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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“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
k1
)|
is bounded for all partitions P of [a, b] given by a = x
0
< x
1
< ···<
x
n
= b. In this case, the length ...
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Publisher Resources

ISBN: 9781299447561Publisher Website