(mathH)  ∂[∂(t): Rpcs(POIo(t),t)]  =  -Vons(PART)*cos(Aθpc(POIo(t),t))
(endMath)

/*????????????????????????????
>>>>>>>>>>>> Confirmation of ∂[∂(t): Rpcs(POIo(t),t)]
12Oct2019 is this wrong?  /%
1369:(mathH)  ∂[∂(t): Rpcs(POIo(t),t)]  =  -Vons(PART)*cos(Aθpc(POIo(t),t))

/*	The work below was done independently of my previous work, to see if I :
	- obtaine the same answers
	- notice [missing issues, factors] 
	The original checks can be found at : 
	- "?documenht?"
	- ?section?

/*	see "$d_PROJECTS""Lucas - Universal Force/Images/Howell - Chapter 4 - dp[dt _ Rpcs(POIo,t)] - cropped.png"
	"Figure "∂[∂(t): Rpcs(POIo(t),t) ]""
In the figure, I have 
/%
	delta(Rpcs) 
	=  Rpcs(POIo(t),t2) - Rpcs(POIo(t),t1)
	=  Vonv(PART)*delta(t)*cos(pi - Aθpc(POIo(t),t2))

/*	Reminders :   /%
	cos(pi - Aθpc(POIo(t),t)) = -cos(Aθpc(POIo(t),t))
	sin(pi - Aθpc(POIo(t),t)) = -sin(Aθpc(POIo(t),t))

	cos(pi/2 - Aθpc(POIo(t),t)) = ?[+,-]?*sin(Aθpc(POIo(t),t))
	sin(pi/2 - Aθpc(POIo(t),t)) = ?[+,-]?*cos(Aθpc(POIo(t),t))

/*	+-----+
Case 1 : the particle is "ahead" of the Point of Interest in the Observer frame (POIo)
	ie the particle is flying away from the POIo on a parallel track
	and Rpcs is GROWING
/%
	Aθpc(POIo(t),t) >= pi/2, so
		cos(Aθpc(POIo(t),t2)) <= 0
		∂[∂(t): Rpcs(POIo(t),t) ] >= 0

	delta(Rpcs) 
	= Rpcs(POIo(t2),t2) - Rpcs(POIo(t1),t1)
/* but there is NO change in the AθPI2pc(POIo(t),t) perpendicular direction  /%
	Rpcs(POIo(t2),t2)*sin(Aθpc(POIo(t2),t2)) = Rpcs(POIo(t1),t1)*sin(Aθpc(POIo(t1),t1))
/* and   /%
	Vons(PART)*(t2 - t1)
	= Rpcs(POIo(t2),t2)*cos(Aθpc(POIo(t2),t2)) - Rpcs(POIo(t1),t1)*cos(Aθpc(POIo(t1),t1))




/* This is the delta form of the Chain Rule for differentiation
	Imagine a box that expands at the advancing edges
/% 
	= Vons(PART)
	  *{	Rpcs(POIo(t1),t1)*{cos(Aθpc(POIo(t2),t2)) - *cos(Aθpc(POIo(t1),t1))}
	  	 +	(t2 - t1)*cos(Aθpc(POIo(t1),t1))
		 - 1/2*(t2 - t1)*{cos(Aθpc(POIo(t2),t2)) - *cos(Aθpc(POIo(t1),t1))}
		}						  	 
/* But as deltas -> 0, ignore the third term, so  /%
	∂[Rpcs(POIo(t),t)] 
	= limit as delta(t) -> 0 { Vons(PART) *d[t2*cos(Aθpc(POIo(t2),t2)) - t1*cos(Aθpc(POIo(t1),t1))] }
	= Vons(PART)*{ ∂[t]*cos(Aθpc(POIo(t),t)) +  t*∂[cos(Aθpc(POIo(t),t))]

	∂[∂(t): Rpcs(POIo(t),t)]
	=  ∂[∂(t): Vons(PART)*{ ∂[t]*cos(Aθpc(POIo(t),t)) +  t*∂[cos(Aθpc(POIo(t),t))] }
	=  Vons(PART)*{ cos(Aθpc(POIo(t),t)) +  t*∂[∂(t): cos(Aθpc(POIo(t),t))] }
/* with 




|cos(Aθpc(POIo(t),t))|

28Oct2019 YIKES!!! I didn't put in the || signs.  

/*	WRONG!!  so this result is the same as  /%
1369:(mathH)  ∂[∂(t): Rpcs(POIo(t),t)]  =  -Vons(PART)*cos(Aθpc(POIo(t),t))

I need to change this!!!! -> huge, cascading] corrections needed in derivations!


/*	Note that for Case 1,  /%  
	Aθpc(POIo(t) >= pi/2,    	
/*	so  /%
	cos(Aθpc(POIo(t),t))			<= 0
	∂[∂(t): Rpcs(POIo(t),t)]	>= 0

/*	+-----+
Case 2 :  POIo is ahead of particle 
	ie the particle is flying toward the POIo on a parallel track
	and Rpcs is SHRINKING!
/%
	Aθpc(POIo(t),t) <= pi/2, so
		cos(Aθpc(POIo(t),t2)) >= 0
		∂[∂(t): Rpcs(POIo(t),t) ] <= 0

	delta(Rpcs) is <= 0
	= (-1)* cos(Aθpc(POIo(t),t1)) *(Vons(PART)*(t2 - t1))
	= (-1)*|cos(Aθpc(POIo(t),t1))|*Vons(PART)*delta(t)

	∂[∂(t): Rpcs(POIo(t),t)] 
	= limit as delta(t) -> 0 : 
	=       cos(Aθpc(POIo(t),t)) *Vons(PART)
	= (-1)*|cos(Aθpc(POIo(t),t))|*Vons(PART)

/*	this result is the same as Case 1 and :  /%
1369:(mathH)  ∂[∂(t): Rpcs(POIo(t),t)]  =  -Vons(PART)*cos(Aθpc(POIo(t),t))

/*	Note that for Case 2,  /%  
	Aθpc(POIo(t) <= pi/2,    	
/*	so  /%
	cos(Aθpc(POIo(t),t))			>= 0
	∂[∂(t): Rpcs(POIo(t),t)]	<= 0

So the original results are confirmed.

28Oct2019 CHANGE!!!
Using :"vector" notation for [+,-] angles :
In order to have a unified expression for both conditions, for clarity I SHOULD use :
	∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*|cos(Aθpc(POIo(t),t))|

This looks the "same" as what I've been using, but check how that affects derivatives!