Chapter 7: Hints and (partial) Solutions
Exercise 7.1
(a) Velocity:
˙
α
(
t
) = (
α
(
t
)
,
(1
,
2
t
)).
Acceleration:
¨
α
(
t
) = (
α
(
t
)
,
(0
,
2)).
Speed:
k
˙
α
k
(
t
) =
k
(
α
(
t
)
,
(1
,
2
t
))
k
=
√
1 + 4
t
2
.
(b) Velocity:
˙
α
(
t
) = (
α
(
t
)
,
(

sin
t,
cos
t
)).
Acceleration:
¨
α
(
t
) =

(
α
(
t
)
,
(cos
t,
sin
t
)) =

(
α
(
t
)
,α
(
t
)).
Speed:
k
˙
α
k
(
t
) =
k
(
α
(
t
)
,
(

sin
t,
cos
t
))
k
=
p
sin
2
t
+ cos
2
t
= 1.
(d) Velocity:
˙
α
(
t
) = (
α
(
t
)
,
(

sin
t,
cos
t,
1)).
Acceleration:
¨
α
(
t
) =

(
α
(
t
)
,
(cos
t,
sin
t,
0)).
Speed:
k
˙
α
k
(
t
) =
k
(
α
(
t
)
,
(

sin
t,
cos
t,
1))
k
=
p
sin
2
t
+ cos
2
t
+ 1 =
√
2.
Exercise 7.2
Hint
: Constant speed implies
d
dt
(
k
˙
α
(
t
)
k
2
)
= 0. But
k
˙
α
(
t
)
k
2
= ˙
α
(
t
)
·
˙
α
(
t
).
Exercise 7.5
Hint
: If
α
is of the from indicated in this problem then argue that
α
(
t
)
∈
S
for any
t
∈
R
(Note: a
point
p
∈
S
iﬀ
p
has coordinates (
x,y,z
) with
x
2
+
y
2
=
r
2
). Now ﬁnd a vector at
α
(
t
) that is orthogonal to
S
at
α
(
t
). (Note that
n
p
= (
p,
(
x,y,
0)) is orthogonal to
S
at
p
= (
x,y,z
), why?). Now argue that ¨
α
(
t
) = (
α
(
t
)
,
¨
α
(
t
)) is a
multiple of
n
α
(
t
)
, i.e., is orthogonal to
S
. Conclude
α
is a geodesic in
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 Spring '09

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