1922. No. 13. ON A SPECIAL POLYADIC OF ORDER // />. 49 



area of the parallelogram on t» and a, and / X CI is the vector a turned 

 one right angle in negative direction, that is in the direction from a to 

 l> if D X CI is a positive scaler. Then (161 only says that two opposite sides 

 of a parallelogram are equal in length. 



§ 1 6. Remarks concerning the divergence and the curl. 

 By the Xabla vector \7 we understand the symbolic vector difterentiator 



d 



e, ^ — . Hence: 



c A,- 



(1) ^-^-i 



In the three-dimensional vector anah'sis the scalar and vector of this 

 dyadic is called the divergence and curl of a respectively. 



As the first of these conceptions only depends] upon the definition 

 of the scalar product of two vectors — which is valid in any space -^ 

 we put also in S«: 



V-; c a d ai 



(?) div a ^ \/ • ci = e, • -r — = - — 



'-' e Xi c Xi 



As in Sg, we will apply this equation also to the case when we instead 

 of a have in general a polyad(ic), whereby the divergence of any polyad(ic) 

 is defined. Particularly we notice that if a polyadic is written as a deter- 

 minant whose first row consists of the unit vectors, the divergence of it is 



a 



obtained by interchanging the first row with the operators -— . 



The generalisation of the curl to S„ is not so obvious. We here want 

 to emphasize that by the term curl we only understand the (special) vector 

 function, such as it is defined in classical vector analysis, not the physical 

 phenomena (the rotation) which this vector ma}' represent. And it is out- 

 side our province to consider whether or not there may be a more suitable 

 mathematical representation for those phenomena (e. g. a skew-symmetric 

 tensor of the second order), r But from this point of view, the curl is 

 nothing but a certain vector product (i. e. : a sum of such ones), and a way 

 of extending the latter to S„ once defined or adopted, necessarily leads to 

 a corresponding generalization of the curl. 



Hence, the quantity which we here will consider to be the generalized 

 „vector" of the dyadic V a, is the following: 



(31 V '^ a = C/ X r— , 



c X'i 



t Weyl, H.; loc. cit. p. 54. 

 Vid.-Selsk. Skrifter. I. M.-N. Kl. 1922. No. 13. 



