Page 153 - Vector Analysis

§4.8 Green’s Theorem 149

()
Proof 2. Let F(x, y, z) = M (x, y), N (x, y), 0 be a vector-valued function defined in a

subset of R3. Then

curlF = (0, 0, Nx ´ My) ;

thus the Stokes theorem implies that

¿ żż ż

(M, N ) ¨ dr = F ¨ T ds = curlF ¨ N dS = (0, 0, Nx ´ My) ¨ (0, 0, 1) dA

BD BD D D

ż

= (Nx ´ My) dA . ˝

D

Proof 3. Let Σ = D ˆ tz = 0u. Then Σ is a surface with boundary and the upward-

pointing unit normal N = (0, 0, 1(). Let F : Σ Ñ R3 and u : D Ñ (R2 be vector-value)d
functions defined by F(x, y, z) = N (x, y), ´M (x, y), 0) and u(x, y) = N (x, y), ´M (x, y) ,

respectively. We note that if B D is parameterized by r(t) = (x(t), y(t), 0), then

T ˆ N = 1 ( 1(t), y 1 (t), ) ˆ (0, 0, 1) = 1 ( 1 (t), ´x 1(t), ) ;
1(t)}R3 x 0 1(t)}R3 y 0
}r }r

thus by the fact that the surface divergence operator divΣ is the same as the 2-d divergence

operator (since Σ is flat), Theorem 4.83 implies that

¿¿ ż żż

(M, N ) ¨ dr = F ¨ (T ˆ N) ds = divΣF dS = divu dA = (Nx ´ My) dA . ˝

BD BD Σ DD

Corollary 4.91. Let R (Ď R2 be a domain enclosed by a simple closed curve C which is
)
parameterized by r(t) = x(t), y(t) for t P [a, b]. Suppose r 1 points in the counter-clockwise

direction. Then żb [ ]
x(t)y y(t)x1(t) dt
the area of R= 1 a 1(t) ´ .
2

Proof. The corollary is concluded by applying Green’s theorem to the special case: M (x, y) =

´y and N (x, y) = x. ˝

Example 4.92. Compute the area enclosed by the Cardioid which has a polar representa-

tion r = (1 ´ sin θ) with θ P [0, 2π]. y

x

Figure 4.7: The Cardioid