18 QUATERNIONS APPLIED TO PHYSICS 



of the complete boundary, the relative senses of the two 

 being as usual. The surface-volume integral is 



f^dv = {{L^Q^db (7) 



where db is . an element of any given volume and dv 

 a rotor element of the complete boundary, pointing away 



from the bounded volume. [That r/v ;= for a closed 



surface follows from U/X = 2J c?v for an open surface.] 



o- being any vector function of position (rotor, rotor- 

 couple or motor ; but generally to be thought of as a 

 rotor), and ^ == S( )(t, (6) and (7) become 



|S,TC?X= f[S.r/v j9 + 2J)(r (8) 



S(Tdv= {{[SQadh (9) 



\l 



(8), (9) and (2) give us the proper forms for curl 

 convergence and gradient (denoted below by crl, cnv, 

 and grd). In Euclidean space, if </ is a quaternion 

 function of position we conveniently define thus 



crl(/ = V'^Vq, cnxq = Sv^', gi'dy = V% 

 so that V7 = (crl + cnv + grd)^'. 



Similarly in non-Euclidean space a symbolic quaternion 

 linit^ A takes the place of the symbolic linity v( )• Let 



Ay= (e + 2,lV)g (10) 



Then crlry = VAVy = V. + 2J)Vr/'j 



cnv^.= SA7= S0r/ ( . 



whence Aq = (crl + cnv + grd)// * 



It is easy to prove that 



(crl + grd + cnv)2 = A^ 



= crP -H cnv . grd + grd . cnv ... (12) 

 the last equation implying the six zero relations 

 = crl . grd ^ crl . cnv ^ grd . crl 



= grd . grd = cnv . crl = cnv . cnv ... (13) 



When q, the quaternion function of a point, consists 



of a rotor through the point and a real scalar we will 



call it local. From (2), (8) and (9) it is obvious that 



