(d) For any vector feld
,
F
in
R
3
all o± whose coordinate ±unctions
have continuous frst and second partial derivatives, we have that
div
(
curl
(
,
F
)) = 0.
TRUE:
This is one o± the things Clairaut’s Theorem tells us;
another is that, ±or a scalar ±unction
f
with continuous frst and
second partial derivatives,
curl
(
grad
(
f
)) = 0.
(e) I± the vector feld
,
F
is conservative on the open region
D
then line
integrals o±
,
F
are path-independent on
D
, regardless o± the shape
o±
D
.
TRUE:
The Fundamental Theorem o± Line Integrals tells us this.
It is necessary ±or
D
to have no holes i± we want to use the ±act
that
curl
(
,
F
)=
,
0on
D
to tell us that
,
F
is conservative on
D
.
(±) I±
,
F
is any vector feld, then
curl
(
,
F
) is a conservative vector feld.
FALSE:
We know the curl o± a conservative vector feld must be
,
0,
so i± this were true then we would always have
curl
(
curl
(
,
F
)) =
,
0
±or any vector feld
,
F
. But this is not true. (Try computing
curl
(
curl
(
xz, yz,
0)).)
2. (a) Find a potential ±unction
f
±or the vector feld
,
F
(
x, y
)=(2
x
+2
y,
2
x
y
)
.
A potential ±unction is just a ±unction
f
such that
,
F
=
∇
f
.
SOLUTION:
There are several ways to go about fnding a po-
tential ±unction. The organized antidi²erentiation methodi
sa
s
±ollows:
I±
∇
f
=
,
F
we must have
f
x
=2
x
y
(1)
f
y
x
y
(2)
Starting with Equation 1 and integrating with respect to
x
(treat-
ing
y
as a constant) we get
f
x
x
y
2