VECTOR ANALYSIS. 35 



64. If u or CD is a function of several scalar or vector variables 

 which are themselves functions of the position of a single point, the 

 value of Vu or V.o> or Vxo> will be equal to the sum of the values 

 obtained by making successively all but each one of these variables 

 constant. 



65. By the use of this principle we easily derive the following 

 identical equations : 



(I) 



(2) 



(3) 

 (4) 



(5) 

 (6) 



The student will observe an analogy between these equations and 

 the formulae of multiplication. (In the last four equations the 

 analogy appears most distinctly when we regard all the factors but 

 one as constant.) Some of the more curious features of this analogy 

 are due to the fact that the V contains implicitly the vectors i, j 

 and k, which are to be multiplied into the following quantities. 



Combinations of the Operators V, V., and Vx. 



66. If u is any scalar function of position in space, 



as may be derived directly from the definitions of these operators. 



67. Conversely, if o> is such a vector function of position in space 

 that 



CD is the derivative of a scalar function of position in space. This will 

 appear from the following considerations : 



The line-integral fu> . dp will vanish for any closed line, since it may 

 be expressed as the surface-integral of Vxo>. (No. 60.) The line- 

 integral taken from one given point P' to another given point P" is 

 independent of the line between the points for which the integral 

 is taken. (For, if two lines joining the same points gave different 

 values, by reversing one we should obtain a closed line for which the 

 integral would not vanish.) If we set u equal to this line-integral, 

 supposing P" to be variable and P' to be constant in position, u will 

 be a scalar function of the position of the point P", satisfying the 

 condition du = w.dp, or, by No. 51, Vu=w. There will evidently be 

 an infinite number of functions satisfying this condition, which will 

 differ from one another by constant quantities. 



