koji-13.dvi

26 13 1, 2, 3, 4, 5, 6, 7 1 18 1. xy D D = {(x, y) y 2 x 4 y 2,y } x + y2 dxdy D 2 y O 4 x 2. xyz D D = {(x, y, z) x 1, y x 2, z 1, y+ z x} D 3. [, 1] [, 1] (, ) 2 f (1) <a 1 a f(x, y) = x2 y 2 (x 2 + y 2 ) 2 f(a, y)dy

(2) (1) ϕ(a) (3) <b 1 b lim t + t ϕ(x)dx f(x, b)dx (4) (3) ψ(b) lim t + t ψ(y)dy 1. x 2. D 1 3. (3) (4) (1) (2) x y

26 13 1, 2, 3, 4, 5, 6, 7 1 18 1 D y y y 2 4 y 2 y 2 y x y 2 x 4 y 2 x y ( 2 ) 4 y 2 x + y2 dxdy = x + y2 dx dy D x t = x + y 2 4 y 2 4 [ 2 4 3] x + y2 dx = tdt = t = 16 y 2 2y 3 2 2y 3 4 2 3 y3 2 D x + y2 dxdy = 2 ( 16 3 4 2 3 y3 y 2 ) dy =. y [ ] 2 16 2 3 y 3 y4 =4 2 2 1 D (x, y, z) x [, 1] x y [,x 2 ] y z [,x y ] x 1 x 2 x y x 2 x y x y y z

13 2 z y x ( 1 x 2 ( x y ) ) ( 1 ) x 2 [ ] x y 1dxdydz = 1dz dy dx = z dy dx D ( 1 ) x 2 = (x y)dy dx = [ 12 ] x 2 (x y)2 dx = ( 12 (x x2 ) 2 + 12 ) x2 dx = (x 3 12 ) x4 dx = 1 4 1 1 = 3 2. 5 3 (1) t f(a, y)dy = = t = [ = y 2 + a 2 (y 2 + a 2 ) 2 dy y 2 + a 2 2y 2 (y 2 + a 2 ) 2 dy (y) (y 2 + a 2 ) y (y 2 + a 2 ) dy (y 2 + a 2 ) 2 y y 2 + a 2 = 1 1+a 2 (2) (Arctan x) =1/(1 + x 2 ) t lim t + ϕ(x)dx = t ] 1 [ ] 1 Arctan x = Arctan 1 Arctan t t ϕ(x)dx = lim t + (Arctan 1 Arctan t) = π 4

13 3 (3) f(y, x) = f(x, y) (1) (4) (2) f(x, b)dx = lim t + t ψ(y)dy = lim f(b, x)dx = 1 1+b 2 = ϕ(b) t + t ϕ(y)dy = π 4 1 8 8 (^^; 8

13 4 1 5 1.1... 5 1.2... 6 1.3... 7 1.4... 9 2 11 2.1... 11 2.2... 13 3 16 4 18 4.1... 18 4.2... 2 4.3... 22 5 23 5.1... 23 5.2... 25 5.3... 25 6 26 6.1... 26 6.2... 28 6.2.1... 28 6.2.2... 3 7 34 7.1... 34 7.2... 35 7.3... 36 7.4... 37 7.5... 38 7.6... 4

13 5 1 1.1 x = x. V V = b a S(x)dx S(x) a x b 1 y = f(x) x x = x f [a 1,b 1 ] [a n,b n ] n b1 { b2 a 1 a 2 { { bn } } } f(x 1,x 2,,x n )dx n dx 2 dx 1 a n f 2 1 2 iterated integral repeated integral

13 6 1.2 1 1 f f(x, y) = ( ) F y x d c f(x, y)dy = F F (x, d) (x, c) x x b { d } f(x, y)dy dx =(F (b, d) F (b, c)) (F (a, d) F (a, c)) a c F y x (1) x y (2) x y (3) x y (4) x (1) (2)(3)(4) 2

