()
P. () AB () AB ³ ³, BA, BA ³ ³ P. A B B A IA (B B)A B (BA) B A ³, A ³ ³ B ³ ³ x z ³ A AA w ³ AA ³ x z ³ x + z +w ³ w x + z +w ½ x + ½ z +w x + z +w x,,z,w ³ A ³ AA I x,, z, w ³ A ³ ³ + + A ³ A A P. () ( ax + b ( ) cx + d ( ) ³ ( az + bw ( ) cz + dw ( ) ( ) d ( ) b ( ) c ( ) a ( ) d ( ) b ( ) c ( ) a adx + bd d bcx + bd acx + bc c acx + ad adz + bdw bcz + bdw b acz + bcw acz + adw a (ad bc)x d d x ad bc (bc ad) c c x bc ad c ad bc (ad bc)z b b z ad bc (bc ad)w a a w bc ad a ad bc () A A ³ x z w ³ a b c d ad bc ad bc ³ d b c a ³ a b c d ³ ad bc bd bd ac + ac bc + ad ³
³ P. () A det (A) ( ) A A ³ ³ ³ () A det (A) () P. x D A A ³ ³ D z D b c b c b c D b c b c a c a c a c a c D a c a b a b a b a b D a b P. () c + c + c D D ½ a c a c a c c a c c a c + c a c c a c c a c c a c ¾ ³ 8 () A det (A) 8 A ³ () A det (A) () A A ³ x D D z D b c b c b c D 8 b c b c a c a c a c a c D a c a b a b a b a b D a b () a z + a z + a z ½ ¾ a b a b a b a D a b a a b + a a b a a b D a a b a a b () b z + b z + b z ½ ¾ a b b a b a D a b b + b a b b a b b a b D b a b b a b () c z + c z + c z ½ ¾ a c b a b a D a b c + b a b c a b c a b D c a b a c b c a b D a c b a c b a b c D a b c a b c
P.7 [] XA I X A X XI X(AA )(XA)A IA A X A () ½ x + 9 P.8 () x + ³ ³ x ³ 9 ³ x ³ ³ 9 ³ ³ 9 ³ 7 ³ 9+ () x, ½ x () x + 9 ³ ³ x ³ 9 ³ x ³ ³ 9 ³ ³ 7 9 ³ ³ 7 () x, P.9 ½ x x x + x +( ) ³ x ³ ³ x () A ³ P. () cos sin sin cos () () cos 9 sin 9 µ sin 9 cos 9 cos sin sin cos () cos θ sin θ sin θ cos θ
P. x r cos(α θ) r cos α cos θ + r sin α sin θ x cos θ + sin θ r sin(α θ) r sin α cos θ r cos α sin θ cos θ x sin θ ³ x ³ x cos θ + sin θ x sin θ + cos θ A cos( θ) sin( θ) cos θ +sin θ ³ cos θ sin θ ³ x sin θ cos θ ³ cos θ sin θ ³ cos θ sin θ sin θ cos θ sin θ cos θ sin( θ) ³ cos θ sin θ cos( θ) sin θ cos θ cos sin () sin cos cos 9 sin 9 () sin 9 cos 9 () ³ cos θ sin θ sin θ cos θ cos sin () sin cos cos sin () sin cos P. ³ x ³ x ³ ³ ³ x x + B, 7 8 7x +8 ½ x x + (x +)+(7x +8) 9x + x + (x +)+(7x +8) x + µ 9 () ³ x ³ x ³ A µ 9 ³ x ³ x ³ x + x + ³ x () AB ³ ³ 7 8 ³ 9 () AB ³ ³ 7 8 ³
