/*	Restating iterative solution : 
(mathL)/*	generative form   /%
	EIods(POIo,t,2nd stage) =  K_1st + f_sphereCapSurf(EIods(POIo,t,1st stage))
(endMath)

/*	using  /% 
1)	
????:(mathL)
	EIods(POIo,t=0,2nd stage)  
	=	+	K_1st 
		+	β^1*Rocs(POIo)^2
1a)			*∫[∂(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)^(-5)	*sin(Aθpc(POIo(t),t))^2 ]
						+ ∂[∂(t): 3	 *β^2			*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t)) - 1) ] 
						- ∂[∂(t): λ(Vons(PART))	*Q(PART)		 			*Rpcs(POIo(t),t)^(-2) ]  
						 }
					  ]
		+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}

??????????????????????????????????

/*	Substitute for (1b) from "Bottom-up (6b)" results below 
2)		EIods(POIo,t=0,2nd stage) 
		=	+	K_1st 
			+	β*Rocs(POIo)^2
1g)		*{	+ 21/2					*β^3*Q(PART)*Rocs(POIo)^2	*Rpcs(POIo(t),t)^(-6)	*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t)) ]
				+ 3						*β^3*Q(PART)*Rocs(POIo)		*Rpcs(POIo(t),t)^(-5)	*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *(cos(Aθpc(POIo(t),t)) - 1)
				- 2	*λ(Vons(PART))	*β	 *Q(PART)/Rocs(POIo)		*Rpcs(POIo(t),t)^(-3)	*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))	*cos(Aθpc(POIo(t),t)) ]
			 }
			+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}


/***********************************
>>>>>> Bottom-up (2b1)

/*	Looking at (1b)  /%
1b)			*∫[∂(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)^(-5)	*sin(Aθpc(POIo(t),t=0))^2 ]
						+ ∂[∂(t): 3	 *β^2			*Q(PART)*Rocs(POIo)^2*Rpcs(POIo(t),t)^(-5)*Vons(PART)*t*(cos(Aθpc(POIo(t),t)) - 1) ] 
						- ∂[∂(t): λ(Vons(PART))	*Q(PART)		 			*Rpcs(POIo(t),t)^(-2) ]  
						 }
					  ]
/*	"Percolate" constant terms up through [derivatives, integrals] in Chapter 4 :  
	see 		"Howell - Background math for Lucas Universal Force, Chapter 4.txt"
	section	"Constants of [derivative, integration] expressions"
	section	"Rpcs(POIo(t),t)" is a constant for integrals like "∫{∂(Aθtc),0 to Aθoc(POIp(t),t=0):" ?"
		[c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)] 
		[Rpcs(POIo(t),t)]
			- is NOT a constant wrt ∂[∂(t):
			- IS		a constant wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
/%
1c)			*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
					*{	+ 3/2	*β^2				*Q(PART)*Rocs(POIo)^3*∂[∂(t): Rpcs(POIo(t),t)^(-5)	*sin(Aθpc(POIo(t),t))^2 ]
						+ 3	*β^2*Vons(PART)*Q(PART)*Rocs(POIo)^2*∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t))] - 1) ] 
						- λ(Vons(PART))		*Q(PART)					*∂[∂(t): Rpcs(POIo(t),t)^(-2) ]
					 }
				  ] 
/*	using   /%
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)
????:(mathH)  
	∂[∂(t):	Rpcs(POIo(t),t=0)^(-5)*(t=0)*(cos(Aθpc(POIo(t),t)) - 1)]
	=	+	Rpcs(POIo(t),t=0)^(-5)*(cos(Aθpc(POIo(t),t)) - 1)
2059:(mathL)  ∂[∂(t): Rpcs(POIo(t),t)^(-2)] = 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3)

