Calculus: Analytic Geometry and Calculus, with Vectors

542 Polar, cylindrical, and spherical coordinates

For the case in which z(t) is identically zero and p(t) > 0, this gives the
formula

(10.275) cos ¢ = P'(t)

IP'(t)J= + IP(t)O'(t)J2

In case p(t) > 0, 4'(t) > 0, and p'(t) s. 0, this enables us to prove the

first formula in

p(t)4'(t)

tan

do p

(10.276) tan ¢ =

P' (t)

' ' = pdp = dp

In appropriate circumstances, the second formula follows from the first.

In accordance with conclusions reached at the end of Section 7.2, we

can put (10.272) in the form

(10.277) [P'(t)]2 + [P(t)41(t)J2 + [z'(t)]2.

where s(t) is the coordinate at time t which is obtained by measuring

distance along the curve C. When z'(t) = 0 for each t and 0(t) = t so

4'(t) = 1, this is often put in the form

(10.278)

ds Vp2+ (dpy.

To-

'0

Matters relating to lengths of curves are of sufficient interest to

justify close scrutiny of the following theorem.

Theorem 10.28 If p and 0 are functions having continuous derivatives

over a < t <_ b, then the integral in the formula

(10.281) L = f

b

a [P'(t)l2 + [P(t)0'(t)I2 dt

is the length of the curve C consisting of the ordered set of points P having

polar coordinates (p(t), 0(t)) for which a <_ t <_ b.

The simplest proof of this theorem is obtained by setting

(10.282) x(t) = p(t) cos 4(t), y(t) = p(t) sin 4(t), z(t) = 0

in the formula (7.26) which was thoroughly discussed and proved in

Section 7.2. The formulas

(10.283) x'(i) -p(t) sin cos O(t)p'(t)

(10.284) y'(t) = p(t) cos ¢(t)4'(t) + sin 0(t)p'(t)