13 7 1.3 1 1 2. [a, b] [c, d] [a, b] [c, d] a = x <x 1 < <x n 1 <x n = b c = y <y 1 < <y m 1 <y m = d [x k 1,x k ] [y l 1,y l ] [a, b] [c, d] [x k 1,x k ] [y l 1,y l ] = max (xk x k 1 ) 2 +(y l y l 1 ) 2 1 k n 1 l m x k 1 ξ kl x k y l 1 η kl y l ξ kl,η kl (ξ kl,η kl ) [x k 1,x k ] [y l 1,y l ] [a, b] [c, d] 2 f(x, y) lim f(ξ kl,η kl )(x k x k 1 )(y l y l 1 ) 1 k n 1 l m f f(x, y)dxdy [a,b] [c,d]

13 8 1 1 3 4 (i) x y y x (ii) x y y (iii) x y y d f(x, y)dy x c (2) (3) (ii) (iii) (ii) (iii) 1 1 2 [, 1] [, 1] f(x, y) x =1/2 2 f(1/2,y) 1 f(1/2,y)dy 3 4

13 9 f(x, y) = {, x =1/2 y Q 1, { f(x, y)dxdy = [,1] [,1] } f(x, y)dx dy =1 f(1/2,y)dy y (iii) g(x, y) = {, x =1/2 y Q x Q y =1/2 1, 1.4 1 11 1 3 3 f lim s t [s,1] [t,1] f(x, y)dxdy s t (s, t) (, ) 3 f (, 1] (, 1] { } f(x, y)dy dx = f(x, y)dxdy = s t [s,1] [t,1] t { s } f(x, y)dx dy [s,1] [t,1] f(x, y)dxdy = π 4 Arctan s Arctan 1 t Arctan s t t + s + 2(2) π/4 s + t + 2(4) π/4 f(x, y)

13 1 f(x, y) 2 (1)(2) f(x, y) (, 1] (, 1] π/4 1 1 1 1 1 1 ( + e x2 dx) 2 = = ( + ( + )( + e x2 dx )( + e x2 dx ) e x2 dx ) e y2 dy = 1 e x2 y 2 dxdy 2 x = r cos θ, y = r sin θ = π 2 + e r2 rdrdθ = + π re r2 2 dr dθ = [ 12 ] + [ e r2 θ 1 (r, θ) e x2 y 2 x = r cos θ, y = r sin θ r 1 b a f(x)dx = β α f ( g(t) ) g (t)dt g (t) r R 2 (r, θ) (r cos θ, r sin θ) R 2 ] π 2 = π 4

13 11 2 2.1 7 1. xy x x =[a, b] x : a = x <x 1 < <x n 1 <x n = b y y =[c, d] y : c = y <y 1 < <y m 1 <y m = d xy D ij (1 i n, 1 j m) D ij =[x i 1,x i ] [y j 1,y j ] ={D ij 1 i n, 1 j m} = x y D ij = ( x ) 2 +( y ) 2 x i 1 ξ ij x i,y j 1 η ij y j (ξ ij,η ij ) D ij 1 {(ξ ij,η ij ) 1 i n, 1 j m} ξ ξ 2. =[a, b] [c, d] f ξ R,ο (f) = f(ξ ij,η ij )(x i x i 1 )(y j y j 1 ) 1 i n 1 j m (, ξ) f

13 12 d = y 3 (ξ 32,η 32 ) y 2 y 1 c = y x x 1 x 2 x 3 x 4 a b 1: =[a, b] [c, d] 3. =[a, b] [c, d] f lim R,ο (f) =J J f J f J = f(x, y)dxdy ε δ <δ ξ R,ξ (f) J <ε 1

13 13. f(x, y) (x, y) 2.2 1 1. =[a, b] [c, d] f M f(x, y) >M (x, y) f. 1 f R D ij f S S > δ x 1 x,..., x n x n 1,..., y 1 y,..., y m y m 1 f(x, y) > R + S δ 2 (x, y) ij f D ij (x,y ) R,ο (f) > S + f(x,y )δ 2 > S + R + S δ 2 = R

13 14 4. f R (f) = sup f(x, y)(x i x i 1 )(y j y j 1 ) (x,y) D ij R (f) = 1 i n 1 j m 1 i n 1 j m inf f(x, y)(x i x i 1 )(y j y j 1 ) (x,y) D ij f f 2. f R (f) R,ξ (f) R (f) (1) lim R (f) = lim R (f) f f(x, y)dxdy f 1 f 3. f lim R (f) = lim R (f) =. lim R (f) = f(x, y)dxdy f(x, y)dxdy