P. ½ x αx + β α(ax + b)+β(cx + d) (αa + βc)x +(αb + βd) γx + δ γ(ax + b)+δ(cx + d) (γa + δc)x +(γb + δd) µ x µ µ αa + βc αb + βd x γa + δc γb + δd () µ µ µ a b α β aα + bγ aβ + bδ AB c d γ δ cα + dγ cβ + dδ µ µ µ α β a b αa + βc αb + βd BA γ δ c d γa + δc γb + δd P. BA µ µ cos β sin β cos α sin α sin β cos β sin α cos α µ αa + βc αb + βd γa + δc µ cos β cos α sin β sin α cos β sin α sin β cos α sin β cos α +cosβ sin α sin β sin α +cosβ cos α γb + δd cos(α + β) cosβ cos α sin β sin α cosα cos β sin α sin β sin(α + β) sinβ cos α +cosβ sin α sinα cos β +cosα sin β P. µ x µ µ µ µ x a b x ax + b A c d cx + d ½ x ax + b cx + d µ x A ad bc µ µ d b x c a d(ax + b) b(cx + d) ad bc c(ax + b)+a(cx + d) (ad bc)x ad bc (ad bc) µ x µ dx b ad bc cx + a ()
P. µ µ µ a b x x λ c d ½ ax + b λx cx + d λ ½ (a λ)x + b cx + (d λ) a λ b c d λ ³ x ³ (a λ)(d λ) bc λ (a + d)λ + ad bc () λ (a + d)λ + ad bc () µ A, λ ( + )λ + λ 7λ + (λ )(λ ) () λ λ µ () A, λ λ + () λ, λ µ () A, λ λ + () λ µ () A, λ λ + () λ ± i µ P.7 () A λ, det (A λi) λ λ 8λ + (λ )(λ ) () λ, λ () A λ det(a λi) λ λ ( λ)( λ)( λ) () λ, λ, λ
7 P.7 () A 9 det(a λi) λ 7 λ 9 λ ( λ)( λ)( λ)+( λ) ( λ){ ( λ)( λ)+} ( λ){ λ λ +} λ λ λ + () λ, λ ± () A 7 det(a λi) λ 7 λ λ ( λ)( λ)( λ) ( λ) λ( λ)(+λ)+8λ λ{ ( λ)(+λ)+8 } λ{ λ +8 } λ(9 λ ) λ( λ)(+λ) () λ, λ, λ µ λ P.8 () A det(a λi) λ λ, λ λ (A λi) x ³ ³ ³ λ x λ x ³ ³ ³ x + λ (A λi) x ³ ³ λ x λ ³ x () λ λ µ () A λ det(a λi) λ ³ x ³ ³ x ³ ³ x ³ ³ x + x + ³ µ ³ x + x ³ λ, λ λ (A λi) x ³ ³ ³ x x ³ x + ³ ³ x λ (A λi) x ³ ³ x () λ λ ³ x ³ ³ x x ³ ³ ³ µ ³ ³ ³,
µ P.8 () A λ det(a λi) λ λ, λ (A λi) x ³ ³ ³ ³ ³ x x ³ x + ³ ³ ³ x λ (A λi) x ³ ³ x () λ λ x ³ x ³ ³ x ³ µ ³ P.9 ³ x x µ µ µ λ x x λ λ λ λ x λ x ³λ x λ x () λ λ µ a b A ³x x c d µ a ³Ax Ax c b d µ x x ax + b ax + b cx + d cx + d µ x µ a b c d µ ax x + b ax + b cx + d cx + d P. A (x x ) (Ax Ax ) (λ x λ x ) (x µ λ x ) λ () () AP µ µ λ P λ () P µ () P AP µ µ µ µ µ µ µ µ µ µ µ µ µ
³ P. () A ³ P P.8 () λ, P λ ³ P AP ³ ³ ³ ³ ³ ³ ³ ³ ³ ( ) ³ P, P ³ ³ P AP ³ ³ ³ ³ ³ ³ ³ ³ () A ³ P ³ () P P ³ () A ³ P.8 () λ ³ λ, P ³ ³, P AP, det(a λi) λ (A λi) x λ (A λi) x ³ P,P ³ ³ ( ) P P ³ ³, P AP λ λ λ λ + λ, µ µ µ µ x x ³, P AP µ µ µ µ x x ³, P AP
³ ³ ( (x ), ³ ( ) P. A () P. () A ³, det (A λi) λ λ ³ ³ ³ x x λ (A λi) ³ λ λ+8 λ, ³ ³ x a ³ ³ ³ x x λ (A λi) ³ ³ x () λ x ³ x (Ax λx) Aa a, Ab b ³ ( a, ³ b () A P. () λ, ³ ³ x λ a ³ ³ x λ b λ Aa a, Ab b ³ ( a, () ³ b () () P (a, b) A P ³ P. () P AP ³ b ³ ³ P P.
