POIo(t),t=0))^8/8 3225:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^8] = sin(Aθpc(POIo(t),t=0))^9/9 3227:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^9] = sin(Aθpc(POIo(t),t=0))^10/10 3229:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^10] = sin(Aθpc(POIo(t),t=0))^11/11 3231:(mathH) ∫[∂(Aθpc),0.to.Aθpc(POIo(t),t=0): cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)))^11] = sin(Aθpc(POIo(t),t=0))^12/12 3257:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-4)] = 1/ 4/Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^1 3259:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t)) *Rpcs(POIo(t),t)^(-5)] = 1/ 5/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^1 3261:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-3)] = 1/ 4/Rocs(POIo)*Rpcs(POIo(t),t)^(-2)*sin(Aθpc(POIo(t),t))^2 3263:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-5)] = 1/ 6/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^2 3265:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-4)] = 1/ 6/Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^3 3267:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-5)] = 1/ 7/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^3 3269:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-4)] = 1/ 7/Rocs(POIo)*Rpcs(POIo(t),t)^(-3)*sin(Aθpc(POIo(t),t))^4 3271:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-6)] = 1/ 9/Rocs(POIo)*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^4 3273:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-7)] = 1/10/Rocs(POIo)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^4 3275:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-5)] = 1/ 9/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^5 3277:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-6)] = 1/10/Rocs(POIo)*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^5 3279:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-5)] = 1/10/Rocs(POIo)*Rpcs(POIo(t),t)^(-4)*sin(Aθpc(POIo(t),t))^6 3281:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-6)] = 1/11/Rocs(POIo)*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^6 3283:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-7)] = 1/12/Rocs(POIo)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^6 3285:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-6)] = 1/12/Rocs(POIo)*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^7 3287:(mathH) ∫[∂(Aθpc),0.to.Aθpcf: cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^7*Rpcs(POIo(t),t)^(-7)] = 1/14/Rocs(POIo)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^8 3500:(mathH)/* at time t when POIo and POIp(t) are the same point /% Rocv(POIo) = constant ≠ Rpcv(POIo(t),t) (mathH)/* at time t when POIo and POIp(t) are the same point /% Aθoc(POIo) = constant ≠ Aθpc(POIo(t),t) 3505:(mathH)/* at time t when POIo and POIp(t) are the same point /% Aφoc(POIo) = constant ≠ Aφpc(POIo(t),t) 3517:(mathH) Rocv(POIp(t),t) = Rpcv(POIp) + Vonv(PART)*t 3546:(mathH) Rocs(POIp(t),t) = { Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) 3563:(mathH) Rθ0ocs(POIp(t),t) = Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t 3583:(mathH) ROPI2ocs(POIp(t),t) = Rpcs(POIp)*sin(Aθpc(POIp)) 3603:(mathH) sin(Aθoc(POIp(t),t)) = Rpcs(POIp)*sin(Aθpc(POIp)) /{ Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) 3628:(mathH) cos(Aθoc(POIp(t),t)) = [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / {Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) 3675:(mathH) R_O0_ocs(POIo) = Rocs(POIo)*cos(Aθocs(POI)) 3697:(mathH) E0odv(POIo,t) = E0pdv(POIo(t),t) = Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) 3723:(mathH) E0odv(POIo,t) = E0pdv(POIo(t),t) = Q(PART)*Rpch(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } 3762:(mathH) E0ods(POIo,t) = |Q(PART)|/Rpcs(POIo(t),t)^2 3790:(mathH) E0ods(POIo,t) = |Q(PART)|/{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 } 3887:(mathH)/* Generalized Ampere's Law : /% BIodv(POIo,t) = Vonv(PART)/c X EOpdv(POIo(t),t) 