13 15 ε δ <δ R (f) f(x, y)dxdy <ε sup f(x, y) f(ξ ij,η ij ) < (x,y) D ij ε 2(b a)(d c) (ξ ij,η ij ) D ij R (f) R,ο (f) < 1 i n 1 j m ε 2(b a)(d c) (x i x i 1 )(y j y j 1 )= ε 2 f(x, y) δ <δ ξ R,ο(f) f(x, y)dxdy < ε 2 <δ R (f) f(x, y)dxdy R (f) R,ο (f) + R,ο(f) f(x, y)dxdy <ε 2, 3 2 3 4. f f lim R (f) = lim R (f) f(x, y)dxdy

13 16 3 5(). =[a, b] [c, d] f(x, y) x [a, b] y f(x, y) y x b ( d ) f(x, y)dy dx = f(x, y)dxdy (2) a c. D ij f M ij,m ij M ij = sup f(x, y), m ij = inf f(x, y) (x,y) D ij (x,y) D ij x i 1 ξ i x i ξ i m ij f(ξ i,y) M ij, (y j 1 y y j ) y y j 1 y j m ij (y j y j 1 ) yj F (x) = y j 1 f(ξ i,y)dy M ij (y j y j 1 ) d c f(x, y)dy j =1 j = m m m ij (y j y j 1 ) F (ξ i ) j=1 m M ij (y j y j 1 ) j=1 (x i x i 1 ) R (f) n F (ξ i )(x i x i 1 ) R (f) (3) i=1

13 17 ξ i x i 1 ξ i x i F (ξ i ) [x i 1,x i ] R (f) R x (F ) R x (F ) R (f) 1 4 f(x, y)dxdy lim R x (F ) = lim R x (F )= f(x, y)dxdy x x F (x) [a, b] 1 b a F (x)dx 1 F b F (x)dx = f(x, y)dxdy a (3) y x lim R (f) = ( lim x lim R (f) y 2. y x y x y y lim R,ο(f) =J lim x lim R,ο(f) =J y 2 )

13 18 d = y 3 η 3 y 2 η 2 y 1 η 1 c = y x ξ 1 x 1 ξ 2 x 2 ξ 3 x 3 ξ 4 x 4 a b 2: y x 4 4.1 4 (1) 4 R (f) f(x, y)dxdy R (f) (4) (1) (4) 1, 2 R 1 (f) R 2 (f) (5) 5. x, y x, y 1

13 19 6. f R (f) R (f) R (f) R (f). x y x (1) x x x [x k 1,x k ] x [x k 1,x ] k [x,x k ] k +1 D ij D kj =[x k 1,x ] [y j 1,y j ], D k+1j =[x,x k ] [y j 1,y j ] sup f(x, y) sup f(x, y), D kj D kj sup f(x, y) sup f(x, y) D k+1j D kj R (f) R (f) = sup f(x, y)(x k x k 1 )(y j y j 1 ) D kj ( ) sup f(x)(x x k 1 )(y j y j 1 ) + sup f(x)(x k x ) (y j y j 1 ) D kj D k+1j (5) 1 2 3 3 1 2 R 1 (f) R 3 (f) R 3 (f) R 2 (f) (4) R (f) f(x, y)dxdy R (f) f(x, y)dxdy R (f) < f(x, y)dxdy R (f) <R (f)

13 2 (4) R (f) R (f) 7. f f ( lim R (f) R (f) ) = (6). (6) 4 R (f) M (5) R (f) M R (f) R (f) R (f) M R (f) (6) R (f) M (6) R (f) M 1 5.2 4.2 7 1 1 2 f(x, y) [ ] ε δ (x, y) (x,y ) (x x ) 2 +(y y ) 2 <δ= f(x, y) f(x,y ) <ε δ (x, y) δ C 1 f(x,y ) f(x, y) =f x (a, b)(x x)+f y (a, b)(y y)

