Suppose you are given a topographical map and want to see how steep it is from a point
that is neither due West or due North. Recall that the slopes due north
and due west are the two partial derivatives. The slopes in other
directions will be called the directional derivatives. Formally, we
be a differentiable function and let u
be a unit vector then the directional derivative of f
in the direction of u
Note that if u is i then the directional derivative is just
if u is j the it is
Just as there is a difficult and an easy way to compute partial derivatives,
there is a difficult way and an easy way to compute directional derivatives.
be a differentiable function, and
u be a unit vector with direction
f(x,y) = 2x + 3y2 - xy
v = <3,2>
fx = 2 - y
fy = 6y -
Dv f(x,y) =
<2 - y, 6y -
x> . <3/,
(2 - y) +
(6y - x)
Find Dv f(x,y)
The gradient has a special place among directional derivatives. The
theorem below states this relationship.
If grad f(x,y) = 0
then for all
f(x,y) = 0
The direction of
grad f(x,y) is the
direction with maximal directional derivative.
The direction of
-grad f(x,y) is
with the minimal directional derivative.
gradf(x,y) = 0
Du f(x,y) =
grad f .
u = 0 . u = 0
Du f(x,y) = grad f . u =
||grad f || cos q
This is a maximum when q
= 0 and a minimum when
q = p. If q
= 0 then grad f and u point in the same direction. If
= p then u and
grad f point in opposite directions.
This proves 2 and 3.
Suppose that a hill has altitude
x2 - y
Find the direction that is the steepest uphill and the steepest downhill at the
grad w = <2x,
-y> = <4, -3>
Hence the steepest uphill is in the direction
while the steepest downhill is in the direction
-<4,-3> = <-4,3>
The Gradient and Level Curves
If f is differentiable at (a,b) and
grad f is
nonzero at (a,b) then grad f is
perpendicular to the level curve through (a,b).
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