30 



VECTOR ANALYSIS. 



46. Differential coefficient with respect to a scalar. The quotient 

 obtained by dividing the differential of a vector due to the variation 

 of any scalar of which it is a function by the differential of that 

 scalar is called the differential coefficient of the vector with respect 

 to the scalar, and is indicated in the same manner as the differential 

 coefficients of ordinary analysis. 



If we suppose the quantities occurring in the six equations of the 



last section to be functions of a scalar t, we may substitute -^ for d in 



those equations since this is only to divide all terms by the scalar dt. 



47. Successive differentiations. The differential coefficient of a 

 vector with respect to a scalar is of course a finite vector, of which 

 we may take the differential, or the differential coefficient with 

 respect to the same or any other scalar. We thus obtain differential 

 coefficients of the higher orders, which are indicated as in the scalar 

 calculus. 



A few examples will serve for illustration. 



If p is the vector drawn from a fixed origin to a moving point at 



any time t, -fr will be the vector representing the velocity of the 



d 2 p 

 point, and -rK the vector representing its acceleration. 



If p is the vector drawn from a fixed origin to any point on a 

 curve, and s the distance of that point measured on the curve from 



any fixed point, -- is a unit vector, tangent to the curve and having 



the direction in which s increases; -,- is a vector directed from a 



ds 2 



point on the curve to the center of curvature, and equal to the 



7 J7O 



curvature ; -f X -T? is the normal to the osculating plane, directed to- 



the side on which the curve appears described counter-clockwise 

 about the center of curvature, and equal to the curvature. The 

 tortuosity (or rate of rotation of the osculating plane, considered as 

 positive when the rotation appears counter-clockwise as seen from 

 the direction in which s increases) is represented by 



dp d 2 p d s p 



ds ds 2 ds s 



ds 2 ds 2 



48. Integration of an equation between differentials. If t and u 

 are two single-valued continuous scalar functions of any number of 

 scalar or vector variables, and 



dt = du, 



then t = u-\-a, 



where a is a scalar constant. 



