“real: chapter_07” — 2011/5/22 — 23:09 — page 17 — #17
Functions of Bounded Variation 7-17
It should be emphasized here that a curve has two components one
f and the other its image. Thus, there may be several curves whose
images are the same but the functions may be different. These are
considered as different curves.
Examples 7.5.2
1. The curve {(t, f (t))/t ∈[0, 2π]}, where f (t) = e
it
describes the
unit circle in the plane.
2. The curve {(t, g(t))/0 ≤ t ≤ 1}, where g(t) = ta + (1 −
t)b, (a, b ∈
R
2
), describes the line segment joining a and b in R
2
.
Definition 7.5.3 Let γ : [a, b]→
R
2
be a curve. We say that γ is
rectifiable if
(γ , P) =
n
k=1
|γ(x
k
) − γ(x
k−1
)|
is bounded for all partitions P of [a, b] given by a = x
0
< x
1
< ···<
x
n
= b. In this case, the length ...