Page 103 - Vector Analysis

§4.1 The Line Integrals 99

Example 4.11. The length of the elliptic helix C parameterized by

γ(t) = (a cos t, b sin t, ct) t P [ π ]
0, 2

can be computed by

ż π ż π
2 2
}γ 1(t)}R3dt a sin2 b2 cos2 c2
ℓ(C ) = = a2 t + t + dt .

00

1. When a ă b, letting k = c b2 ´ a2 , then
+ c2
b2

? ż π
b2 2
c2 a k2 sin2
ℓ(C ) = + 1 ´ t dt .

0

2. When a ą b, letting k = c a2 ´ b2 , then
+ c2
a2

? ż π ? ? ż π
a2 2 1 a2 2
c2 k2 cos2 c2 a k2 sin2
ℓ(C ) = + ´ t dt = + 1 ´ t dt .

00

The integral E(k, ϕ) ” żϕ? 1 ´ k2 sin2 t dt, where 0 ă k2 ă 1, is called the elliptic integral

function of the 0 kind, and E(k) ” ( π ) is called the complete el liptic
E k,
second 2

integral of the second kind.

Definition 4.12. Let C Ď Rn be a curve with finite length. An arc-length parametriza-
tion of C is an injective parametrization γ : [a, b] Ñ Rn such that the length of the curve
γ([a, s]) is exactly s ´ a; that is,

()
ℓ γ([a, s]) = s ´ a @ s P [a, b] .

Example 4.13. Let C be the circle centered at the origin with radius R. Then the

parametrization ( )
R cos
γ(s) = s , R sin s s P [0, 2πR] ,

RR

is an arc-length parametrization of C. To see this, we note that

( ) = żs ››γ 1(t)››R2 dt = żs ››(´sin s , cos s )› dt = żs dt = s @ s P [0, 2πR] .
ℓ γ([0, s]) R R ›R2
0 0 0