52 VECTOR ANALYSIS. 



The Maxwellian product of a vector and a scalar function of 

 position in space is defined by the equation 



/ 

 Max (w, u) ~ffffff^ ^ T^ . a>' dv dv' fffu Max wdv - New (u, w). 



It is of course supposed that u, w, 0, w are such functions of position 

 that the above expressions have definite values. 



104. By No. 97, 



4>7ruPotw= V.NewuPotiy = V.[NewuPotit/] + New u. New w. 



The volume-integral of this equation gives 



4-7T Pot (u, w) ==/J^New u. New w dv, 

 if the integral 



ffdv. New u Pot w, 



for a closed surface, vanishes when the space included by the surface 

 is indefinitely extended in all directions. This will be the case when 

 everywhere outside of certain assignable limits the values of u and w 

 are zero. 



Again, by No. 102, 



4 < 7Tft).Pot = VxLapw.Pot V Max a). Pot 

 = V.[Lap ft>xPot 0] -f- Lap to. Lap <j> 



V. [Max w Pot 0] -h Max w Max 0. 



The volume-integral of this equation gives 



4-7T Pot (0, ft)) ==/5f7*Lap -Lap cfa; -f j^JOVIax Max w cfo, 

 if the integrals 



ffd<r . Lap w x Pot (f>, ffd<r Pot Max w, 



for a closed surface vanish when the space included by the surface is 

 indefinitely extended in all directions. This will be the case if every- 

 where outside of certain assignable limits the values of and co are 

 zero. 



CHAPTER III. 

 CONCERNING LINEAR VECTOR FUNCTIONS. 



105. Def. A vector function of a vector is said to be linear, when 

 the function of the sum of any two vectors is equal to the sum of the 

 functions of the vectors. That is, if 



f unc. [p + p] = f unc. [p] -f f unc. [p] 



for all values of p and p, the function is linear. In such cases it is 

 easily shown that 



f unc. [ap -f bp + cp" -f etc.] = a f unc. [p] -f b f unc. [p] + c f unc. [p"] + etc. 



, 



