Page 152 - Vector Analysis

148 CHAPTER 4. Vector Calculus

thus

(F ˝ r)(t) ¨ r 1(t) = ´ sin2(sin t) sin2 t cos t + sin(sin t) cos(sin t) sin t cos t
´ cos2(sin t) sin2 t cos t ´ sin(sin t) cos(sin t) sin t cos t

= ´ sin2 t cos t .

As a consequence,

¿ ż 2π 1 ˇt=2π
´ ´ tˇ
F ¨ dr = sin2 t cos t dt = sin3 = 0.
3 ˇt=0
C0

On the other hand,

ż ż π ż π´sin ϕ

curlF ¨ N dS = (0, 0, ´2) ¨ (cos θ sin ϕ, sin θ sin ϕ, cos ϕ) sin ϕ dθdϕ

Σ 0 sin ϕ

żπ

= ´2 sin ϕ cos ϕ(π ´ 2 sin ϕ) dϕ

= (π 0 + 4 sin3 )ˇϕ=π = 0.
ϕˇ
cos 2ϕ
2 3 ˇϕ=0

4.8 Green’s Theorem

In most of materials Green’s theorem is introduced prior to the divergence theorem and the
Stokes theorem; however, we treat Green’s theorem as a corollary of the divergence theorem
(Theorem 4.75), the Stokes theorem (Theorem 4.86) and Theorem 4.83.

Theorem 4.90 (Green’s theorem). Let D be a bounded domain whose boundary B D is
piecewise smooth, and M, N : D Ñ R be of class C 1. Then

¿ż

(M, N ) ¨ dr = (Nx ´ My) dA ,

BD D

where the line integral (on the left-hand side of the identity above) is taken so that the curve

is counter-clockwise oriented.

Proof 1. Let u(x, y) = () be a vector-valued function defined on the 2-
N (x, y), ´M (x, y) ()

dimensional domain D. Suppose that B D is parameterized by r(t) = x(t), y(t) for t P [a, b],

where r 1 points in the counter-clockwise direction. Then with N denoting the outward-

pointing unit normal of B D, the divergence theorem implies that

¿ ¿ żż

(M, N ) ¨ dr = u ¨ N ds = divu dA = (Nx ´ My) dA . ˝

BD BD D D