x
y
z
(0,1,1)
x
y
z
(1,1,0)
(0,0,1)
parallel translates of the one shown in the next Fgure.
C15S01.008:
The vector Feld
F
(
x, y, z
)=
h
1
,
1
,
−
1
i
is a constant vector Feld. All vectors in this Feld
are parallel translates of the one shown in the next Fgure.
C15S01.009:
Each vector in the Feld
F
(
x, y, z
h−
x,
−
y
i
is parallel to the
xy
plane and reaches from
its initial point at (
x, y, z
) to its terminal point (0
,
0
,z
)onthe
z
axis.
C15S01.010:
Each vector in the Feld
F
(
x, y, z
h
x, y, z
i
points directly away from the origin and its
length is the same as the distance from the origin to its initial point.
C15S01.011:
The vector Feld
∇
(
xy
h
y, x
i
is shown in ±ig. 15.1.8. To verify this, evaluate the gradient
at (2
,
0).
C15S01.012:
The gradient vector Feld
∇
(2
x
2
+
y
2
h
4
x,
2
y
i
is shown in ±ig. 15.1.9. To verify this,
evaluate the gradient at (2
,
2).
C15S01.013:
The gradient vector Feld
∇
(
sin
1
2
(
x
2
+
y
2
)
)
=

x
cos
1
2
(
x
2
+
y
2
)
,y
cos
1
2
(
x
2
+
y
2
)
®
is shown in ±ig. 15.1.10. To verify this, evaluate the gradient at (1
,
1) and at (0
,
1).
4