PIEZOELECTRIC CRYSTALS IN TENSOR FORM 



101 



There are single quantities such as mass and distance, that are the same 

 for all systems of coordinates. These are called tensors of the zero rank 

 or scalars. 



Consider now two tensors of the first rank «, and Vk ■ Suppose that each 

 component of one is to be multiplied by each component of the other, then 

 we obtain a set of nine quantities expressed by Ui Vk , where i and k are 

 independently given all the values 1. 2, 3. The components of «; Vk with 

 respect to the Xj set of axes are Uj V( , and 



tijVi = (aijtii) (aicfk) = anQkiUiVk 



(65) 



The suffixes / and k are repeated on the right. Hence (65) represents nine 

 equations, each with nine terms. Each term on the right is the product 

 of two factors, one of the. form a ijOki, depending only on the orientation of 

 the axes, and the other of the form UiVk , representing the products of the 

 components referred to the original axes. In this way the various Uj Vf can 

 be obtained in terms of the original UiVk . But products of vectors are not 

 the only quantities satisfying the rule. In general a set of nine quantities 

 IV ik referred to a set of axes, and transformed to another set by the rule 



^';Y = OijQki u>ik 



(66) 



is called a tensor of (he second rank. 



Higher orders tensors can be formed by taking the products of more 

 vectors. Thus a set of n quantities that transforms like the vector product 

 XiXj • • • Xp is called a tensor of rank /?, where n is the number of factors. 



On the right hand side of (66) the / and k are dummy suffices; that is, 

 they are given the numbers 1 to 3 and summed. It, therefore, makes no 

 difference which we call i and which k so that 



^^'j7 



jakfiCik — OkjaifCkf 



(67) 



Hence Wk( transforms by the same rule as u' ik and hence is a tensor of the 

 second rank. The importance of this is that if we have a set of quantities 



which we know to be a tensor of the second rank, the set of quantities 



(68) 



(69) 



is another tensor of the second rank. Hence the sum (idk + i^'ki) and the 

 difference (^c',k — iVk,) are also tensors of the second rank. The first of 



