78 VECTOR ANALYSIS. 



transformation of the expression which would be allowable if the V 

 were a vector (viz., by changes in the order of the factors, in the 

 signs of multiplication, in the parentheses written or implied, etc.), by 

 which changes the V is brought into connection with one particular 

 factor, the expression thus transformed will represent the part of 

 the value of the original expression which results from the variation 

 of that factor. 



167. From the relations indicated in the last four paragraphs, may 

 be obtained directly a great number of transformations of definite 

 integrals similar to those given in Nos. 74-77, and corresponding to 

 those known in the scalar calculus by the name of integration by 

 parts. 



168. The student will now find no difficulty in generalizing the 

 integrations of differential equations given in Nos. 78-89 by applying 

 to vectors those which relate to scalars, and to dyadics those which 

 relate to vectors. 



169. The propositions in No. 90 relating to minimum values of the 

 volume-integral fffuoo.wdv may be generalized by substituting 

 u)&.(0 for uw.oi, 3? being a given dyadic function of position in space. 



170. The theory of the integrals which have been called potentials, 

 Newtonians, etc. (see Nos. 91-102) may be extended to cases in which 

 the operand is a vector instead of a scalar or a dyadic instead of a 

 vector. So far as the demonstrations are concerned, the case of a 

 vector may be reduced to that of a scalar by considering separately 

 its three components, and the case of a dyadic may be reduced to that 

 of a vector, by supposing the dyadic expressed in the form <j>i + x3 ' + w & 

 and considering each of these terms separately. 



CHAPTER V. 

 CONCERNING TRANSCENDENTAL FUNCTIONS OF DYADICS. 



171. Def. The exponential function, the sine and the cosine of a 

 dyadic may be defined by infinite series, exactly as the corresponding 

 functions in scalar analysis, viz., 



These series are always convergent. For every value of < there is 

 one and only one value of each of these functions. The exponential 

 function may also be defined as the limit of the expression 