3894:(mathH)/* (4-13) Generalized Ampere's Law : /% BTpdv(POIo(t),t) = Vonv(PART)/c X [E0pdv(POIo(t),t) + EIpdv(POIo(t),t)] 3943:(mathH) BTpdv(POIo(t),t) = Vons(PART)/c*sin(Aθpc(POIo(t),t))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[Q(PART)/Rpcs(POIo(t),t)^2 - EIpds(POIo(t),t)] 4033:(mathH) BTodv(POIo,t) = Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo))*Rodh(Vonv_X_Rpcv(POIo(t),t)) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) - EIods(POIo(t),t) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) ] 4081:(mathH) EIods(POIo,t) = EIpds(POIo(t),t) ≠ EIpds(POIp) = 0 4113:(mathH)/* Following Lucas p67h0.6 Eqn (4-13) & (4-41) /% E0pdv(POIo(t),t) + EIpdv(POIo(t),t) = ETodv(POIo,t) = E0odv(POIo,t) + EIodv(POIo,t) 4137:(mathH) E0pds(POIo(t),t) = Q(PART)/Rpcs(POIp)^2 4146:(mathH)/* seems wrong???? /% ETpds(POIo(t),t) = Q(PART)/Rpcs(POIp(t),t)^2 - EIpds(POIo(t),t) 4191:(mathH) ∂[∂(t): Rocv(POIo)] = 0 4194:(mathH) ∂[∂(t): Aθoc(POIo)] = 0 4197:(mathH) ∂[∂(t): Aφoc(POIo)] = 0 4215:(mathH) ∂[∂(t): Rocv(POIp(t),t) = Vonv(PART)] 4252:(mathH) ∂[∂(t): Rocs(POIp(t),t)] = Vons(PART)*cosAθoc(POIp(t),t) 4352:(mathH) ∂[∂(t): Rocs(POIp(t),t)] = {Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + Vons(PART)^2*t} / {Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + (Vons(PART)*t)^2 }^(1/2) 4458:(mathH) ∂[∂(t): sin(Aθoc(POIp(t),t))] = -Rpcs(POIp)*sin(Aθpc(POIp)) * Vons(PART) * [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ] / { [Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART)*t ]^2 + (Rpcs(POIp)*sin(Aθpc(POIp)))^2 }^(3/2) 4594:(mathH) ∂[∂(t): cosAθoc(POIp(t),t)] = Vons(PART) / {Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART) *t]^2}^(1/2) - { Rpcs(POIp)*cos(Aθpc(POIo(t),t)) + Vons(PART) *t} *{ + Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART) + [Vons(PART)^2*t] } /{Rpcs(POIp)^2 + 2*Rpcs(POIp)*cos(Aθpc(POIo(t),t))*Vons(PART)*t + [Vons(PART) *t]^2}^(3/2) 4625:(mathH) ∂[∂(t): Rocs(POIp(t),t)*sin(Aθoc(POIp(t),t))] = ∂[∂(t): ROPI2ods(POIp(t),t)] = 0 4731:(mathH) ∂[∂(t): Rocs(POIp(t),t)*cosAθoc(POIp(t),t)] = ∂[∂(t): Rθ0ocs(POIp(t),t)] = Vons(PART) 4976:(mathH) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)] 5065:(mathH) ∂[∂(t): E0pdv(POIo(t),t)] = Q(PART)*Vons(PART) * {Rocs(POIo)*sin(Aθoc(POIo))*RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t)] } / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 5106:(mathH) ∂[∂(t): E0pds(POIo(t),t)] = ∂[∂(t): |E0pdv(POIo(t),t)|] 5255:(mathH) ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 5351:(mathH) ∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)] = 2*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 5408:(mathH) ∂[∂(t): E0pds(POIo(t),t)] = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp(t),t)^3 5466:(mathH) ∂[∂(t): E0ods(POIo,t)] = 2*|Q(PART)|*Vons(PART) *{ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 5504:(mathH) ∂[∂(t): E0pdv(POIo(t),t)] = E0pds(POIo(t),t)*∂[∂(t): Aθpc(POIo(t),t)]*RDEpdh(POIo(t),∂(t)) + ∂[∂(t): |E0pdv(POIo(t),t)|] *Rpch(POIo(t),t) 5630:(mathH) ∂[∂(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(PART)/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))} 5789:(mathH)/* Note that I keep the direction vector Rodh(Vonv_X_Rpcv(POIo)) for simplicity!! /% ∂[∂(t): BTodv(POIo,t)] = Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo))* { - Rocs(POIo)*sin(Aθoc(POIo))*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART) *t ] /[ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + (Vons(PART)*t)^2 ]^(3/2) *[ Q(PART) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2)^2 - EIods(POIo(t),t) ] + sin(Aθpc(POIo(t),t)) *[ Q(PART)*-2*Vons(PART) *[ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 - ∂[∂(t): EIods(POIo(t),t)] ] } 5918:(mathH)/* (4-31) /% EI_LENZods(POIo,t)|(Aθpc(POIo(t),t=0))*Rpch(POIo(t),t) = -λ(Vons(PART))*E0ods(POIo)*Rpch(POIo(t),t) 5972:(mathH) EI_LENZpdv(POIo(t),t) = -λ(Vons(PART))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) 