P. A A 7 9 7 7, A, A, A +7+ +9+ ++7 7, A 7 9 A (,, ) (,, ), (,, ) (,, ), (,, ) (7, 9, ), (,, ) (,, 7 ), (,, ) (, 7, ) P. () x x cos θ sin θ x sin θ + cos θ () x z cos θ sin θ sin θ cos θ x z P. a a cos θ b sin θ b a sin θ + b cos θ cos θ z sin θ z sin θ + z cos θ x z cosθ sin θ sinθ cos θ x z P.7 z z cos θ x sin θ x z sin θ + x cos θ x x cos θ + z sin θ z x sin θ + z cos θ x z cos θ sinθ sin θ cosθ x z
P.8 x x cos ϕ sin ϕ (xcos θ + z sin θ)cosϕ sin ϕ (cosθcos ϕ)x (sin ϕ) +(sinθcos ϕ)z x sin ϕ + cos ϕ (xcos θ + z sin θ)sinϕ + cos ϕ (cosθsin ϕ)x +(cosϕ) +(sinθsin ϕ)z z z ( sin θ)x +(cosθ)z x z cos θ cos ϕ sin ϕ sin θ cos ϕ x cos θ sin ϕ cos ϕ sin θ sin ϕ sin θ cosθ z cos ϕ sin ϕ BA sin ϕ cos ϕ cos θ sinθ sin θ cosθ cos θ cos ϕ sin ϕ sin θ cos ϕ cos θ sin ϕ cos ϕ sin θ sin ϕ sin θ cosθ P.9 () A t ³, () B t 7 8 9 () p t (p p p ), () r t ( ) () Ax a b c a b c a b c () x t A t (x z ) Ax x z a a a b b b c c c a x + b + c z a x + b + c z a x + b + c z (a x + b + c z a x + b + c z a x + b + c z) a x + b + c z a x + b + c z a x + b + c z (a x + b + c z) +(a x + b + c z) +(a x + b + c z) a x + b + c z (a x+b +c z a x+b +c z a x+b +c z) a x + b + c z a x + b + c z x t A t A x
P. p ³ cos θ sin θ, p ³ sin θ cos θ () p cos θ +sin θ, p ( sin θ) +(cosθ) p p cosθ( sin θ)+sinθ cos θ µ cos θ sin θ () P (p p ) sin θ cos θ P () P. p p µ cos θ sin θ P P t sin θ cos θ - -, p p ( ) +(- ), p ( ) +( ), p p ( )( )+(- )( ) P (p p ) - P P t - P t P - - P P t P, p 9 + -, p - + 9 7 7 7 µ () p +, p +9+, p 9++ 7 p p, p p, p p () P (p p p ) P t P - - 7 7 7-7 7-7
P. () P P t P. () P t P - - 7 7 7 x z x x x x z x z z z z x + + z x x + + z z x x + + z z x x + + z z x + + z x x + + z z x x + + z z x x + + z z x + + z p p p p p p p p p p p p p p p () () p t, t p p t p p t p p t p P t P t p ( p p p ) p t p p t p p t p t p p t p p t p p t p p p p p p p p p p p p p p p p () x z P P t x z x z () x x x x z x + x + x x + x + x x z + x z + x z PP t x z x + x + x + + z + z + z z z z x z z x + z x + z x z + z + z z + z + z () x + x + x, + +, z + z + z x + x + x, z + z + z, x z + x z + x z
P. () x () ³ x x x, ³, z x,, z ³ z z z x, z, x z P t (x z ) P. B { x,, z } P t (x z ) () Q P t Q t (P t ) t P Q t Q PP t PP I Q t Q Q () cos ϕ sin ϕ sin ϕ cos ϕ (b b b ) cos ϕ sin ϕ b sin ϕ, b cos ϕ, b () b cos ϕ +sin ϕ, b ( sin ϕ) +cos ϕ, b b b cos ϕ sin ϕ +sinϕ cos ϕ, b b, b b cos ϕ cos θ c sin ϕ cos θ, c sin θ sin ϕ cos ϕ, c cos ϕ sin θ sin ϕ sin θ cos θ () c cos ϕ cos θ +sin ϕ cos θ +sin θ cos θ +sin θ c sin ϕ +cos ϕ c cos ϕ sin θ +sin ϕ sin θ +cos θ sin θ +cos θ c c cos ϕ cos θ sin ϕ +sinϕ cos θ cos ϕ c c sin ϕ cos ϕ sin θ +cosϕ sin ϕ sin θ c c cos ϕ sin θ cos θ +sin ϕ sin θ cos θ sin θ cos θ sinθ cos θ sin θ cos θ () () { b, b, b } { c, c, c } B (b b b ) C (c c c ) cos ϕ sin ϕ cos ϕ cos θ sin ϕ cos θ sin θ B B t sin ϕ cos ϕ, C C t sin ϕ cos ϕ cos ϕ sin θ sin ϕ sin θ cos θ