13 21 a b x x y y f x f y M f(x,y ) f(x, y) f x (a, b) x x + f y (a, b) y y 2M (x x) 2 +(y y) 2 8.. 1 (x, y) x 2 x y x y D f D ε ε ε <max D f(x) min D f(x) D x r D B r (x) D x r(x) B r (x) =D {y x y <r} y, z B r (x) f(y) f(z) <ε r ε r(x) r(x) > f r B r (x) y f(x) f(y) < ε 2 B r (x) 2 y, z f(y) f(z) f(y) f(x) + f(x) f(z) <ε (7) r(x) y x y <r(x) r(x) x y r B r (y) B r(x) (x) B r (y) 2 z, z f(z) f(z ) <ε r(y) r(x) x y r(x)+ x y r B r (y) B r(x) (x) B r (y) 2 z, z f(z) f(z) ε

13 22 r(y) r(x)+ x y r(x) r(y) x y r(x) r(x) D r r > x y <r x, y D r r(x) f(x) f(y) <ε B r (x) 4.3 ε-δ ε δ <δ R (f) R (f) <ε R (f) R (f) R (f) R (f) = 1 i n 1 j m sup (f(x, y) f(x,y ))(x i x i 1 )(y j y j 1 ) (x,y),(x,y ) D ij (x, y) (x,y ) D ij f(x, y) f(x,y ) f(x, y) f(x,y ) < ε (b a)(d c) R (f) R (f) ε 5 5 (8)

13 23 9.. f f ε δ 2 (x, y), (x,y ) (x x) 2 +(y y) 2 <δ f(x, y) f(x,y ) < ε (b a)(d c) <δ R (f) R (f) = max f(x, y)(x i x i 1 )(y j y j 1 ) (x,y) D ij 1 i n 1 j m = 1 j n 1 j m < 1 i n 1 j m 1 j n 1 j m min f(x, y)(x i x i 1 )(y j y j 1 ) (x,y) D ij max (f(x, y) f(x,y ))(x i x i 1 )(y j y j 1 ) (x,y),(x,y ) D ij ε (b a)(d c) (x i x i 1 )(y j y j 1 )=ε f 5 5.1 1? Yes 2.2 4 lim R (f) = lim R (f) R (f) R (f)

13 24 1 4 7 2 R (f) R (f) R (f) R (f) R (f) R (f) R (f) R (f) R (f) R (f) 6. f f(x, y)dxdy f(x, y)dxdy f(x, y)dxdy = inf R (f) f(x, y)dxdy = sup R (f) inf f(x, y)(b a)(d c) R (f) R (f) sup f(x, y)(b a)(d c) 1 ( ). f lim R (f) = f(x, y)dxdy, lim R (f) = f(x, y)dxdy 1

13 25 5.2 3 2 11 ( ). f n 1, 2,... ( lim R n (f) R n (f) ) = (9) n 12 (ε-δ ). [a, b] f ε R (f) R (f) <ε 5.3 f a < α < b, c < β < d 2 α, β 1 =[a, α] [c, β], 2 =[α, b] [c, β], 3 =[a, α] [β, d], 4 =[α, b] [β, d] fdxdy = fdxdy + 1 fdxdy + 2 fdxdy + 3 fdxdy 4 8 1

13 26 6 6.1 =[a, b] [c, d] x y D D D f(x, y) D f D D 7. f D D f f(x, y), (x, y) D f(x, y) =, (x, y) D f D f(x, y)dxdy := D J f(x, y)dxdy f f D 6 f 8. R 2 D D 1 D 1 6 Jordan

13 27 y D 14 D 13 D 12 D 11 D 1 D 15 D 9 D 16 D 8 D 3 D 1 D 2 D 7 D 4 D 5 D 6 x 3: D 1 D D 1 D D = 1dxdy. D D χ D D 1 D 1dxdy = χ D (x, y)dxdy D D ij D D D k (1 k N) 3 R (χ D )=D D ij R (χ D )=D D ij D R (χ D ) R (χ D )= N D k k=1

13 28 y D O x 4: D k D D C 1 ϕ(x) >ψ(x) D = {(x, y) R 2 ψ(x) y ϕ(x), a x b} 4 D ( b ) ϕ(x) f(x, y)dy dx a ψ(x) 6.2 6.2.1 (x, y) =ϕ(s, t) =(ξ(s, t), η(s, t))

