Problem 1.
It is easy to see that the tangent vector to the curve (
r
(
t
)
,θ
(
t
)) in polar coordi
nates has components ˙
r
and
r
˙
θ
. So, the length of a curve is
l
=
Z
b
a
q
(˙
r
(
t
))
2
+ (
r
(
t
)
˙
θ
(
t
))
2
dt.
Using this formula we compute length of the curve (2sin
πt,πt
):
l
=
Z
1
0
p
4
π
2
cos
2
πt
+ 4
π
2
sin
2
πtdt
= 2
π.
Problem 2.
Let
P
=
{
x
0
,...,x
n
}
be a partition of [
a,b
]. Then we can construct partition
Q
=
{
y
1
...,y
n
}
of [
c,d
] such that
x
i
=
φ
(
y
i
). Also, for any partition of [
c,d
] we can
construct corresponding partition of [
a,b
]. Since
g
=
f
◦
φ
, values of
f
on [
x
i

1
,x
i
]
are the same as values of
g
on [
y
i

1
,y
i
]. So, we have
U
(
Q,g,β
) =
U
(
P,f,α
)
,
L
(
Q,g,β
) =
L
(
P,f,α
)
.
The statement follows from these equalities and Theorem 6.6 from [Rudin].
Problem 3.
We deﬁne
d
(
E,F
) =
m
((
F
∪
E
)
\
(
F
∩
E
)). Let
d
(
E,F
) = 0 and
d
(
F,G
) = 0.
First, we see that
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 Winter '08
 Zinchenko,M
 Topology, Equivalence relation, Metric space, Topological space, polar coordi˙ nates

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