Calculus: Analytic Geometry and Calculus, with Vectors

7.1 Curves and lengths 409

For an accurate treatment of matters relating to lengths of curves, we
need more information about things that are sometimes called curves
and are sometimes called oriented curves. A bumblebee can give us
preliminary ideas by starting at 4, flying to a rose at B, flying on to
another rose at C, flying back over the same route to B, and then flying
to D and on toE as in Figure 7.13. His
total path is a curve, and we get the idea A
that a curve is not determined by a point

set. We cannot know what the curve is
until we know the order in which points on
the curve were visited. For example, if
the bumblebee flies from E back to 4 by

Figure 7.13

traversing his route in reverse, his path is a new curve which can be called

the negative of the original one. While the matter has psychological

rather than logical importance to us now, we can feel quite confident that

if the bumblebee makes a flight over the interval t1 < t < t2, and if we

introduce an x, y, z coordinate system, then at each time in the interval

he is surely someplace and that if we denote his coordinates by x(t),

y(t), z(t), then these coordinate functions are continuous functions of t.

Of course, the curve does not uniquely determine the coordinate functions

because the bumblebee can fly his course at different speeds, but any one

set of appropriate coordinate functions does determine the intrinsic

curve. The above brief discussion of curves leaves many unanswered

questions. It will have served its purpose if it provides a hazy feeling

that the following definition uses words in a reasonable way.

Definition 7.14 If x, y, z are continuous functions of t over an interval,

then the ordered set of points

(7.141) P(t) = P(x(t), Y(t), z(t))

for which t lies in the interval, and for which P(t') is said to precede P(t") if

t' < t", is a curve (or oriented curve) C.

Professional creators of complicated curves can give an example of a

curve in E3 that is clever enough to "pass through" each pointin a given

cube or even in the whole E3. Such curves are space-filling curves. It

must not be presumed that all curves are complicated things, however.

For example, when z(t) = 0 for each t, the curve lies in the xy plane and

we omit the z(t) when no confusion can result. We always have the

possibility of setting x(t) = t and replacing t by x. This shows that if f

is continuous over a < x <_ b, then the set of points (x, f(x)) on the graph

of y = f(x) becomes a curve when we so order the points that (x1, f(xi))

precedes (x2, f (X2)) when a < x1 < x2 < b. Unless an explicit statement

to the contrary is made, it is presumed that the points are ordered inthis

way whenever the graph of y = f(x) is called a curve. Finally, such

things as circles, ellipses, rectangles, and triangles become curves as soon