13 29 f(x, y) f ϕ(s, t) =f(ξ(s, t),η(s, t)) 1 b a f(x)dx = β α f ϕ(s)ϕ (s)ds f(x) f ϕ(s)ϕ (s) 2 f(x, y)dxdy = f ϕ(s, t) det J ϕ (s, t) dsdt D E ( ) = f(ξ(s, t),η(s, t)) det ξ s (s, t) ξ t (s, t) η s (s, t) η t (s, t) dsdt E E ϕ D st ( ) ξ s (s, t) ξ t (s, t) J ϕ (s, t) = η s (s, t) η t (s, t) ϕ 7 ϕ ξ(r, θ) =r cos θ, η(r, θ) =r sin θ ( ) ( ) det ξ r ξ θ = det cos θ r sin θ sin θ rcos θ = r = r η r η θ f(x, y)dxdy = f(r cos θ, r sin θ)rdrdθ D E 7 7

13 3 y D ij 1 x 5: 6.2.2 1 xy rθ x y 5 ε-δ xy 5 rθ 6 x = r cos θ, y = r sin θ 5 f(x, y) 6 g(r, θ) 6 ij 5 D ij ij (ρ ij,ϑ ij ) ξ ij = ρ ij cos ϑ ij,η ij = ρ ij sin ϑ ij D ij (ξ ij,η ij )

13 31 θ 2π ij 1 r 6: g(ρ ij,ϑ ij )=f(ξ ij,η ij ) ij g(ρ ij,ϑ ij ) D ij f(ξ ij,η ij ) ij D ij ij ij ij [r i 1,r i ] [θ j 1,θ j ] D ij = 1 2 (r i 2 r i 1 2 )(θ j θ j 1 ) = 1 2 (r i + r i 1 )(r i r i 1 )(θ j θ j 1 ) = r i + r i 1 2 ij f(ξ ij,η ij ) D k = i,j i,j g(ρ ij,ϑ ij ) r i + r i 1 (r i r i 1 )(θ j θ j 1 ) 2 f(x, y)dxdy, D D

13 32 r θ (r i + r i 1 )/2 ρ ij,ϑ ij h(ρ ij,ϑ ij ) g(r, θ)h(r, θ) g(r, θ)h(r, θ)rdrdθ h(r, θ) (r i + r i 1 )/2 r i r i 1 ρ ij (r i + r i 1 )/2 ρ ij h(r, θ) =r i,j g(ρ ij,ϑ ij ) r i + r i 1 (r i r i 1 )(θ j θ j 1 ) 2 g(ρ ij,ϑ ij )ρ ij (r i r i 1 )(θ j θ j 1 ) i,j f(x, y)dxdy D r i ρ ij <, r i 1 ρ ij < r i + r i 1 2 ρ ij = r i ρ ij + r i 1 ρ ij 2 r i ρ ij + r i 1 ρ ij 2 < + 2 =

13 33 θ y θ φ h r φ h r r r O φ x 7: [r,r + h] [θ,θ + φ] xy r g(ρ ij,ϑ ij ) r i + r i 1 (r i r i 1 )(θ j θ j 1 ) 2 i,j g(ρ ij,ϑ ij )ρ ij (r i r i 1 )(θ j θ j 1 ) i,j ( ) ri + r i 1 = g(ρ ij,ϑ ij ) ρ ij (r i r i 1 )(θ j θ j 1 ) 2 i,j i,j g(ρ ij,ϑ ij ) (r i r i 1 )(θ j θ j 1 ) f(x, y) ξ(r, θ) = r cos θ, η(r, θ) =r sin θ g(r, θ) g(r, θ) i,j g(ρ ij,ϑ ij ) (r i r i 1 )(θ j θ j 1 ) g(r, θ)rdrdθ, =[, 1] [, 2π] g(r, θ) drdθ = g(r, θ) r g(r, θ) =f(r cos θ, r sin θ) r f(x, y)

