/* Reference frame notations for Chapter 4 (constant Vonv(PART)) - most commonly used forms in (4-30) to (4-37) : 
/* constants    :  c,Q,Vons,beta,lambda(v)=beta^2
	particle RFp :  t  , Rocs(POIp(t),t),Rpcs(POIp)			,Aθoc(POIp(t),t),Aθpc(POIp),EIpds(POIp)  ,[EI,BX] =0 in RFp
	observer RFo :  t  , Rocs(POIo)		,Rpcs(POIo(t),t)  ,Aθoc(POIo)		 ,Aθpc(POIo(t),t),EIods(POIo)  ,BIods(POIo)
	frozen   RFt :  t=0, Rocs(POIo)		,Rpcs(POIo(t),t), t=0),Aθtc(RFt)	 ,EItds(notused),BItds(notused) 
/* POIo context : 
integration  ∫[∂(Aθpc),0.to.Aθpc(POIo(t), t=0):  expressions with [Rocs(POIo),Rpcs(POIo(t), t=0),Aθtc(RFt)]} 
	result gives																[Rocs(POIo),Rpcs(POIo(t),t),Aθoc(POIo),Aθpc(POIo(t),t)]  } 

/* For integrals in next step : 
 derivative ∂[∂(t): expressions with [Rocs(POIo),Rpcs(POIo(t),t)    ,Aθoc(POIo),Aθpc(POIo(t),t),E0ods(POIo,t)]
 result gives								 [Rocs(POIo),Rpcs(POIo(t),t=0),Aθtc(RFo) ,Aθtc(RFt)		  ,E0ods(POIo,t=0)] 

/* General formulae for Chapter 4 
/$	dp[dt: E0ods(POIo,t=0)*Rpcs(POIo(t),t)^( - b)*sin(Aθoc(POIo))^a]
=  dp[dt: E0ods(POIo,t=0)] *Rpcs(POIo(t),t)^(-b)*sin(Aθoc(POIo))^a   +  E0ods(POIo,t=0)  *dp[dt: Rpcs(POIo(t),t)^( - b)*sin(Aθoc(POIo))^a]

/*From "Derivatives & Integrals adapted to Chapter 4 - Summary" above  
/%	∂[∂(t): Rpcs(POIo(t),t)^( - β)*sin(Aθpc(POIo(t),t))^a]	=    β*Vons(PART)*Rpcs(POIo(t),t)^(-β-1)*cos(Aθtc(POIp(t),t=0))*sin(Aθtc(RFt))^(a  ) 
	  +  a     *Rpcs(POIo(t),t)^(-β  )*cos(Aθtc(POIp(t),t=0))*sin(Aθtc(RFt))^(a-1)
/* From "Howell - Background math for Lucas Universal Force, Chapter 4.odt" 
/%	section "∂[∂(t): E0ods(POIo,t)] = ∂[∂(t): E0pds(POIo(t),t)]  - proper E0odv(POIo,t) vector approach" 
13*)  ∂[∂(t): E0ods(POIo,t)] 
      = 2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3
      = E0ods(POIo,t)*2*Vons(PART)*Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t))
/* Put into RFt form to be fed into integral
/*	∂[∂(t):  E0ods(POIo,t)] = E0ods(POIo(t)=0)*2*Vons(PART)*Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t))
/*Click to see  

Therefore : 
/%   ∂[∂(t): E0ods(POIo,t=0)*Rpcs(POIo(t),t)^( - β)*sin(Aθoc(POIo))^a]
=  E0ods(POIo,t=0)*
   (+ 2  *Vons(PART)*Rpcs(POIo(t),t)^(-1  )*cos(Aθtc(POIp(t),t=0)) *Rpcs(POIo(t),t)^(-β)*sin(Aθtc(RFt))^a   
    + β  *Vons(PART)*Rpcs(POIo(t),t)^(-β-1)*cos(Aθtc(POIp(t),t=0))*sin(Aθtc(RFt))^(a  ) 
    + a       *Rpcs(POIo(t),t)^(-β  )*cos(Aθtc(POIp(t),t=0))*sin(Aθtc(RFt))^(a-1)
   ()
=  E0ods(POIo,t=0)*
   (+ 2  *Vons(PART)*Rpcs(POIo(t),t)^(-1  )*cos(Aθtc(POIp(t),t=0)) *Rpcs(POIo(t),t)^(-β)*sin(Aθtc(RFt))^a   
    + β  *Vons(PART)*Rpcs(POIo(t),t)^(-β-1)*cos(Aθtc(POIp(t),t=0))*sin(Aθtc(RFt))^(a  ) 
    + a       *Rpcs(POIo(t),t)^(-β  )*cos(Aθtc(POIp(t),t=0))*sin(Aθtc(RFt))^(a-1)
   ()

