340 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Of all the numerical values which the derivative of u can have at a 

 given point, the greatest is to be found by making s normal to the level 

 surface of u which passes through the point. This maximum value, 



[(=)' + (S v 



day 

 dz~) J 



(3) 



is usually regarded as the value at the point of a vector point function 

 called the gradient vector of u, the lines of which cut orthogonally the 

 level surfaces of u, and the components of which parallel to the coord- 

 inate axes are 



du du du . 



dx by dz 



This vector is, of course, lamellar. 



The value of the tensor of the gradient vector is often called simply 

 the "gradient" of u and is denoted by h u . If at any point a straight 

 line be drawn in the direction (n) normal to the level surface of u, 

 in the sense in which u increases, and if a length h u be laid off on this 

 line, the projection, 



h u ■ cos (n, s), (5) 



of this length on any other direction (s) is numerically equal to the 

 derivative of u in the direction s. 



Most physical quantities — such as temperature, barometric pres- 

 sure, density, inductivity — present themselves to the investigator as 

 single valued point functions, which, except perhaps at one or more 

 given surfaces of discontinuity, are differentiable in the sense just 

 considered. 



It is often desirable to differentiate a scalar function, u, at a point, 

 in the direction in which another scalar function, v, increases fastest, 

 and if (u, v) represents the angle between the gradient vectors of u and 

 v at the point, the derivative is evidently equal to 



h u • cos (u, v). (6) 



It frequently happens that in a question of maxima and minima, 

 one wishes to determine the greatest (or the smallest) value which a 

 quantity U may have, subject to the condition that another quantity 

 V shall have a given value ( T ). If these quantities can be represented 

 by point functions, the problem geometrically considered requires one 

 to find the parameter of a surface of the constant U family, which is 

 tangent to the surface of the V family upon which V is everywhere 



