Page 122 - Vector Analysis

118 CHAPTER 4. Vector Calculus

and

ˇm n c( i ´ 1 L, b + j ´1M)L W ż ?g ˇ ε if n, m ě N .
ga ´ dAˇ
ÿ ÿ

ˇ + ă

ˇ n m nm [a,a+L]ˆ[b,b+W ] ˇ2

j=1 i=1

Then for n, m ě N, with (h, k) denoting (L, W ) (4.11) implies that
m
n

ˇż ?g dA ˇ
ˇ the surface area of ψ([a, a + L] ˆ [b, b + W ]) ´ ˇ

ˇˇ

[a,a+L]ˆ[b,b+W ]

ˇm n
= ˇ ÿ ÿ the surface area of ψ([a + (i ´ 1)h, a + ih] ˆ [b + (j ´ 1)k, b + jk])

ˇ

j=1 i=1

ż ?g dA ˇ
´ ˇ

ˇ

[a,a+L]ˆ[b,b+W ]

ˇ m n b( ) ż ?g ˇ
ˇ ga 1)k hk ˇ
ÿ ÿ dA

ď ˇ + (i ´ 1)h, b + (j ´ ´ ˇ

j=1 i=1 [a,a+L]ˆ[b,b+W ]

mn ˇ
ˇÿÿ

+ ˇ f (a + (i ´ 1)h, b + (j ´ 1)k; h, k)hk ˇ
ˇˇ

j=1 i=1

ăε+ ε mn

2 2LW ÿÿ
hk = ε .

j=1 i=1

The discussion above verifies the following

Theorem 4.47. Let Σ Ď R3 be a regular C 1-surface, tV, ψu be a local C 1-parametrization
of Σ at p, and g be the first fundamental form associated with tV, ψu. Then

ż ?g dA .
the surface area of ψ(V) =

V

Example 4.48. Recall from Example 4.46 that the first fundamental form g of the parametriza-

tion tV, ψu of the 2-sphere centered at the origin with radius R, where

ψ(θ, ϕ) = (R cos θ sin ϕ, R sin θ sin ϕ, R cos ϕ)

and V = (0, 2π) ˆ (0, π), is given by g(θ, ϕ) = R4 sin2 ϕ. Therefore,

( )ż R2 sin ϕ d(θ, ϕ)
the surface area of ψ (0, 2π) ˆ (0, π) =

(0,2π)ˆ(0,π)

ż 2π ż π

= R2 sin ϕ dϕdθ = 4πR2 .

00

()
Since the difference of the 2-sphere and ψ (0, 2π) ˆ (0, π) has zero area, we find that the

surface area of the 2-sphere with radius R is 4πR2.