ALM AR NÆSS. M.-N. Kl. 



^ 2. Preliminaries. 



Firstly we lay down a lew delinitions: 



In an ordinary «-dimensional space .S« be j^iven a fixed set of rect- 

 angular (i. e. mutually perpendicular) axes o x^, Xj . . . . x„ defining a coor- 

 dinate system. To any given set of ;/ real numbers 



1 ' 1 J . . . . X/^ 



corresponds a point in this space. Further let 



Ci, e., e„ 



designate a uoriiia/ system of unit vectors in this coordinate system, i. e. : 

 n vectors of length one, originating from any point in S» and parallel to 

 the coordinate axes respectivel}', i. e. each of them is at right angles to the 

 other {n — 1). These vectors, therefore, determine the coordinate system. 



Any scalar function ï' of // variables x-^, Xo, .... Xn determines for each 

 set of the variables a scalar quantity. Hence : to each point in S„ is thus 

 made to correspond a scalar ; v defines a scalar field. 



The e's are ;/ linearly independent vectors. Any other vector in S,t is 

 expressible by them. This contains our axiom of dimensions. A vector 

 is then a quantity of the form 



(1) t) =-- Ci t'l + Co î'.2 + . . . . + e„ Vn =■ ZiVi. 



Summation with respect to a subscript appearing twice is always under- 

 stood. The î''s are called the coniponciits of the vector t>. Supposing 



the 7''s are functions of the variables .r^, .v«, v«. With each point 



(x^, x^.... x„) in .S„ is then associated a set of the î''s, that is a vector. 

 The point is called its origin. An expression as (1) thus determines in 

 each point a vector. ï> is a vector function of position in space, defining a 

 vector field, but is in what follows nevertheless usually spoken of as a vector. 

 If t> = ^iVi and t>' =-^ e,î'/ then the scalar quantity v,v,' is called the 

 scalar product of l^ and !">' and denoted by V • v'. If v • ï»' vanishes, the two 

 vectors are said to be perpendicular on one another, i*» • v is the square 

 of the length of t>.' The fundamental properties of the unit vectors can 

 thus be written : 



(2) e,-.ey=^,-; 



where <5,y is a symbol equal to one for / = / and equal to zero for / ^ j. 

 The definition of Gibbs's indeterminate product of vectors (dN'ads, triads 

 and in general polyads) can evidendy be extended to S„ without further 

 explanation, as there is nothing in the mathematical nature of those concep- 

 tions which limits them to three-space onl}'. This is simply a consequence 

 of the fact that a dyad (and a dyadic) is expressible as (Bôcher says: 

 identical with) a square matrix. Here may brieflN- be mentioned: 



