12.5 Gradiants

12.5 Gradiants -...

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Unformatted text preview: 12.5
Directional
Derivatives
and
Gradients
 
 Now
we
want
to
calculate
the
slope
at
any
point
moving
in
any
 direction.

(So
far
we
have
only
moved
in
the
+
x
and
+
y
 directions.)
 x






















y
 ‐.2
 0
 .2
 
 ‐2.2
 271.8
 270.8
 269.8
 ‐2
 278
 277
 276
 ‐1.8
 280.2
 279.2
 278.2
 Find
the
slope
moving
in
the
northeast
direction
from
(0,
‐2).
 
 
 
 
 
 
 
 
 In
general
the
slope
from
(x1,
y1)
to
(x2,
y2)
is
 m= f ( x 2 , y 2 ) − f ( x1, y1 ) ( x 2 − x1) + ( y 2 − y1) 2 2 
 Ex.

 
 1.0 0.8 0.6 0.4 0.2 0.0 0.0 
 
 
 
 
 
 
 0.2 0.4 0.6 0.8 1.0 
 Find
the
line
through
P(a,
b)
in
the
direction
of
 ˆ u = u1i + u2 ˆ = u1, u2 
which
is
a
unit
vector.
 j 
 
 
 
 
 
 Def:

The
gradient
vector
of
f(x,
y)
at
a
point
P0(a,
b)
is:
 ∇f = f x , f y .
 Def:

The
directional
derivative
of
f
in
the
direction
of
any
unit
 vector
 u 
is
 Du f ( x, y ) = f x ( x, y ) u1 + f y ( x, y ) u2 = ∇f ⋅ u 
 
 1 Ex.


 z = 2 + 2 x − y, 
 
 
 P (2, 0) v = 1, 2 
 Ex.
 f ( x, y, z) = xy + ln z P (2, − 1, 1) v = 1, 0, 4 
 
 
 
 
 
 
 
 
 
 Note:


 Du f ( x, y ) = ∇f ⋅ u = ∇f u cosθ 
 1. The
function
increases
most
rapidly
when
 cosθ = 1 or θ = 0 .

Then
 u 
is
in
the
direction
of
the
 gradient.

Therefore
the
gradient
vector
points
in
the
 direction
in
which
f
increases
most
rapidly.
 2. 
The
function
decreases
most
rapidly
when
 cosθ = −1 or −∇f 3. θ = π .

Then
 u 
is
in
the
direction
of
 .
 π 

When
 cosθ = 0 or θ = 2 , Du f ( x, y ) = 0 .

The
 gradient
is
perpendicular
to
the
contour.
 Ex.

 2 1 0 -1 -2 -2 -1 0 1 2 
 
 
 Gradient
Vector
 1. The
gradient
vector
points
in
the
direction
of
maximum
 increase
at
(a,
b)
 2. 
The
vector’s
magnitude
is
the
rate
of
change
in
that
 direction.
 3. If
the
directional
derivative
of
f
at
(a,
b)
is
zero
in
every
 direction,
then
 ∇f = 0 .
 
 
 Ex.

Suppose
T(x,
y)
=
70
+
xy
represents
level
curves
around
a
 heat
source.

Let
P(2,
‐1)
be
the
point
where
you
are
sitting.
 1. How
hot
is
it?
 2. 
What
is
the
equation
of
the
level
curve
you
are
on?
 3. If
you
move
in
the
direction
of
 A = −1, 1 
is
it
getting
 hotter
or
cooler?
 4. Find
the
direction
of
maximum
temperature
increase.

 What
is
the
rate
of
change
in
this
direction?
 5. Find
the
direction
of
maximum
temperature
decrease.
 6. Find
the
direction
to
stay
at
your
current
temperature.
 7. Find
the
direction
in
which
Duf
=
1
 
 
 
 
 
 
 
 
 
 
 
 
 
 Note:

The
gradient
vector
doesn’t
necessarily
point
to
the
 steepest
point
(peak).

It
points
to
the
next
contour
in
the
 direction
of
maximum
increase
to
that
contour.
 The
magnitude
of
the
gradient
is
large
when
the
contours
are
 close
together
and
small
when
they
are
far
apart.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 F(x,y)
=
cos(x)sin(y)
 
 
 
 
 
 ...
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