34 VECTOR ANALYSIS 



Now 



and 



where the summation relates to the coordinate axes and connected 

 quantities. Substituting these values in the preceding equation, 

 we get 



or by No. 30, 



. [Spxdp] = 



But Spxdp represents an element of the surface generated by the 

 motion of the element dp, and the last member of the equation is 

 the surface-integral of Vxw for the infinitesimal surface generated 

 by the motion of the whole line. Hence, if we conceive of a closed 

 curve passing gradually from an infinitesimal loop to any finite form, 

 the differential of the line-integral of co for that curve will be equal 

 to the differential of the surface integral of Vxo> for the surface 

 generated : therefore, since both integrals commence with the value 

 zero, they must always be equal to each other. Such a mode of 

 generation will evidently apply to any surface closing any loop. 



61. The line-integral of w for a closed line bounding a plane surface 

 da- infinitely small in all its dimensions is therefore 



This principle affords a definition of Vx which is independent 

 of any reference to coordinate axes. If we imagine a circle described 

 about a fixed point to vary its orientation while keeping the same 

 size, there will be a certain position of the circle for which the line- 

 integral of ft> will be a maximum, unless the line-integral vanishes for 

 all positions of the circle. The axis of the circle in this position, 

 drawn toward the side on which a positive motion in the circle 

 appears counter-clockwise, gives the direction of Vxo>, and the 

 quotient of the integral divided by the area of the circle gives the 

 magnitude of 



V, V., and Vx applied to Functions of Functions of Position. 



62. A constant scalar factor after V, V., or Vx may be placed 

 before the symbol. 



63. If f(u) denotes any scalar function of u, and f'(u) the derived 



function, _. _ 



Vf(u)=f'(u)Vu. 



