6 x x 6.1 t P P = P t P = I P P P 1 0 1 0,, 0 1 0 1 cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ x θ x θ P x P x, P ) = t P x)p ) = t x t P P ) = t x = x, ) 6.1) x = Figure 6.1 Px = x, P=, θ = θ P θ x P Px θ Figure 6.1: 81
6. n P j j j = 1,,, n P = ) 1 n t t 1 1 t 1 1 t 1 n t t ) t t 1 t n P P = 1 n =....... t t n n t 1 n t n n t P P i, j) i j t i j = i, j ) t P P = I i, j ) = { 1 i = j) 0 i j) 6.) 1 n 1 n P i u i i = 1,,, n P =. u 1 u t 1 u 1 u t 1 u u t 1 u n P t u P = t ) u t 1 u t u t u 1 u t u u t u n u n =....... u n u t n u 1 u t n u u t n u n P t P i, j) u i u j u t i u j = u i, u j ) P t P = I u 1 u u n u i, u j ) = { 1 i = j) 0 i j) 6.3) u 1 u u n P = ) p q r s p + r = 1 p = cos θ r = sin θ θ q + s = 1 q = cos α s = sin α α P 1 0 = pq + rs = cos θ cos α + sin θ sin α = cosθ α) = cosα θ), 8
α = θ ± π + nπ n = 0, ±1, ±, α = θ + π + nπ cos α = cosθ + π ) = sin θ sin α = sinθ + π ) = cos θ ) cos θ sin θ P = sin θ cos θ 6.4) α = θ π + nπ cos α = cosθ π ) = sin θ sin α = sinθ π ) = cos θ P ) cos θ sin θ cos θ sin θ 1 0 P = = sin θ cos θ sin θ cos θ 0 1 6.) 6.4) 6.) ) 0.8 0.6 r s r s θ 6.4) 6.) ) ) ) 1) 1 3 1 ) 1 3 1 1 3 1 3) 1 3 1 3 1 3 ) 4) 1 3 1 1 3 ) 1 ) 3 1 1 3 6) 1 ) 3 1 1 3 7) 1 ) 3 1 1 3 8) 1 ) 3 1 1 3 9) 1 0 0 1 ) 1 0 ) ) 0 1 ) 0 1 10) 0 1 11) 1 0 1) 1 0 83
6.3 A = deta λi) = λ a)λ d) b ) a b b d λ a + d)λ + ad b = 0 6.6) D = a + d) 4ad + 4b = a d) + 4b 0 0 ) a = d b = 0 a 0 A = 0 a 1 0 1 = 0 = 1 A λ 1 λ 1 λ 1 t 1 ) = λ 1 t 1 ) = t λ 1 1 ) = t A 1 ) = t 1 t A) = t 1 A) = t 1 A ) = t 1 λ ) = λ t 1 ) 6.7) λ 1 λ 6.7) t 1 = 0 1 1 6.4 1 P = ) 1 P = ) 1 1) 3 3 4 ) ) ) 0 3) 3 1 ) 1 1 1 A 1) deta λi) = λ 3 = 0 3 4 λ λ )λ 4) 3 = λ 6λ + = λ )λ 1) = 0 84
λ 1 = λ = 1 λ 1 = ) 3 3 1)/ 3 ) 3 1 A λ 1 I = 3 1 )+1) 0 0 ) 1 = 1 1 3 1 1 = 1 λ = 1 ) 1 3 ) ) 3 1) A λ I = 1 3 3 3 0 0 = 1 ) 3 1 1 1 = 1 A P = 1 ) 1 3 = 3 1 cos π 3 sin π ) 3 sin π cos π 0 t P AP = 0 1 3 3 ) deta λi) = λ 3 λ = 0 λ3 λ) 4 = λ 3λ 4 = λ 4)λ + 1) = 0 λ 1 = 4 λ = 1 λ 1 = 4 4 1 A λ 1 I = 1 0 0 1)/ )+1) 1 = 1 ) 1 1 1 = 1 λ = 1 1 ) 1) 1 A λ I = 4 0 0 = 1 ) 1 1 1 = 1 A P = 1 1 cos α sin α = 1 sin α cos α ) 4 0 t P AP = α cos α = 1 sin α = 0 1 8