13 34 rθ xy x = r cos θ, y = r sin θ g(r, θ) det J ϕ (s, t) ϕ(s, t) 7 7.1 8 1 1 f (, ) f F (x) (, ) f(x) x + f(x)dx = lim r + r R f(x)dx + lim f(x)dx = lim F (R) lim F (r) R 1 R r + 1 2 f (, ) 1 f(x) [a, b) r lim f(x)dx = S r b a S r n b (n ) r n rn lim n a f(x)dx = S 2 8

13 35 R 2 A [,n] n [, ) 2 2 9. A 1,A 2,... A A n A n <m n, m A n A m A B n B A n A n A A n = A n=1 n A n A n=1a n A A (x, y) (x, y) {(x, y)} = {(x, y)} A n n (x, y) n=1 A n f A A f O.K. 1. A A n lim f(x, y)dxdy = S n A n S f A S 13. f A f A 7.2 A n 3 f(x, y) = x2 y 2 (x 2 + y 2 ) 2

13 36 (x, y) (, ) [, 1] [, 1] (, ) f [, 1] [, 1] A n A n f(x, y) A n =[1/n, 1] [, 1] lim n A n f(x, y)dxdy = π 4 A n =[, 1] [1/n, 1] lim f(x, y)dxdy = π n A 4 n f(x, y) [, 1] [, 1] 1 r n =1/(nπ) f(x) = 1 x 2 cos 1 x /π lim f(x)dx = sin π lim sin nπ = n r n n r + 1 [a, b) r b A A A n 7.3 A A n A 1 A 2 A 3 A 4

13 37 1dxdy 1dxdy 1dxdy 1dxdy A 1 A 2 A 3 A 4 A f f fdxdy fdxdy fdxdy fdxdy A 1 A 2 A 3 A 4 B n B n A N B n fdxdy lim fdxdy lim fdxdy n B n n A n A n B n lim n fdxdy lim fdxdy B n n A n A n A n fdxdy B n B n fdxdy f A f f A A n 7.4 f,g A a, b (af + bg)dxdy = a fdxdy + b gdxdy A f,g A A A n A n (af + bg)dxdy = a fdxdy + b gdxdy A n A n A n f g n 1 2 A A

13 38 1. A x,y e x y cos xdxdy x, y e x y >, 1 + cos x A e x y cos x = e x y (1 + cos x) e x y (1) 1 2 1 2 A A n A n [,n] [,n] ( n )( n ) e x y (1 + cos x)dxdy = e x (1 + cos x)dx e y dy A n ( = [ e x 1+ ( = ( e n 1+ cos x sin x 2 cos n sin n 2 )] n [ ] e y n ) + 3 ) ( e n +1 ) 2 lim e x y (1 + cos x)dxdy = 3 n A n 2 lim e x y dxdy =1 n A n (1) A (1) A A e x y cos xdxdy = 3 2 1=1 2 7.5 1 cos x

13 39 f f + = f + f, f = f f 2 2 f + f f + f = f A n lim n A n fdxdy lim n A n f dxdy f fdxdy = lim fdxdy A n A n f + f 11 1 f f 9 11 4 f f 14. f A g A (x, y) A f(x, y) g(x, y) f. A n A n fdxdy gdxdy gdxdy A n A n A A n fdxdy f A f f + f g f 15. f A f + f A f A 1 11. f A f 1 2. e x y cos x e x y e x y A 14 e x y cos x A A A n A n =[,n] [,n] ( n )( n ) e x y cos xdxdy = lim e x cos xdx e y dy = 1 n 2 A 9

13 4 7.6 1 dx 1+x 2 = π/2 x = tan θ 2 f A ϕ: B A B A A n ϕ 1 A n B n B n fdxdy = gdudv (g = f ϕ J ϕ ) A n B n g B lim gdudv = gdudv n B n B n 1 1 3. Γ(x) = + dθ e t t x 1 dt (x >)

