88 ni. GENERAL PRINCIPLES. WORK AND ENERGY. 



and B, and not on the path of integration. Consequently, as stated 

 without proof in 28, the conditions 38) are sufficient as well as 

 necessary. 



If A is given, I is a point -function 1 ) of its upper limit _B, let 

 us say (p. If B is displaced a distance s in a given direction to .Z?', 

 the change in the function cp is 



B' 



VB' <PB = I (Xdx 4- Ydy + Zdi), 



/ 

 and the limit of the ratio of the change to the distance, 



38a) 



= , 



g=0 s ds ds ds ds 



is the derivative of cp in the direction s. 



If we take s successively in the directions of the axes of coordinates, 

 onrx d<p y d<f> v dy 7 



dx = x > dj ==Y > Tz = z ' 



A vector whose components are thus derived from a single scalar 

 function qp is called the vector differential parameter of (p. 



Accordingly the three equations of condition 38), equivalent to 

 curl R = 0, are simply the conditions that X, Y, Z may be represented 

 as the derivatives of a scalar point -function. In this case the 

 expression 



Xdx + Ydy + Zdz = d ^dx + d 2ydy + d ^dz = d<p 



is called a perfect differential. 



From the definition of the parameter of a scalar point -function 

 38 b), we see that the components of the vector E at any point are 

 proportional to the direction cosines of the normal to the surface 

 qp = const, passing through the point in question, that is R is 

 perpendicular to the surface. A surface for which a scalar point- 

 function is constant is called a level surface of that function. Since 



dx dy dz 



~~^ ? "~^ y ~^ } 

 ds ds ds 



are the direction cosines of the tangent to the arc ds, we see that 

 equation 38 a) states that the derivative of cp in any direction is the 

 projection of its vector parameter on that direction. Since a vector 

 is the maximum value of any of its projections, we see that the 

 direction of the normal to the level surface of <p at any point is the 

 direction of fastest increase of qp at that point. Also if we take 

 for y B ' and cp B in 38 a) the constant values belonging to two infinitely 



1) A function of the coordinates of a point. 



