25Sep2015 revision 1 started
  14Sep2015 start verification, 23Sep2015 corrections & cleanup, 27May2016 reworked
  From 04_28a Faradays_law_spherical_coordinates - full form, 1st expression  

/$		∇´Ei(r´,t)  
		=  + r´h/rs/sin(θ)*[ ∂[∂(θ): Ei(r´,t)•φ´hat*sin(θ)) -  ∂[∂(φ): Ei(r´,t)•θ´hat)    ]  
			+ θ´hat/rs     *[ ∂[∂(φ): Ei(r´,t)•r´h)/sin(θ) -  ∂[∂(r): Ei(r´,t)•φ´hat*rs) ]   
			+ φ´hat/rs     *[ ∂[∂(r): Ei(r´,t)•θ´hat*r)    -  ∂[∂(θ): Ei(r´,t)•r´h)    ]  

/*  By symmetry, the following partial derivatives are zero (with respect to the scalar result!!): 
/$ 		∂[∂(φ): Ei(r´,t)•θ´hat)
		∂[∂(φ): Ei(r´,t)•r´h)
		∂[∂(θ): Ei(r´,t)•r´h)
/* The following are zero because Ei(r´,t) & Ph are at right angles :  
/$ 		∂[∂(θ): Ei(r´,t)•φ´hat*sin(θ)) 
		∂[∂(r): Ei(r´,t)•φ´hat*rs)
/* Therefore, the ONLY partial derivative that remains is : 
/$		∂[∂(θ): Ei(r´,t)•r´h)
/* yielding the expression : 
/$		∇´Ei(r´,t) =  -φ´hat/rs*∂[∂(θ): Ei(r´,t)•r´h)
/* but rh is a unit vector in the same direction as Ei, so for a partial derivative along angle O : 
/$		∂[∂(θ): Ei(r´,t)•r´h) = ∂[∂(θ): |Ei(r´,t)|)
/*so 
/$		∇´Ei(r´,t) =  -φ´hat/rs*∂[∂(θ): |Ei(r´,t)|)
/*	which is the same as Lucas's LHS expression.  

/*	Earlier approach, essentially the same, but clouded by ambiguous wording in Lucas's explanation 
	Extract the "dp[dO  : " terms : 

/$		∂[∂(θ): Ei(r´,t)) 
		= + r´h/rs/sin(θ)*[ ∂[∂(θ): Ei(r´,t)•φ´hat*sin(θ)) ] - φ´hat/rs      *[ ∂[∂(θ): Ei(r´,t)•r´h)       ]  
		= + r´h/rs/sin(θ)*[ ∂[∂(θ): Ei(r´,t)•φ´hat*sin(θ)) ] - φ´hat/rs      *[ ∂[∂(θ): Ei(r´,t)•r´h)       ]  

/*zz)see "Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt" :
zz)"Pseudo-constant variables/expressions" within integrals wrt O 
zz)Remove : [q,v,L(v),r´,c,b]
zz)Retain : [sinO,O] are the key retained variables for Chapter 4 verifications 

/$1)	∂[∂(θ): Ei(r´,t))  
		= + r´h/rs/sin(θ)•φ´hat*[ ∂[∂(θ): Ei(r´,t)*sin(θ)) ]  
		  - φ´hat/rs         *[ ∂[∂(θ): Ei(r´,t)•r´h)  ]  
/*  Now 
/$		∂[∂(θ): Ei(r´,t)*sin(θ)) = ∂[∂(θ): Ei(r´,t))*sin(θ) + Ei(r´,t)*∂[∂(θ): sin(θ))  
		∂[∂(θ): Ei(r´,t)*sin(θ)) = ∂[∂(θ): Ei(r´,t))*sin(θ) + Ei(r´,t)*cos(θ)  
/*  But from symmetry 
/$ 		∂[∂(θ): Ei(r´,t)*sin(θ)) = ∂[∂(θ): Ei(r´,t))*sin(θ) + Ei(r´,t)*cos(θ)  
		∂[∂(θ): Ei(r´,t)) = 0  
/*  therefore 
/$1a)	∂[∂(θ): Ei(r´,t)*sin(θ)) = Ei(r´,t)*cos(θ)  
1b)	∂[∂(θ): Ei(r´,t)•r´h)   = 0 given symmetry  
/*  Sub (1a),(1b) into (1)
/$1)	∂[∂(θ): Ei(r´,t))  
		= + r´h/rs/sin(θ)•φ´hat*[ ∂[∂(θ): Ei(r´,t)*sin(θ)) ]	- φ´hat/rs*[ ∂[∂(θ): Ei(r´,t)•r´h)   ]  
		= + r´h/rs/sin(θ)•φ´hat*Ei(r´,t)*cos(θ)						- φ´hat/rs*0  
		= + r´h/rs/sin(θ)•φ´hat*Ei(r´,t)*cos(θ)    

/*  The RHS of (4-29a) is the exact same as (4-28b) and (4-16) so no 
   further work is required on that part.

/$		-1/c*∂[∂(t): Bi(r - v*t,t)]  
		=  -vs/c*rs*sin(θ)*φ´*[  3*q*vs/c*(rs*cos(θ) - vs*t)/|rs - vs*t|^5 + 1/rs/c*∂[∂(t): Eis(r - v*t,t)] ] 

/*++++++++++++++++++++++++++++++++++++++
/*add_eqn  "no_issue 
   04_29arev1 
   Faradays_law_spherical_coordinates - reduced, integral form  

/$L		-1/rs*∂[∂(θ): Ei(r - v*t,t))*φ´  = -vs/c*rs*sin(θ)*φ´*[  3*q*vs/c*(r*cos(θ) - v*t)/|r - v*t|^5  + 1/rs/c*∂[∂(t): Eis(r - v*t,t)] ]    
/$H		-1/rs*∂[∂(θ): Ei(r - v*t,t))*φ´  = -vs/c*rs*sin(θ)*φ´*[  3*q*vs/c*(r*cos(θ) - v*t)/|r - v*t|^5  + 1/rs/c*∂[∂(t): Eis(r - v*t,t)] ] 

/*	OK - straightforward, but clouded by some of the text explanations & omissions. 
   
/%		-1/Rocs(POIo)*∂[∂(θ): EIods(POIo,t)]*Rφpdh  
		= -Vons(PART)/c*Rocs(POIo)*sin(Aθoc(POIo))   *Rφpdh  
		  *{  3*Q(particle)*Vons(PART)/c*(Rocs(POIo)*cos(Aθoc(POIo)) - Vons(PART)*t)/Rpcs(POIo(t),t)^5  
		    + 1/Rocs(POIo)/c*∂[∂(t): EIods(POIo,t)] 
		   }