/home/bill/Lucas - Universal Force/individual formulae developments/d-dt Rpcs^5*t*_cos - 1.txt
www.BillHowell.ca  24Sep2019 initial based on past work

15Oct2019 -  WRONG!!!!!!  I cannot drop :
		∂[∂(t):	Rpcs(POIo(t),t=0)^(-5)*(t=0)*(cos(Aθpc(POIo(t),t=0)) - 1)]
			=	+	Rpcs(POIo(t),t=0)^(-5)*(cos(Aθpc(POIo(t),t=0)) - 1)
This file is used for the "correct aqs far as I can tell" derivations of equations (4-[32-37]).

Cool!  derivative = original expression without t at t=0 !!!

/*********************  24Sep2019 
>>>>>> ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] =  0  
/% 
1)		∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)]
a)		= 		∂[∂(t): Rpcs(POIo(t),t)^(-5)]*t*cos(Aθpc(POIo(t),t))]
b)		 	-	∂[∂(t): Rpcs(POIo(t),t)^(-5) *t]

/*+-----+
/* looking at (1a) /%
2)				∂[∂(t): Rpcs(POIo(t),t)^(-5)			  *t			  *cos(Aθpc(POIo(t),t))]
a)		=		∂[∂(t): Rpcs(POIo(t),t)^(-5)]			  *t			  *cos(Aθpc(POIo(t),t)) 
b)			+			  Rpcs(POIo(t),t)^(-5)	*∂[∂(t): t]			  *cos(Aθpc(POIo(t),t))
c)			+			  Rpcs(POIo(t),t)^(-5)			  *t	*∂[∂(t): cos(Aθpc(POIo(t),t))]

/*	from section "∂[∂(t): Rpcs(POIo(t),t)^(-α)]" : /%
		∂[∂(t): Rpcs(POIo(t),t)^(-α)]  =  α*Vons(PART)*Rpcs(POIo(t),t)^(-α - 1)*cos(Aθpc(POIo(t),t))
/* therefore /%
d)		∂[∂(t): Rpcs(POIo(t),t)^(-5)] =  5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t))
/*	noting  /%
e)		∂[∂(t): t] = 1
f)		∂[∂(t): cos(Aθpc(POIo(t),t))]  =  -sin(Aθpc(POIo(t),t))*∂[∂(t): Aθpc(POIo(t),t))]
/*	from "Howell - independent math for Lucas Universal Force, Chapter 4.txt" 
		section "∂[∂(t): Aθpc(POIo(t),t))]" : /%
g)		∂[∂(t): Aθpc(POIo(t),t)] =  Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)
/*	therefore  /%
h)		∂[∂(t): cos(Aθpc(POIo(t),t))]  
		=  -sin(Aθpc(POIo(t),t))*Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)
		=	(-1)*Vons(PART)*sin(Aθpc(POIo(t),t))^2/Rpcs(POIo(t),t)
				
/*	substitute (2d,e,h) into (2a-c)  /%
3)				∂[∂(t): Rpcs(POIo(t),t)^(-5)			  *t			  *cos(Aθpc(POIo(t),t))]
a)		=		5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t))	*t	  *cos(Aθpc(POIo(t),t)) 
b)			+			  Rpcs(POIo(t),t)^(-5)			  *1			  *cos(Aθpc(POIo(t),t))
c)			+			  Rpcs(POIo(t),t)^(-5)			  *t		*(-1)*Vons(PART)*sin(Aθpc(POIo(t),t))^2/Rpcs(POIo(t),t)
/*	collecting & rearranging terms  /%
4)				∂[∂(t): Rpcs(POIo(t),t)^(-5)		*cos(Aθpc(POIo(t),t))		*t ]
a)		=		5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t))^(2)	*t
b)			+			  		 Rpcs(POIo(t),t)^(-5)*cos(Aθpc(POIo(t),t))
c)			-	  Vons(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^(2)	*t

/*+-----+
/* looking at (1b) /%
5)				∂[∂(t): Rpcs(POIo(t),t)^(-5)				*t]
a)		=		∂[∂(t): Rpcs(POIo(t),t)^(-5)]				*t
b)			+			  Rpcs(POIo(t),t)^(-5)	*∂[∂(t):  t]
/*	sub (2d,e) into (5a,b)  /% 
6)				∂[∂(t): Rpcs(POIo(t),t)^(-5)				*t]
a)		=		5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t))	*t
b)			+			  		 Rpcs(POIo(t),t)^(-5)								*1

/*+-----+
	sub (4,6) into (1)
/%
7)		∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)]
a)		=	+	5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t))^(2)	*t
b)			+			  		 Rpcs(POIo(t),t)^(-5)*cos(Aθpc(POIo(t),t))
c)			-	  Vons(PART)*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^(2)	*t
d)			-	5*Vons(PART)*Rpcs(POIo(t),t)^(-6)*cos(Aθpc(POIo(t),t))		*t
e)			-			  		 Rpcs(POIo(t),t)^(-5)

/*	At time t=0  /%
8)		∂[∂(t):	Rpcs(POIo(t),t=0)^(-5)*(t=0)*(cos(Aθpc(POIo(t),t=0)) - 1)]
			=	+	Rpcs(POIo(t),t=0)^(-5)* cos(Aθpc(POIo(t),t=0))
				-	Rpcs(POIo(t),t=0)^(-5) 

/*	finally   /%
(mathH)
	∂[∂(t):	Rpcs(POIo(t),t=0)^(-5)*(t=0)*(cos(Aθpc(POIo(t),t=0)) - 1)]
	=	+	Rpcs(POIo(t),t=0)^(-5)*(cos(Aθpc(POIo(t),t=0)) - 1)
(endMath)