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

/*	using  /% 
1)		EIods(POIo,t=0,2nd stage) 
		=	+	K_1st 
			+	β^1*Rocs(POIo)^2
1b)				*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t)) 
						*{
							+ ∂[∂(t): K_1st]
							+ ∂[∂(t): f_sphereCapSurf(EIods(POIo,t)) ]
						 }
					  ]
			+ f_sphereCapSurf{∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage))]}

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

/*	Substitute for (1b) from "Bottom-up (6b)" results below 
		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)) ]
						- 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)) ]
				  }
				 *{	+ 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)) ]
						- 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{∂[∂(t): f_sphereCapSurf(EIods(POIo,t,0th stage))]}

7)		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)) ]
						- 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)  /%
2b)				*∫[∂(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 
							- λ(Vons(PART))*Q(PART)					/Rpcs(POIo(t),t)^2
						  ]
					  ] 

/*	Distribute the derivative  /%	
3b)				*∫[∂(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): 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):
/%
4b)				*∫[∂(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 ]
							- λ(Vons(PART))*Q(PART)					*∂[∂(t): Rpcs(POIo(t),t)^(-2) ]
						 }
					  ] 

/*	using   /%
2059:(mathL)  ∂[∂(t): Rpcs(POIo(t),t)^(-2)] = 2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3)
2069:(mathL)  ∂[∂(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)

/*	substitute into (4b)  /%
5b)				*∫[∂(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)
							- λ(Vons(PART))*Q(PART)					*2*Vons(PART)*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3)
						 }
					  ] 
/* [collect, rearrange] terms
/%					*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): 1/c/Rocs(POIo)*sin(Aθpc(POIo(t),t))
						*{	+ 3/2*7*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)
						 }
					  ] 
/* 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):
/%
			*{	+ 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 ]
				- 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)) ]
			 }
			*{	+ 21/2*β^2*Rocs(POIo)^2	*Q(PART)*Vons(PART)	*1/c				*Rpcs(POIo(t),t)^(-6)	*∫[∂(Aθpc),0 to Aθpc(POIo(t),t=0): sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t)) ]
				- 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  /%
6b)		*{	+ 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)) ]
				- 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 

	EIods(POIo,t,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): sin(Aθpc(POIo(t),t))^3	*cos(Aθpc(POIo(t),t)) ]
					- 2	*λ(Vons(PART))	*β	 *Q(PART)/Rocs(POIo)		*Rpcs(POIo(t),t)^(-3)*∫[∂(Aθpc),0 to Aθpc(POIo(t),t): sin(Aθpc(POIo(t),t))		*cos(Aθpc(POIo(t),t)) ]
			 }
		+ f_sphereCapSurf{f_sphereCapSurf(EIods(POIo,t,0th stage))}

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