3) deta λi) = 1 λ 1 1 1 λ = 0 λ 1) 1 = λ λ = λλ ) = 0 λ 1 = λ = 0 λ 1 = 1 1 1 1 1 A λ 1 I = 1 1 0 0 ) 1) 1) 1) 1 = 1 ) 1 1 1 1 = 1 λ = 0 1 1 )+1) 1 1 A λ 1 I = 1 1 0 0 = 1 ) 1 1 1 1 = 1 ) A P = 1 1 1 = 1 1 ) ) cos sin π 4 π ) 4 sin cos π 4 ) π 4 t P AP = ) 0 0 0 ) 7 3 3 3 3 13 ) 0 1 1 0 ) 1 4 86
6. x x x ax + bx + c = 0 a 0) 6.8) 6.8) 6.8) 6.8) 6.8) D = b ac D = 0 1 D < 0 1 x + c = 0 6.9) c < 0 6.9) x = c x = c c 0 c = 0 1 c > 0 6.9) c > 0 6.9) 0 6.9) x 1 6.6 x 0 x x ax + bx + c + dx + e + f = 0 6.10) a b c d e f a b c 1 0 x 6.10) a f α β 1) 4) 7) x α + β = 1 ) x α + β = 0 3) x α + β = 1 x α β = 1 ) x α β = 0 6) x α = 1 x α = 0 8) x α = 1 9) x α = 87
x 0 x 0 ax + c + dx + e + f = 0 6.11) 6.11) 1 x a > 0 6.11) 3 E) a > 0, c > 0 H) a > 0, c < 0 P) a > 0, c = 0 E) H) P) 1.) x + 10x + + f = 0 6.1) x x 1) + + 1) = 6 f x 1) + 6.1) + 1) f = 9 Figure 6. = 6 f f < 6 f 6 ) 1 f = 6 1 1 f > 6 E) 4x + 8x + + f = 0 6.13) x 4x + 1) 1) = 3 f x + 1) 6.13) 88 1) 4 = 3 f 4
f = 1 Figure 6. f < 3 x f 3 f = 3 Figure 6. f > 3 H) + 1) x 1) + = 3 1) 1 x+ 1) = 4 1) x+ 1) = 0 4 = x+ 1) 3 Figure 6.: x + 4x + e 4 = 0 6.14) x e) = x + 1) 6 6.14) e = Figure 6. 89
e < 0 e 0 x + 1) = 3 e = 0 x + 1) = 3 e > 0 P) E) 6.11) a > 0 c > 0 6.11) a x + d ) + c + e ) d = a c 4a + e f a > 0, c > 0) 4c x 1 = d a 1 = e d p = c 4a + e 4c f ax x 1 ) + c 1 ) = p a > 0, c > 0) p p > 0 x x 1) α + 1) = 1 β α = p/a, β = ) p/c p = 0 x x 1) α + 1) = 0 1 β α = 1/a, β = ) 1/c 6.1) p < 0 x x 1) + 1) α β = 1 α = p/a, β = ) p/c H) 6.11) a > 0 c < 0 6.11) a x + d ) + c + e ) d = a c 4a + e f a > 0, c < 0) 4c x 1 = d a 1 = e d p = c 4a + e 4c f ax x 1 ) c) 1 ) = p a > 0, c > 0) p 90
p > 0 x x 1) 1) α β = 1 x α = p/a, β = ) p/c p = 0 x x 1) 1) α β = 0 α = 1/a, β = ) 1/c 6.16) p < 0 1) x x 1) β α = 1 α = p/a, β = ) p/c P) 6.11) a > 0 c = 0 a > 0 6.11) a x + d ) + e = d f a > 0) a 4a x 1 = d d p = a 4a f ax x 1 ) + e = p a > 0) e 0 p e = 0, p > 0 x x 1) = 1 α α = ) p/a e = 0, p = 0 x x 1) = 0 1 α α = ) 1/a 6.17) e = 0, p < 0 x x 1) = 1 α α = ) p/a e 0 = p e a e x x 1) x 0 1 x 6.1) 6.16) 6.17) 91
