C15S01.014:
The gradient vector field
∇
(
sin
1
2
(
y
2
−
x
2
)
)
=
−
x
cos
1
2
(
y
2
−
x
2
)
, y
cos
1
2
(
y
2
−
x
2
)
is shown in Fig. 15.1.7. To verify this, evaluate the gradient at the point (1
,
1).
C15S01.015:
If
F
(
x, y, z
) =
x, y, z
, then
∇·
F
= 1 + 1 + 1 = 3
and
∇×
F
=
i
j
k
∂
∂x
∂
∂y
∂
∂z
x
y
z
= 0
,
0
,
0 =
0
.
C15S01.016:
If
F
(
x, y, z
) = 3
x,
−
2
y,
−
4
z
, then
∇·
F
= 3
−
2
−
4 =
−
3
and
∇×
F
=
i
j
k
∂
∂x
∂
∂y
∂
∂z
3
x
−
2
y
−
4
z
= 0
,
0
,
0 =
0
.
C15S01.017:
If
F
(
x, y, z
) =
yz, xz, xy
, then
∇·
F
= 0 + 0 + 0 = 0
and
∇×
F
=
i
j
k
∂
∂x
∂
∂y
∂
∂z
yz
xz
xy
=
x
−
x, y
−
y, z
−
z
=
0
.
C15S01.018:
If
F
(
x, y, z
) =
x
2
, y
2
, z
2
, then
∇·
F
= 2
x
+ 2
y
+ 2
z
and
∇×
F
=
i
j
k
∂
∂x
∂
∂y
∂
∂z
x
2
y
2
z
2
= 0
,
0
,
0 =
0
.
C15S01.019:
If
F
(
x, y, z
) =
xy
2
, yz
2
, zx
2
, then
∇·
F
=
y
2
+
z
2
+
x
2
and
∇×
F
=
i
j
k
∂
∂x
∂
∂y
∂
∂z
xy
2
yz
2
zx
2
=
−
2
yz,
−
2
xz,
−
2
xy .