P. x, x Ax (BA)x B(Ax) Bx () A, B [] Ax x, Bx x () (), () (BA)x Bx x Ax x (BA)x x BA [] BA () P. A A A t det(a t )det(a) (deta) (deta) (deta) (deta t ) (deta) det(a t A)det(I) deta ± () { a, b, c } (a b) c () deta (a b) c deta ± deta () D deta, x b c b c, x b c b c, x b c b c a c a c, a c a c, a c a c z a c a c, z a b a b, z a b a b x x x a a a () A A t b b b () z z z c c c a b a b a b z c a b b c c a a a b b b c c c b b c a b a b a b a b b c b c c c b c b c b c b c a c a c a a a c a c a c a c a z z x x x c c a a a b b b c a b
P. P.7 µ µ µ µ µ p a a p a p + a p p a p b b p b p + b p p b ³ ³ ( x x cos θ p a x cos θ + sin θ sin θ ³ ³ x -sinθ p b x sin θ + cos θ cos θ p a a a p b b b p c c c p p p a p + a p + a p b p + b p + b p c p + c p + c p p a p b p c P.8 () x p a x sin θ cos ϕ + sin θ sin ϕ + z cos θ p b x cos θ cos ϕ + cos θ sin ϕ z sin θ z p c x sin ϕ + cos ϕ i +j +k, () i +k, () i +j k () i i i, () j j j, () k k k () i j, () j k, () k i () a i a, () a j a, () a k a () (i +j) (i k) i i +j i i k 8j k () (i j +k) (i +j k) +( ) + ( ) 8 () (a i + a j + a k) (b i + b j + b k)a b + a b + a b () i j p +( ) () i + j k p + +( ) () a i + a j + a k a + a + a
i j k i j k P.9 () i i () j j i j k i j k () j k i () k i j i j k () (i + j k) (i j +k) - - - - i - j + - k i 9j 7k i j k () (i +j + k) (i + j +k) i j + k 7i j k () P. a () i+ j + k, b i+ j k, c i+ j a b +, b c 8 + 8, c a 8 + 8 () r a + + r, b + + r, c + () a b i j k i j + k ( 8 8 )i ( 8 i j k b c 8 )j i + j c i j + k ( )i ( i j k c a )j +( 8 + 8 )k i + j + k a i j + k ( )i ( )j +( 8 8 )k i + j k b
P. a i + j + k, b i + j k, c i + j () det(a) ( 8 8 + 8 ) () p p i + p j + p k, q q i + q j + q k ( ) p p a + p b + p c, q q a + q b + q c ( ) () p p a p + p p c p + p + p, p p b p, q q a p + q + p q + p q q q b q + q q, q q c q + q () a a, a b, a c, b b, b c, c c (p a + p b + p c) (q a + q b + q c) p q (a a)+p q (a b) +p q (a c) +p q (b a)+p q (b b) p q + p q + p q + p q (b c) +p q (c a) +p q (c b) +p q (c c) (7) (), () ( (p a + p b + p c) (q a + q b + q c)p q + p q + p q p + p + p )( q + q + q )+( p + p p )( q + q q ) +( p + p )( q + p q + p q + 8 p q + p q + p q + 8 p q + 8 p q + 8 p q + p q + p q + p q 8 p q + p q + p q 8 p q 8 p q 8 p q + p q q ) + p q p q p q + p q ++ p q + + p q + + p q + ++ p q + + p q p q + p q + p q
P. () (), () () () () (),() { a, b, c } () det(a) { a, b, c } () b c a, c a b () () () det(a) (a b) c c c c () A (a, b, c ), A (p a + p b + p c) (q a + q b + q c)a p A p p p q q q q p q p q + p q + p q p q (7) p,p,p,q,q,q ( ) p,p,p,q,q,q ( ) (p a + p b + p c) (q a + q b + q c)p q (p i + p j + p k) (q i + q j + q k) p q + p q + p q