Calculus: Analytic Geometry and Calculus, with Vectors

432 Curves, lengths, and curvatures

Since we want a formula for 0'(t), it is reasonable to differentiate (7.385)

with respect tot and to try to use the result. We divide by jv(t)I and

then differentiate to obtain

I v(t) I a(t) - v(t)dtIv(t) I

(7.386) [- sin 4(t)i + cos O(t)j]O'(t) = Iv(t)12

The coefficient of O'(t) is the unit vector n for which

(7.3861) n = [- sin 0(t)i + cos O(t)j] _

-y'(tI i + x'(t)j

V(t)1

Since 0, we can equate the scalar products of n and the mem-

bers of (7.386) to obtain

(t) =

[x"(t)i + y"(t)j].[-y'(t)i + x'(t)jl

Iv(t)12 lo(t)12

and hence

(7.387)

x,(t)y,"(t) - x"(t)y'(t).

(t) _ [x'(t)12 + [y'(t)]2

For some applications of (7.387), we can set x(t) = t so that x'(t) = 1

and x"(t) = 0. In such cases, we can replace t by x to obtain

(7.3871) (x) Y" (x)

1 + [y (x)12

In this, and in any other case in which x'(t) s 0, we can eliminate prac-

tically all of the work of this section by writing

(7.3872) tan ¢ =y,(t)

X, (t)(t)

with or without the aid of (7.3851) and then differentiating with respect

to t to obtain (7.387).

Returning to (7.387), we note that 0'(t), which might be measured in

radians per minute, gives the time rate of change of ¢ with respect to t.

In case the point P(t) traverses C with unit speed, which might be 1

kilometer per minute, 0'(t) becomes also a number of radians per unit

distance measured along C, and this is called the curvature K or K(t) of

C at P(t). To give curvature an additional leg upon which to stand, we

introduce upon C a coordinate system like that in Figure 7.27 with the

stipulation that the coordinate s of P(t) increases as t increases. The

curvature of C at P(t) can then be defined by the formula

(7.388) K = K(t) = ds-