7.2 Lengths and integrals 41717 Sketch some figures to obtain preliminary ideas about matters relating to
the following definition. Let S be a given nonempty set in the xy plane and let
P' be a point of S. Let S' be the set of points P" in S that can be connected to
P' by curves lying in S. This means that P" is a point of S' if and only if there
exist functions x(t) and y(t) continuous over an interval a 5 t S b such that
P(x(a), y(a)) is P', P(x(b), y(b)) is P", and P(x(t), y(t)) is a point in S whenever
a < t 5 b. The set S' is a connected set because if P, and P2 lie in S', then
Pi and P2 can be connected to P' by curves C, and C2 lying in S' and these two
curses can be combined to give a single curve C connecting P, to P2 and lying in
S'. This set S' is the maximal connected subset of S that contains P' or, briefly, the
component of S that contains P'.
18 The curve C of Problem 15 is said to have a multiple point at the point Q
if there exist two numbers t, and t2 such that a <_ ti < t2 < b (or a < tl < to 5 b)
and the two points P(ti) and P(t2) coincide with Q. The curve is said to be simple
(free from multiple points) if it has no multiple points. Give some examples of
closed curves that have multiple points and some examples of simple closed
curves.
19 The French mathematician Camille Jordan (1838-1922) was the first
person to give serious attention to the difficult question whether each simple
closed plane curve "separates the plane" into exactly two nonempty components
one of which constitutes the outside of the curve and the other of which con-
stitutes the inside. Such curves are called Jordan curves. The following
theorem is known as the Jordan curve theorem.
Theorem If C is a simple closed plane curve (or Jordan curve), then the set S of
points of the plane which are not points of C contains exactly two (no more and no
fewer) nonempty components S, and S2.
The author now owes his readers an explanation. It never has been and need
not be expected that students of elementary calculus know anything about the
Jordan curve theorem. Nevertheless, the author insists that each student should
have opportunities to pick up ideas about mathematics. We can know that the
Jordan curve theorem is so difficult that Jordan never succeeded in proving it
and that we must learn more mathematics before we can undertake to under-
stand proofs that have been given or to construct new proofs. We can know
that many persons who never generated much interest in additions of fractions
become intensely interested in problems involving sets and curves.
7.2 Lengths and integrals Let x, y, and z be given functions having
continuous derivatives over the closed interval a _<_ t:_!9 b. Let(7.21) P(t) = P(x(t), y(t), z(t))so that, for each t, P(t) is the point having coordinates displayed in (7.21).
Let r(t) denote the vector running from the origin to P(t) so that, for
each t,(7.22) r(t) = x(t)i + y(t)j + z(t)k.
We propose to set up an integral for the length ICI of the curve C traversed