Math2011, Vector Calculus
5.1
Vector Fields
Definition 5.1
A vector field in the
n
dimensional space is a function in which
a vector in the
n
dimensional space is plugged in and a vector in the
n
dimen
sional space is returned.
Example 5.2
Let
φ
be a function in
n
variables.
∇
φ
is a vector field in the
n
dimensional space.
Recall that
∇
φ
(
x
1
, ..., x
n
) = (
∂φ
∂x
1
(
x
1
, ..., x
n
)
, ...,
∂φ
∂x
n
(
x
1
, ..., x
n
)). We plug in a
vector in the
n
dimensional space into
∇
φ
and get a vector in the
n
dimensional
space. Thus,
∇
φ
is a vector field in the
n
dimensional space.
Definition 5.3
A vector field
F
in the
n
dimensional space is conservative if
there exists a function
φ
in
n
variables such that
F
=
∇
φ
. In this case,
φ
is
often called a potential function of
F
also.
Example 5.4
Let
F
(
x, y, z
) =

1
(
x
2
+
y
2
+
z
2
)
3
/
2
(
x, y, z
)
be a vector field in the
three dimensional space. Show that
F
is a conservative vector field.
proof:
Let
φ
(
x, y, z
) =
1
√
x
2
+
y
2
+
z
2
be a function in three variables.
∇
φ
(
x, y, z
)
= (

x
(
x
2
+
y
2
+
z
2
)
3
/
2
,

y
(
x
2
+
y
2
+
z
2
)
3
/
2
,

z
(
x
2
+
y
2
+
z
2
)
3
/
2
)
=

1
(
x
2
+
y
2
+
z
2
)
3
/
2
(
x, y, z
) =
F
(
x, y, z
)
.
Thus
F
is a conservative vector field.
Example 5.5
Let
F
(
x, y, z
) = (0
, x,
0)
be a vector field.
Show that
F
is not
conservative.
proof:
Assume that
F
is conservative. i.e. there exists a function
φ
in 3 variables such
that
F
=
∇
φ
. So,
(0
, x,
0) =
F
(
x, y, z
) =
∇
φ
(
x, y, z
) = (
∂φ
∂x
,
∂φ
∂y
,
∂φ
∂z
)(
x, y, z
)
.
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