1) 9 x 4 + 4x + 4 = 0 ) 8 x + 4 + 4x + 4 = 0 3) x 4 + 4 31 = 0 4) 13 4 4 = 0 0 x 0 ax + c + dx + e + f = 0 a + c > 0 E) ac > 0 1 H) ac < 0 P) ac = 0 1 6.7 x 6.10) ) a b A = 1 B = d e ) f b c ) ) x x A + B A P = ) x + f = 0 6.18) q = sin θ ) t λ1 0 P AP =, λ 0 λ 1, λ : A ) p q p = cos θ q p 6.10) 6.18) x x u = P 6.19) 6.10) 6.18) 6.18) u ) ) t u P AP + BP u ) λ 1 0 0 λ ) ) u + BP ) u + f = 0 ) u + f = 0 9
u 0 u λ 1 u + λ + dp + eq)u + dq + ep) + f = 0 6.0) u 6.0) x 6.10) 6.18) P θ 6.0) u θ 6.10) 6.18) x ) a b a + b + c > 0 λ b c 1 λ ax + bx + c + dx + e + f = 0 E) λ 1 λ > 0 1 H) λ 1 λ < 0 P) λ 1 λ = 0 1 1) x + 3x + 4 + 3)x + 1 3) 9 = 0 ) x + 3x + 4 + 3)x + 1 3) + 6 = 0 3) x + 3x + 4 + 3)x + 1 3) + 11 = 0 A = 3 3 4 ) B = + 3) 1 3 ) ) ) x x A + B ) x + f = 0 1) f = 9 ) f = 6 3) f = 11 93
. 1) A π/3 P = ) ) 1 1 3 0 t P AP = 3 1 0 1 ) x = P ) u BP = + 3) 1 3 ) ) 1 1 3 = 10 ) 3 1 u + 10u + + f = 0 u 1) + + 1) = 6 f 6.1) 1) ) 1 f = 9 6.1) ) 1 3) 1 u 1) + + 1) = 3 Figure 6.3 π/3 1) x Figure 6.3 u x Figure 6.3: 1) u x 1) 7x 6 3x + 13 + 4 3x + 4 1 = 0 ) 7x 6 3x + 13 + 4 3x + 4 + 4 = 0 3) 7x 6 3x + 13 + 4 3x + 4 + = 0 94
1) 4x + 3 + 4 x + 18 + 1 = 0 ) 4x + 3 + 4 x + 18 + 3 = 0 3) 4x + 3 + 4 x + 18 + = 0 A = ) 0 B = 3 4 18 ) ) ) x x A + B ) x + f = 0 1) f = 1 ) f = 3 3) f =. ) ) A P = 1 1 = 1 ) ) cos α sin α 4 0 t P AP = α sin α cos α 0 1 cos α = 1 sin α = ) x = P BP = ) u ) 4 18 1 1 1 ) = 8 ) 4u + 8u + + f = 0 4u + 1) 1) = 3 f 6.) 1) 3) ) f = 1 6.) u + 1) 1) = 1 4 Figure 6.4 α 1) x Figure 6.4 f = 3 6.) u + 1) 1) = 0 4 Figure 6. α ) x Figure 6. 9
1) f = 6.) u + 1) = 1 4 Figure 6.6 α 3) x Figure 6.6 u cosα= sinα= 1 x Figure 6.4: 1) u x u cosα= sinα= 1 x Figure 6.: ) u x u cosα= sinα= 1 x Figure 6.6: 3) u x 1) x + 4 x + + 7 = 0 ) x + 4 x + + 8 = 0 3) x + 4 x + + 9 = 0 96
1) x + x + x = 0 ) x + x + x + = 0 3) x + x + x + 4 = 0 A = 4) x + x + x 3 4 = 0 ) 1 1 B = 1 1 ) ) x x A + B ) 1) ) 3) ) x + f = 0 1) f = 0 ) f = 3) f = 4. 3) A π/4 P = 1 ) 1 1 1 1 ) 0 t P AP = 0 0 x u = P BP = ) ) 1 1 1 = 4 0 ) 1 1 u + 4u + f = 0 u + 1) = f u + 1) = 1 f/ 6.3) 6.3) f = 0 u = u = 0 f = 1 u = 1 f = 4 1) ) 3) A B = 3 ) f = 4 4) ) x x x A + B + f = 0 97
) x = P ) u BP = 3 ) ) 1 1 1 = 4 ) 1 1 u + 4u 4 = 0 u + 1) = 6 u + 1) = 3 6.4) 4) Figure 6.7 π/4 4) x Figure 6.7 u x π / 4 Figure 6.7: 4) u x 1) x + 4x + 4 x + = 0 ) x + 4x + 4 = 0 3) x + 4x + 4 = 0 4) x + 4x + 4 + = 0 98