Calculus: Analytic Geometry and Calculus, with Vectors

436

shows that the approximate formula

(2) 2R

Curves, lengths, and curvatures

gives good results when x is small in comparison to R. It is a quite remarkable

fact that if h and R are measured in miles and if the height H of the object is

measured in feet so that H = 5280h, then we can put R = 3960 in (2) and multi-

ply by 5280 to obtain

H=-2x2.

Putting H = 6 shows that only the hair on the top of the head of a man 6 feet

tall is visible from a point on the earth 3 miles away. Putting x = 30 gives

H = 600 and shows that less than the top half of the Empire State Building

can be seen by persons on a ship 30 miles away.

16 It is sometimes useful to have formulas obtainable from (7.361) and

(7.362). Letting N [where the N can make us think of a normal to C at P(t)] be

the vector running from P(t) to the center (X,Y) of curvature, we see that

IXI(t)]2 + [y'(t)12 [-Y'(t)i + x'(t)j]-

(1) N=x'(t)y"(t) - x"(t)Y'(t)

Let b (where the b can make us think of binormal or "second normal") be the

unit vector in the direction of v(t) X a(t) so that, as (7.32) shows, b is the coordi-

nate vector k in our work. Then, with the aid of (7.32) and the fact that

1, we can put (1) in the intrinsic form

(2) N =

v(t) v(t)

[v(t) X a(t)]-b(t)b(t) X v(t)

in which coordinates do not appear. The formula (2) is valid when C is a curve

in E3 for which x,y,z are functions having continuous second derivatives and

r(t) = x(t)i + y(t)j + z(t)k

v(t) = x'(t)i + y'(t)j + z'(t)k

a(t) = x"(t)i + y"(t)j + z"(t)k

whenever t is such that v(t) x a(t) 3-1 0. We shall not prove this, but remark

that the point analogous to the point (t,q) of Figure 7.33 is the intersection

of three planes and that (X,Y,Z) is the limit as At --+ 0 of

17 It is possible to obtain very informative formulas by considering the

motion of a particle which moves along a plane curve C, endowed with coordinates

as in Figure 7.27, in such a way that s increases as t increases. We assume

existence and continuity of all the derivatives we want to use, and we assume that

dx/dt > 0. Let t be the unit forward tangent vector to C at time t so that

(1) t = cos 4i + sin dij,

where ¢ is, as in the discussion of Figure 7.384, an angle giving the direction of t

at time t. Show that differentiating (1) gives

(2)

A

= [- sin ¢i + cos Oj] - _ n

d- ds_ds

dt dt ds dt WtKn,