5992:(mathH) EI_LENZodv(POIo(t),t) = -λ(Vons(PART))*Q(PART)*Rpch(POIo(t),t) /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} 6021:(mathH) ET_LENZpdv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)/Rpcs(POIo(t),t)^2*Rpch(POIo(t),t) 6058:(mathH) ET_LENZodv(POIo(t),t) = (1 - λ(Vons(PART)))*Q(PART)*Rpch(POIo(t),t) /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} 6105:(mathH) EI_LENZpds(POIo(t),t) = λ(Vons(PART))*|Q(PART)|/Rpcs(POIo(t),t)^2 6124:(mathH) EI_LENZods(POIo(t),t) = EI_LENZpds(POIo(t),t) = λ(Vons(PART))*|Q(PART)| /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} 6157:(mathH) ET_LENZpds(POIo(t),t) = |(1 - λ(Vons(PART)))|*|Q(PART)|/Rpcs(POIo(t),t)^2 6181:(mathH) ET_LENZods(POIo(t),t) = |ET_LENZodv(POIo(t),t)| = |(1 - λ(Vons(PART)))|*|Q(PART)| /|{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}| 6212:(mathH) BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t) = (1-λ(Vons(PART)))*Q(PART)*Vons(PART)/c*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^2*Rodh(Vonv_X_Rpcv(POIo(t),t)) 6243:(mathH)/* is there a mistake here? /% BT_LENZpdv(POIo(t),t) = BT_LENZodv(POIo(t),t) = [1-λ(Vons(PART))]*Q(PART)*Vons(PART)/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) *Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(3/2) 6260:(mathH) ∂[∂(t): EIodv(POIo)] = ∂[∂(t): EI_LENZpdv(POIo(t),t)] = -λ(Vons(PART))*∂[∂(t): E0odv(POIo)] 6280:(mathH)/* where : λ(Vons(PART)) is a positive constant, Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t), RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /% ∂[∂(t): EI_LENZpdv(POIo(t),t)] = -λ(Vons(PART))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),∂(t)) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)] 6329:(mathH)/* where : λ(Vons(PART)) is a positive constant, Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t), RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /% ∂[∂(t): EI_LENZodv(POIo(t),t)] = -λ(Vons(PART))*Q(PART)*Vons(PART) *[ Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),∂(t)) + 2*[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rpch(POIo(t),t) ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 6381:(mathH) ∂[∂(t): EI_LENZpds(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*cos(Aθpc(POIo(t),t))*Vons(PART)/Rpcs(POIo(t),t)^3 6415:(mathH) ∂[∂(t): EI_LENZods(POIo(t),t)] = 2*λ(Vons(PART))*|Q(PART)|*Vons(PART) *[ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t ]^2 }^2 6453:(mathH) ∂[∂(t): E0ods(POIo,t)] = 2*λ(Vons(PART))*Q(PART)*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 6535:(mathH)/* Note that RDEpdh(POIo(t),t) & Rpch(POIo(t),t) are NOT the same unit vector! /% ∂[∂(t): ET_LENZpdv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *{sin(Aθpc(POIo(t),t))*RDEpdh(POIo(t),t) + 2*cos(Aθpc(POIo(t),t))*Rpch(POIo(t),t)} 6669:(mathH)/* where RDEpdh(POIo(t),t) is anchored at end of Rpch(POIo(t),t) and is at angle Aθpc(POIo(t),t) + PI/2, ie perpendicular to Rpch(POIo(t),t), angle Aφpc(POIo(t),t) doesn't change /% ∂[∂(t): ET_LENZodv(POIo(t),t)] = (1 - λ(Vons(PART)))*Q(PART)*Vons(PART) * { Rocs(POIo)*sin(Aθoc(POIo)) *RDEpdh(POIo(t),t) - 2*{ - Rocs(POIo)*cos(Aθoc(POIo)) + Vons(PART)*t} *Rpch(POIo(t),t) } / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo)) *Vons(PART)*t + [Vons(PART)*t]^2}^2 6739:(mathH) ∂[∂(t): BT_LENZpdv(POIo(t),t)] = 3*(1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rodh(Vonv_X_Rpcv(POIo(t),t)) 6916:(mathH)/* !!!Note : problem with |Q(PART)| versus Q(PART) ignored here!!! /% BT_LENZodv(POIo(t),t)] = 3*(1 - λ(Vons(PART)))*|Q(PART)|*Vons(PART)^2/c*Rodh(Vonv_X_Rpcv(POIo(t),t)) * Rocs(POIo)*sin(Aθoc(POIo))* [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t ] / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) 7076:(mathH) ETodv(POIo,t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*E0odv 7127:(mathH)/* where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2), β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 /% ET_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)/Rpcs(POIp)^2*Rpch(POIo(t),t) 