46 



ALMAR NÆSS. 



M.-X. Kl. 



^ 15. The skew-symmetric dyadic (tensor) of two vectors expressed 



as a space complement. 



Fi-om two f^ivcn vectors a and b we can derive a .skew-symmetric 

 tensor dcfuicd by the following scalars : 



1) 



dj — a i hj — (7/ hi 



, . /H// 1 I . , , , ^ , 'TM • 



involvmt;' independent scalars, as Ca ^ Ü and Cij — — f/,-. 1 his 



tensor (by some authors called the vector product of a and bt| is in 

 vector analysis notations : 



e,- ey Cij = ab — b a 



a b 

 a b 



the multiplication of the vectors being indeterminate. 



This tensor (dyadic) and the space complement of a and b are very 

 closely related to one another, as either of them in a simple way can be 

 derived from the other. We will here show that the tensor c,j can be 

 obtained as the space complement of CI X b times a scalar. 



By definition we get as an expression for a X b the sum of all possible 

 terms (when i <^j) of the following form: 



(3) 



a X b = 2" - 



-ir'£,; 



hi hj 



So we take the space complement of this. We get by § 7 (2) and 

 p 

 § 14 (1), putting p - 2,Su^. = / +y: 



<«-2£,y = (7/ — 2)! <" -2 Ci . . ,e'/ . . ,e'/ . . e„ 



(;/-2)! (- 1) 



/ + /-1 



e, e; 



t Hkrm.\n!>i Wkyl, R.au.n, Z-it, M;it:?ri-, p. \o. 



