Transcript Section 17.1 - Gordon State College
Section 17.1
Vector Fields
VECTOR FIELDS
Definition:
Let
D
vector field on
2 2 is a (vector-valued) function
F
assigns to each point (
x
,
y
) in
D
that a two-dimensional vector
F
(
x
,
y
).
Since
F
(
x
,
y
) is a two-dimensional vector, we can write it in terms of its
component functions
P
and
Q
as follows:
F
(
x
,
y
)
P
(
x
,
y
)
i
Q
(
x
,
y
)
j
P
(
x
,
y
),
Q
(
x
,
y
)
F
P
i
Q
j
The functions
P
and
Q
are sometimes called
scalar functions
.
EXAMPLES
Sketch the following vector fields.
1.
F
(
x
,
y
) = − yi + xj 2.
F
(
x
,
y
) = 3 xi + yj
VECTOR FIELDS (CONCLUDED)
Definition:
Let
E
3
vector field on
3 is a function
F
that assigns to each point ( three-dimensional vector
F
(
x
,
y
,
z
).
x
,
y
,
z
) in
E
a We can express
F
in terms of its component functions
P
,
Q
, and
R
as
F
(
x
,
y
,
z
) =
P
(
x
,
y
,
z
)
i
+
Q
(
x
,
y
,
z
)
j
+
R
(
x
,
y
,
z
)
k
Three dimensional vector fields can be sketched in space. See Figures 9 through 12 on page 1094.
PHYSICAL EXAMPLES OF VECTOR FIELDS
• •
Velocity fields
describe the motions of systems of particles in the plane or space.
Gravitational fields
are described by
Newton’s Law of Gravitation
, which states that the force of attraction exerted on a particle of mass
m
located at
x
=
x
,
y
,
z
by a particle of mass
M
located at (0, 0, 0) is given by
F
(
x
)
mMG
x
|
x
| 3 where
G
is the gravitational constant.
PHYSICAL EXAMPLES OF VECTOR FIELDS (CONTINUED)
•
Electric force fields
are defined by
Coulomb’s Law
, which states that the force exerted on a particle with electric charge
q
located at (
x
,
y
,
z
) by a particle of charge
Q
located at (0, 0, 0) is given by
F
(
x
)
x x
3 where
x
= xi + yj + zk and
ε
is a constant that depends on the units for |
x
|,
q
, and
Q
.
INVERSE SQUARE FIELDS
Let
x
(
t
) =
x
(
t
)
i
+
y
(
t
)
j
+
z
(
t
)
k
be the position vector. The vector field
F field
if is an
F
(
x
)
k
|
x
| 3
x inverse square
where
k
is a real number.
Gravitational fields and electric force fields are two physical examples of inverse square fields.
GRADIENTS AND VECTOR FIELDS
Recall that the gradient of a function
f
(
x
,
y
,
z
) is a vector given by
f
(
x
,
y
,
z
)
f x
(
x
,
y
,
z
)
i
f y
(
x
,
y
,
z
)
j
f z
(
x
,
y
,
z
)
k
Thus, the gradient is an example of a vector field and is called a
gradient vector field
.
NOTE: There is an analogous gradient field for two dimensions.
CONSERVATIVE VECTOR FIELDS
Definition:
A vector field
F
is called a
conservative vector field
if it is the gradient of some scalar function
f
differentiable function , that is if there exists a
f
such that . The function
f
F
.
is called the
potential function
for
EXAMPLES
1. Show that
F
(
x
,
y
) = 2 xi + yj is conservative.
2. Show that any inverse square field is conservative.