Math 200, Spring 2010
Handout 13
Section 14-5
The Gradient and Directional Derivatives
Definition of the Gradient Vector
If
f
is a function of
x
and
y
, then the
gradient
of
f
is the vector function
f
∇
(“del
f
” or
grad
f
) defined by
(
)
(
)
(
)
,
,
,
,
x
y
f
f
f
x y
f
x y
f
x y
x
y
∂
∂
∇
=
=
+
∂
∂
i
j
The gradient of a function
(
)
,
f
x y
at a point
(
)
,
P
a b
=
is the vector
(
)
(
)
(
)
,
,
,
,
P
x
y
a b
f
f
f
a b
f
a b
∇
= ∇
=
.
In three variables, if
(
)
, ,
P
a b c
=
,
(
)
(
)
(
)
, ,
,
, ,
,
, ,
P
x
y
z
f
f
a b c
f
a b c
f
a b c
∇
=
.
The gradient
f
∇
“assigns” a vector
P
f
∇
to each point in the domain of
f
.
1.
If
(
)
2
,
f
x y
y
x
=
,
(a) find the gradient of
f
and
(b) evaluate the gradient at point
(
)
1,2
P
2.
Use the chain rule for gradients to find the gradient of
(
)
,
,
xyz
g x y z
e
=
Theorem:
Chain Rule for Paths
If
f
is a differentiable function and
( )
( )
( )
( )
(
)
,
,
t
x t
y t
z t
=
c
is a differentiable path, then
( )
(
)
( )
( )
t
d
f
t
f
t
dt
′
= ∇
c
c
c
i
Explicitly, in the case of two variables, if
( )
( )
( )
(
)
,
t
x t
y t
=
c
is a differentiable path in
2
R
, then
( )
(
)
( )
( )
( )
( )
,
,
t
d
f
f
f dx
f dy
f
t
f
t
x
t
y
t
dt
x
y
x dt
y dt
∂
∂
∂
∂
′
′
′
= ∇
=
=
+
∂
∂
∂
∂
c
c
c
i
i