7166:(mathH)/* where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2), β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 /% ET_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Rpch(POIo(t),t) / {Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(1/2)^2 7218:(mathH)/* where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2), β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 /% ET_BARNpds(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|/Rpcs(POIp)^2 7257:(mathH)/* where : f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) = (1 - β^2)/{1 - [β*sin(Aθpd(POIo(t),t))]^2}^(3/2), β = Vons(PART)/c = (λ(Vons(PART)))^(1/2) in the context of Chapter 4 /% ET_BARNods(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)| /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2} 7332:(mathH)/* where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for /% BTI_BARNpdv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*sin(Aθpc(POIo(t),t))/Rpcs(POIp)^2*Rodh(Vonv_X_Rpcv(POIo(t),t)) 7410:(mathH)/* where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for /% BT_BARNodv(POIo(t),t) = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo)) *Rodh(Vonv_X_Rpcv(POIo(t),t)) /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(3/2) 7466:(mathH) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) 7516:(mathH) ∂[∂(t): f_BARNES(Vonv(PART),Aθpd(POIo(t),t))] = 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2}*f_BARNES(Vonv(PART),Aθpd(POIo(t),t)) *Vons(PART)*Rocs(POIo)*sin(Aθoc(POIo)) *[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] /{Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2}^(3/2) 7643:(mathH)/* where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),∂(t)) is at angle Aθpd(RDEpdh(POIo(t),∂(t))) = Aθpc(POIo(t),t) + PI/2 /% ∂[∂(t): ET_BARNpdv(POIo(t),t)] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART)/Rpcs(POIo(t),t)^3 *[ Rpch(POIo(t),t) *cos(Aθpc(POIo(t),t))*{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2}*sin(Aθpc(POIo(t),t))^2 + 2 } + RDEpdh(POIo(t),∂(t)) *sin(Aθpc(POIo(t),t)) ] 7797:(mathH)/* where : Rpch(POIo(t),t) is at angle Aθpc(POIo(t),t) RDEpdh(POIo(t),dt) is at angle Aθpd(RDEpdh(POIo(t),dt)) = Aθpc(POIo(t),t) + PI/2 /% ∂[∂(t): ET_BARNodv(POIo(t),t)] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*Q(PART)*Vons(PART) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^2 *{ Rpch(POIo(t),t) *[Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *[ 3*beta^2/{1 - [beta*sin(Aθpd(POIo(t),t))]^2} * Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + 2 ] + RDEpdh(POIo(t),dt) *Rocs(POIo)*sin(Aθoc(POIo)) } 7946:(mathH)/* where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for /% ∂[∂(t): BT_BARNpdv(POIo(t),t)] = 3*f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIp)^3 *{ { [β*sin(Aθpc(POIo(t),t))]^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} + 1} *Rodh(Vonv_X_Rpcv(POIo(t),t)) 8065:(mathH)/* where : Aθpd(Vonv(PART),Rpch(POIo(t),t)) = Aθpc(POIo(t),t) is the Aθ (theta) angle between the Vonv(PART) & E vectors Aφpd(Vonv_X_Rpcv(POIo(t),t)) = APod(Vonv_X_Rpcv(POIo(t),t)) is the Aφ (phi) direction of the B = Vonv(PART) X E vector (perpendicular to the Vonv(PART) X E plane) Rodh(Vonv_X_Rpcv(POIo(t),t)) is the unit vector in the direction of Aφpd(Vonv_X_Rpcv(POIo(t),t)), anchored at (POIo) Aφpd(Vonv_X_Rpcv(POIo(t),t)) & Rodh(Vonv_X_Rpcv(POIo(t),t)) are CONSTANT for /% ∂[∂(t): BT_BARNodv(POIo(t),t)] = f_BARNES(Vonv(PART),Aθpd(POIo(t),t))*|Q(PART)|*Vons(PART)^2/c*Rocs(POIo)*sin(Aθoc(POIo)) * [ Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t] *Rodh(Vonv_X_Rpcv(POIo(t),t)) / { Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(5/2) *{ 3*β^2/{1 - [β*sin(Aθpd(POIo(t),t))]^2} *Rocs(POIo)*sin(Aθoc(POIo)) /{ Rocs(POIo)^2 - 2*Rocs(POIo)*cos(Aθoc(POIo))*Vons(PART)*t + [Vons(PART)*t]^2 }^(1/2) + 3 } 8176:(mathH) ∂[∂(Aθpc): Rpcs(POIo(t),t)] = (-1)*Rocs(POIo)*sin(Aθoc(POIo))*sin(Aθpc(POIo(t),t))^(-2)*cos(Aθpc(POIo(t),t)