Page 154 - Vector Analysis

150 CHAPTER 4. Vector Calculus

Given the polar representation r = (1 ´ sin θ), a parametrization of the Cardioid is

( )( )
r(t) = x(t), y(t) = (1 ´ sin t) cos t, (1 ´ sin t) sin t t P [0, 2π] .

Then Corollary 4.91 implies that the area enclosed by the Cardioid is

1 ż 2π [ ´ ( )
2 (1 (1 sin t) cos t ´ cos t sin t + (1 ´ sin t) cos t
0 ´
( )]
´ sin t) sin t´ cos2 t ´ (1 ´ sin t) sin t dt

= 1 ż 2π ´ sin [ cos2 t ´ 2 sin t cos2 t + sin t cos2 t + sin2 t ´ sin3 ]
2 (1 t) t dt

0

= 1 ż 2π ´ sin t)(1 ´ sin t cos2 t ´ sin3 t)dt = 1 ż 2π ´ sin t)2dt = 3π .
2 (1 2 (1 2

0 0

Before finishing this chapter, we would like to establish an unproven theorem: Theorem
4.33. We recall Theorem 4.33 as follows.

Theorem 4.33. Let D Ď R2 be simply connected, and F = (M, N ) : D Ñ R2 be of class
C 1. If My = Nx, then F is conservative.

¿

Proof of Theorem 4.33. By Theorem 4.30, it suffices to show that F ¨ dr = 0 for all
piecewise smooth closed curve C P D. Nevertheless, if C is a piecewiseC closed curve and R

is the region enclosed by C, by the fact that D is simply connected, we must have B R = C.

Therefore, Green’s theorem implies that

¿ż ˝

(M, N ) ¨ dr = (Nx ´ My) dA = 0 .

CR