/* General formulae for Chapter 4 
   ∂[∂(t): E0ods(POIo(t)=0)  *Rpcs(POIo(t),t)^(-b)*sin(Aθoc(POIo))^a]
=  E0ods(POIo(t)=0)*
   (+ 2  *Vons(PART)*Rpcs(POIo(t),t)^(-1-b)*cos(AOtc(RFt))*sin(AOtc(RFt))^a   
    + b  *Vons(PART)*Rpcs(POIo(t),t)^(-b-1)*cos(AOtc(RFt))*sin(AOtc(RFt))^a 
    + a       *Rpcs(POIo(t),t)^(-b  )*cos(AOtc(RFt))*sin(AOtc(RFt))^(a-1)
   ()
=  (+ 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)

(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)
(endMath)


/*-----+
/* Looking at "∂[∂(t): E0ods(POIo(t)=0)*Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4]"  

   ∂[∂(t): E0ods(POIo(t)=0)  *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4]
=  ∂[∂(t): E0ods(POIo(t)=0)] *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4   +  E0ods(POIo(t)=0)*∂[∂(t): Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4]

From "Derivatives & Integrals adapted to Chapter 4 - Summary" above  
∂[∂(t): Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4 ]	=    1*Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4 
	  +  4     *Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3
/*	using  /%
2462:(mathH)  ∂[∂(t): E0ods(POIo,t)] = 2*Q(PART)*Vons(PART)*Rpcs(POIo(t),t)^(-3)*cos(Aθpc(POIo(t),t))

   ∂[∂(t): E0ods(POIo(t)=0)			  *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4]
=  ∂[∂(t): E0ods(POIo(t)=0)]			  *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4 
	+		  E0ods(POIo(t)=0)	*∂[∂(t): Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4]
=   2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4
	+  E0ods(POIo(t)=0) 
			*[		1*Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4 
				+	4     *Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3
	       ]
(1)
=   2*|Q(PART)|*Vons(PART)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)^3 *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4 
	+	E0ods(POIo(t)=0) 
			*[		1*Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4 
			  +	4     *Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3
			 ]

/* From "E0ods(POIo,t) = E0pds(POIo(t),t)"
2*)   E0ods(POIo,t) = |Q(PART)|/Rpcs(POIp(t),t)^2 

/* Subbing (2*) into (1) and consolidating
   ∂[∂(t): E0ods(POIo(t)=0)  *Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4]
=   E0ods(POIo,t)*
   (+  2*Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(Aθpc(POIo(t),t))*sin(Aθoc(POIo))^4   
    +  1*Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4 
    +  4     *Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3
   )
=   E0ods(POIo,t)*
   (+  3*Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(Aθpc(POIo(t),t))*sin(Aθoc(POIo))^4   
    +  4     *Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3
   )


/*----+
Check on general model : 
∂[∂(t): E0ods(POIo(t)=0)*Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4]  
=   E0ods(POIo(t)=0)*
   (+ 2  *Vons(PART)*Rpcs(POIo(t),t)^(-1-1)*cos(AOtc(RFt))*sin(AOtc(RFt))^4   
    + 1  *Vons(PART)*Rpcs(POIo(t),t)^(-1-1)*cos(AOtc(RFt))*sin(AOtc(RFt))^4 
    + 4       *Rpcs(POIo(t),t)^(-1  )*cos(AOtc(RFt))*sin(AOtc(RFt))^(4-1)
   ()
=   E0ods(POIo(t)=0)*
   (+ 2  *Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(AOtc(RFt))*sin(AOtc(RFt))^4   
    + 1  *Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(AOtc(RFt))*sin(AOtc(RFt))^4 
    + 4       *Rpcs(POIo(t),t)^(-1)*cos(AOtc(RFt))*sin(AOtc(RFt))^3
   ()
=   E0ods(POIo(t)=0)*
   (+ 3  *Vons(PART)*Rpcs(POIo(t),t)^(-2)*cos(AOtc(RFt))*sin(AOtc(RFt))^4   
    + 4       *Rpcs(POIo(t),t)^(-1)*cos(AOtc(RFt))*sin(AOtc(RFt))^3
   ()

OK!!  This is the same as the earlier hand derivation : 
∂[∂(t): E0ods(POIo(t)=0)*Rpcs(POIo(t),t)^(-1)*sin(Aθoc(POIo))^4] 
=  E0ods(POIo,t)*
   (+  3*Vons(PART)*Rpcs(POIo(t)=0)^(-2)*cos(AOtc(RFt))*sin(AOtc(RFt))^4   
    +  4     *Rpcs(POIo(t)=0)^(-1)*cos(AOtc(RFt))*sin(AOtc(RFt))^3
   ()