108, 109, 110] 



DIFFERENTIAL PARAMETER. 



331 



lim 



AV _dV 

 As ~ Js 



is defined as the derivative of V in the direction s. We may lay off 



x} T7" 



on a line through M in the direction of s a length M Q = -* and 



OS 



as we give s successively all possible directions, 

 we may find the surface that is the locus of Q. 

 Let M N (Fig. 115) be the direction of 

 the normal to the level surface at M, and 

 let MP 7 drawn toward the side on which V 

 is greater, represent in magnitude the derivative 

 in that direction. Let M ' and N be the inter- 

 sections of the same neighboring level sur- 

 face, for which V=V, with MQ and HP. 

 Then 



AV 



AV MN 



MM' MN MM' 

 As MM' approaches zero, we have 



lim 



AV 



dF 



MM' 



,. AV dV 

 lim -v-,= 5; 

 en 



MN 



Hence 



oV 

 ds 



that is, the derivative in any direction at any point is equal to the 

 projection on that direction of the derivative in the direction of the 

 normal to the level surface at that point. Accordingly all points Q 

 lie on a sphere whose diameter is M P. 



The derivative in the direction of the normal to the level surface 

 was called by Lame l ) the first differential parameter of the function V, 

 and since it has not only magnitude but direction, we shall call it 

 the vector differential parameter, or where no ambiguity will result, 

 simply the parameter, denoted by P or Py. The above theorem 

 may then be stated by saying that the derivative in any direction is 

 the projection of the vector parameter on that direction. The theorem 

 shows that the parameter gives the direction of the fastest increase 

 of the function V. 



If V is a function of a point -function #, F= /*(#), its level 

 surfaces are those of g, and 



and if 



T~> dV- dVdq // \$<7 



P == __ = *. = f' (Q) _i, 



^^ dq on ' **? on 



dn 



k, P = 



1) G. Lame. Lemons sur Us coordonnees curvilignes et leurs diverses appli- 

 cations. Paris, 1859, p. 6. 