/*	substitute into (1b)  /%
1d)			*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
					*{	+ 3/2	*β^2				*Q(PART)*Rocs(POIo)^3	*7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6)
						+ 3	*β^2*Vons(PART)*Q(PART)*Rocs(POIo)^2	*Rpcs(POIo(t),t)^(-5)*(cos(Aθpc(POIo(t),t)) - 1)
						- λ(Vons(PART))		*Q(PART)						*2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3)
					 }
				  ] 
/* [collect, rearrange] terms  /%
1e)				*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
					*{	+ 3/2*7*β^2			*Vons(PART)*Q(PART)*Rocs(POIo)^3	*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))
						+ 3	 *β^2			*Vons(PART)*Q(PART)*Rocs(POIo)^2	*Rpcs(POIo(t),t)^(-5)*(cos(Aθpc(POIo(t),t)) - 1)
						- 2*λ(Vons(PART))	*Vons(PART)*Q(PART)					*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t))
					 }
				  ] 
/* distribute the integral 
/*	"Percolate" constant terms up through [derivatives, integrals] in Chapter 4 :  
	see 		"Howell - Background math for Lucas Universal Force, Chapter 4.txt"
	section	"Constants of [derivative, integration] expressions"
	section	"Rpcs(POIo(t),t)" is a constant for integrals like "∫{∂(Aθtc),0 to Aθoc(POIp(t),t=0):" ?"
		[c, β, λ(Vons(PART)), Q(PART), Rocs(POIo), Vons(PART)] 
		[Rpcs(POIo(t),t)]
			- is NOT a constant wrt ∂[∂(t):
			- IS		a constant wrt ∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0):
/%
1f)			*{	+ 21/2*β^2*Rocs(POIo)^3	*Q(PART)*Vons(PART)		*1/c/Rocs(POIo)*Rpcs(POIo(t),t)^(-6)	*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2 ]
				+ 3	*β^2*Vons(PART)	*Q(PART)*Rocs(POIo)^2	*1/c/Rocs(POIo)*Rpcs(POIo(t),t)^(-5)	*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*(cos(Aθpc(POIo(t),t)) - 1)
				- 2	*λ(Vons(PART))		*Q(PART)*Vons(PART)		*1/c/Rocs(POIo)*Rpcs(POIo(t),t)^(-3)	*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t)) ]
			 }
/* set Vons(PART)/c = β, collect terms  /%
1g)		*{	+ 21/2					*β^3*Q(PART)*Rocs(POIo)^2	*Rpcs(POIo(t),t)^(-6)	*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t)) ]
				+ 3						*β^3*Q(PART)*Rocs(POIo)		*Rpcs(POIo(t),t)^(-5)	*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t)) *(cos(Aθpc(POIo(t),t)) - 1)
				- 2	*λ(Vons(PART))	*β	 *Q(PART)/Rocs(POIo)		*Rpcs(POIo(t),t)^(-3)	*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))	*cos(Aθpc(POIo(t),t)) ]
			 }

/*	insert this result in "Top-down" section above  


/*++++++++++++++++++++++++++++++++++++++
/*add_eqn  "Likely_Lucas_error_or_omission    
   04_34
   F therefore E balance - taking partial derivatives wrt time  

/$L		Eis(r - v*t,t) APPLY |t=0 TO EACH TERM 
      =+ 3/2*β^2*q*rs^3/|r - v*t|^5*sin(θ´)^2 - λ(v)*q*rs/|r - v*t|^3  
      +      β    *rs^2*∫[∂(θ´),0 to θ´f: sin(θ´)*
     (+ 15/2*β^3*q*rs^4/|r - v*t|^7*sin(θ´)^2*cos(θ´)  
      -  3  *β  *q*rs^2/|r - v*t|^5*λ(v)     *cos(θ´)   
     ))   
      +      β^2  *rs^4*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): 
                        ∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): Eis(r - v*t,t)])])   

/$H		Eis(r - v*t,t) APPLY |t=0 TO EACH TERM  
     =+  3/2*β^2*q*rs^3/|r - v*t|^5*sin(θ´)^2 - λ(v)*q*rs/|r - v*t|^3  
      +      β    *rs^2*∫[∂(θ´),0 to θ´f: sin(θ´)*
     (+ 15/2*β^3*q*rs^3/|r - v*t|^7*sin(θ´)^2*cos(θ´)  
      -  3  *β  *q*rs^1/|r - v*t|^5*λ(v)     *cos(θ´)  
     ))  
      +      β^2  *rs^4*∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): 
         	            ∫[∂(θ´),0 to θ´f: 1/rs/c*sin(θ´)*∂[∂(t): Eis(r - v*t,t)])])  

/%H 
(mathL)/* 4-34  /%
	EIods(POIo,t=0,2nd stage) 
	=	+	K_1st 
		+	β*Rocs(POIo)^2
			*{	+ 21/2					*β^3*Q(PART)*Rocs(POIo)^2	*Rpcs(POIo(t),t)^(-6)	*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3	 *cos(Aθpc(POIo(t),t)) ]
				+ 3						*β^3*Q(PART)*Rocs(POIo)		*Rpcs(POIo(t),t)^(-5)	*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))		*(cos(Aθpc(POIo(t),t)) - 1)
				- 2	*λ(Vons(PART))	*β	 *Q(PART)/Rocs(POIo)		*Rpcs(POIo(t),t)^(-3)	*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))		 *cos(Aθpc(POIo(t),t)) ]
			 }
		+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))]}
(endMath)

/*	iterative form, see "Howell - independent math for Lucas Universal Force, Chapter 4.txt"  /% 
1049:(mathH)  EIods(POIo,t=0,ith stage) =  K_1st + f_sphereCapSurf(EIods(POIo,t=0,(i-1) stage)))}
1046:(mathH)  EIods(POIo,t=0,2nd stage) =  K_1st + f_sphereCapSurf(EIods(POIo,t=0,1st stage)

/*	for symbols & HIGHLY restrictive conditions : 
	see "textIn - relativistic factor, intermediate symbols, restrictive conditions.txt"  
/%  

/*	for a detailed comparison, see above "Comparison of Lucas (4-34) and previous (4-34) derivation to (5)"