/*	USUALLY, I WORK FROM LUCAS'S RESULTS RATHER THAN MY OWN, BUT IN THIS CASE I WILL START WITH MINE
	22Aug2019 Howell's version of (4-32) :  
/%
(mathL)/*	generative form  /%
	EIods(POIo,t,2nd stage) =  K_1st + f_sphereCapSurf(EIods(POIo,t,1st stage))
(endMath)



/*  1. So what is the next step?  
   It is interesting to directly compare (4-32c), as labelled (a), with (4-30).  
	(4-32) does not have the same integral that was replaced in (4-30), so the next target appears to be the integral term in the 2nd expression on the RHS : 
/% 
2)		∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*∂[∂(t): EIods(POIo,t)]*sin(Aθpc(POIo(t),t)) ] 
/*	But all that I really need to do is to take the derivative of (1) and directly put that into the integral in (1)
	Taking the partial derivative of EIods(POIo,t) :  
/%
3) 		 ∂[∂(t): EIods(POIo,t,1st stage)]  
		= + ∂[∂(t): 3/2*β^2			*Q(PART)*Rocs(POIo)^3	*Rpcs(POIo(t),t=0)^(-5)					* sin(Aθpc(POIo(t),t=0))^2 ]
		  + ∂[∂(t): 3	*β^2			*Q(PART)*Rocs(POIo)^2	*Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t=0)) - 1) ] 
		  - ∂[∂(t): λ(Vons(PART))	*Q(PART)		 				*Rpcs(POIo(t),t=0)^(-2) ]  
		  + ∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage)) ]

/*  Putting (3) into (1) Yields : /%
4)			EIods(POIo,t=0)  
			=	+	K0 + K1 + K2 
				+	β^1*Rocs(POIo)^2
						*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t)) 
							*{
								+ ∂[∂(t): 3/2*β^2			*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)					* sin(Aθpc(POIo(t),t=0))^2 ]
								+ ∂[∂(t): 3	 *β^2			*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t=0)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t=0)) - 1) ] 
								- ∂[∂(t): λ(Vons(PART))	*Q(PART)		 			*Rpcs(POIo(t),t=0)^(-2) ]  
								+ ∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage)) ]
							 }
						  ]
/*	Lucas instructs iterative substitutions for Ei(ro - vo*t,t) at t=0 implicitly, and dropping v*t*(cosO - 1) terms as we go.  
	But why is it still in Equation (4-32)??
/*	using  /% 
	
	t=0
	∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] =  0
2590:(mathH)  ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)]  =  0  
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)

/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"
/*	Also, as the f_sphereCapSurf(EIods(POIo,t)) term is dropped after integration, it is convenient to show it separately
	This also makes the meaning of the 
/%
5)		EIods(POIo,t=0,2nd stage)  
		=	+	K_1st 
			+	β^1*Rocs(POIo)^2
					*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t)) 
						*{
							+ ∂[∂(t): 3/2*β^2			*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)	 *sin(Aθpc(POIo(t),t=0))^2 ]
							- ∂[∂(t): λ(Vons(PART))	*Q(PART)		 			*Rpcs(POIo(t),t=0)^(-2) ]  
						 }
					  ]
			+ f_sphereCapSurf{∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage))]}

/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"

/*	|--???--> (4-33) I DON'T GET THIS! : 	E0ods(POIo,t=0) = Q(PART)*Rocs(POIo)/Rpcs(POIo(t),t=0)^3 
 It appears that, from (4-8) Bi(r,v,t), Lucas has replaced E0(ro - vo*t,t) in the last term with 
	The Grassman form of the generalized Ampere force law is based on derivations in Appendix A (eq (A19).  
	(4-08) is the derivation of (4-01) from the Grassman/Biot-Savart form of Amperes Law
 	This is derived in Appendix A...
/$		q/c*(vr´)/rs'^3  =  (v/c)E0(r',t')   
/*	reference : Jackson 1999 p?? Eqn ?? (I lost the reference location, cant find!!
	such that (in Gaussian coordinates?)
	This does NOT follow! : 
/$		E0(r,t) = q*r´/r´s^3 = q*r´/|r - v*t|^3  
/*	BUT - in (4-33), Lucas has r rather than r' in numerator, WHICH SEEMS WRONG :
/$		E0(r,t) = q*r /r´s^3 = q*r /|r - v*t|^3  
/*	translate reference frame : 
/%		E0ods(POIo,t=0) 
		=  Q(PART)*Rocs(POIo)/Rpcs(POIo(t),t=0)^3 
/*	<--???--|


/*++++++++++++++++++++++++++++++++++++++
/*add_eqn  question with respect to E0ods(POIo,t=0) expression!  
   04_33 22Aug2019 start revision, 27Aug2019 finished revision  
   F therefore E balance - iterations on (4-32)  
 
/$L	Eis(r - v*t,t) APPLY |t=0 TO EACH TERM  
		=   K0 + K2  
		  + β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K0 + K2])  
		  + β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K3])  

/*	03Sep2019  This is old!
/$H	Eis(r - v*t,t) APPLY |t=0 TO EACH TERM   
		=   K0 + K2   
		  + β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K0 + K2])   
		  + β*rs^2*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): K3]) 

/*	OK - works great by using a blend of Lucas & Howell expressions for (4-32). This assumes a Lucas typo in 4-30, dropping a power of r  
	EXPLAIN : 
		Lucas states p71h0.25 that the v*t*(cosO - 1) are dropped, Presumably, at t=0 cosθ = 1, so (cosO - 1)|t=0 = 0.

/*	Result 14Sep2019 - Compact form /% 
	EIods(POIo,t=0,2nd stage)  
	=	+	K_1st 
		+	β^1*Rocs(POIo)^2
				*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t)) 
					*{
						+ ∂[∂(t): 3/2*β^2			*Q(PART)*Rocs(POIo)^3*Rpcs(POIo(t),t=0)^(-5)	*sin(Aθpc(POIo(t),t=0))^2 ]
						- ∂[∂(t): λ(Vons(PART))	*Q(PART)		 			*Rpcs(POIo(t),t=0)^(-2) ]  
						 }
					  ]
		+ f_sphereCapSurf{∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage))]}

/*_file_insert_path "$d_Lucas""relativistic factor, intermediate symbols.txt"