Properties of the Gradient   

If we let q denote the angle between Ñf and a given unit vector u, then
Duf = || Ñf||  || u|| cos( q) = || Ñf||  cos( q)
since || u|| = 1. Thus, the directional derivative is largest when q = 0, which is in the direction of Ñf, and is smallest when q = p, which is in the direction of -Ñf.  

 

Theorem 7.2: The slope ( Duf) ( p) of the tangent line to a vertical slice of z = f(x,y) through ( p,q,f( p) ) is greatest in the direction of Ñf( p) and is smallest in the direction of f( p) .

We say that Ñf( p,q) is the direction of steepest ascent for the surface z = f( x,y) at (p,q) . Analogously, f( p,q) must the direction of steepest descent or direction of least resistance at the point ( p,q) .

Moreover, when u is the direction vector for Ñf, then u = Ñf/||Ñf||, so that
Duf  =  Ñf ·  Ñf
||Ñf||
 =   ||Ñf||2
||Ñf||
 =  ||Ñf||
Similarly, if u is in the direction of f, then Duf = -||Ñf||. Also, the corresponding tangent vectors to the surface are
fastest increase
:
vinc =
 fx
||Ñf||
,  fy
||Ñf||
,||Ñf||
fastest decrease
:
vdec =
- fx
||Ñf||
, -fy
||Ñf||
,-||Ñf||  
This is important in many numerical routines.
LiveGraphics3d Applet

         

EXAMPLE 4    How fast is f( x,y) = x2+y3 increasing at ( 2,1) when it is increasing the fastest?       

Solution: To begin with, Ñf = á2x,3y2 ñ . Moreover, f is increasing the fastest when u is in the direction of Ñf( 2,3) = á4,3 ñ , and in this direction, we have
( Duf) ( 2,1) = || á4,3 ñ || = 5
(The notation ( Duf) ( p,q) means the directional derivative Duf evaluated at the point (p,q) . )

       

Moreover, given a collection of increasing constants k1 < k2 < k3 < ¼, the set of level curves
f( x,y) = k1,    f( x,y) = k2,    f(x,y) = k3,    and so on
forms a family of level curves in the xy-plane in which the levels are increasing. It then follows that Ñf( p,q) points in the direction in which the levels are increasing the fastest, and -Ñf( p,q) points in the direction in which the levels are decreasing the fastest.

       

EXAMPLE 5    In what direction are the levels of f(x,y) = x3y increasing the fastest at the point ( 2,3) ?       

Solution: The levels of f are increasing the fastest in the direction of Ñf( 2,3) . Since Ñf = á3x2y,x3 ñ , we have Ñf( 2,3) = á 36,8 ñ . Thus, the levels of f are increasing the fastest in the direction
u  Ñf( 2,3)
|| Ñf(2,3) ||
 =    á 36,8 ñ
|| á 36,8 ñ ||
 = 

 9
85
,  2
85


        Check Your Reading: How fast are the levels increasing at right angles to Ñg?