13 41 1 B(x, y)= t x 1 (1 t) y 1 dt (x >,y >) 1 2 B(x, y)= Γ(x)Γ(y) Γ(x + y) t = r 2 Γ(x) =2 + e r2 r 2x 1 dr t = sin 2 θ ( Γ(x)Γ(y) = 2 =4 =4 = ( 2 π/2 B(x, y) =2 sin 2x 1 θ cos 2y 1 θdθ + )( e u2 u 2x 1 du 2 [,+ ) [,+ ) [,+ ) [,2π) + =Γ(x + y)b(x, y) + e u2 v 2 u 2x 1 v 2y 1 dudv ) e v2 v 2y 1 dv e r2 r 2(x+y) 2 sin 2x 1 θ cos 2y 1 θrdrdθ e r2 r 2(x+y) 1 dr) ( 2 π/2 sin 2x 1 θ cos 2y 1 θdθ 3 4. 2Γ ( ) = e x2 dx = π 2 )

13 42 e ax2 2bxy cy2 e dxdy R 2 ax 2 +2bxy + cy 2 =(xy) ( a b )( b c 2 A P P t PAP = ( P 1 = t P λ µ t P P u, v ( ) ( ) x u = P (11) y v ( ax 2 +2bxy + cy 2 =(xy)a x y =(uv) (t PAP ) ( u v = λu 2 + µv 2 ) x y ) ) = ( (uv) t P ) ( ( A P ) ( =(uv) u v λ µ x, y u, v P P ( ) p q P = r s (11) P det P =1 )) )( u v )

13 43 3 ax2 2bxy cy2 e dxdy = R 2 λu2 µv2 e det P dudv = R 2 e λu2 du λ t = λu e λu2 du = e 1 π t2 dt = λ λ µ> e µv2 dv = π µ λ, µ A ax2 2bxy cy2 π π e dxdy = R λ µ = π 2 det A e µv2 dv 2 n A =(a ij ) e P n i,j=1 a ijx i x j dx 1 dx n = R n π n det A 2 5. B(α, β) = (α, β, γ) = D x α 1 (1 x) β 1 dx x α 1 y β 1 (1 x y) γ 1 dxdy

13 44 α, β, γ D D = {(x, y) x>, y>, x+ y<1} α, β, γ 1 D = {(x, y) <x<1 y, <y<1} ( y ) (α, β, γ)= y β 1 x α 1 (1 x y) γ 1 dx dy x x =(1 y)t y x α 1 (1 x y) γ 1 dx =(1 y) α+γ 1 t α 1 (1 t) γ 1 dx (α, β, γ) =B(α, γ) =(1 y) α+γ 1 B(α, γ) y β 1 (1 y) α+γ 1 dy = B(α, γ)b(β,α + γ) 3 B(s, t)= Γ(s)Γ(t) Γ(s + t) (α, β, γ)= Γ(α)Γ(β)Γ(γ) Γ(α + β + γ) x α 1 1 1 x α 2 1 2 x αn 1 (1 x 1 x 2 x n ) α 1 dx 1 dx 2 dx n D = Γ(α )Γ(α 1 ) Γ(α n ) Γ(α + α 1 + + α n ) α i n D = {(x 1,x 2,...,x n ) x 1 >, x 2 >,...,x n >, x 1 + x 2 + + x n < 1}

13 45 6. 5 1 1 1+x α dx α > 1 t = xα 1+x α 1 1+x α dx = (1 xα t 1 α 1 1+x α ) dx = (1 t) dx dt dt = 1 dt = 1 t 1 α (1 t) 1 α 1 (1 t) (1 α) 1 1 dt α α = 1 ( 1 α B α, 1 1 ) = 1 ( ) ( 1 α α Γ Γ 1 1 ) α α Γ(1) = 1 1 1+x α + y dxdy α 2 R 2 + R 2 + = {(x, y) x>, y>}, α > 2 1 s = x α 1+x α + y α, t = y α 1+x α + y α ( x = s 1 s t ) 1 α, y = ( t 1 s t ) 1 α ( ) ( x x 1 s 1 α 1 (1 t) s t = α(1 s t) 1 α 1 t 1 α y s y t s 1 α (1 s)t 1 α 1 )

13 46 R 2 + (st) 1 α 1 α 2 (1 s t) 2 α +1 1 1+x 2 + y dxdy = (st) 1 α 1 (1 s t) dsdt 2 D α 2 (1 s t) 2 +1 α = 1 s 1 α 2 α 1 t 1 α 1 (1 s t) (1 α) 1 2 dsdt R 2 + D D = {(s, t) s>, t>, s+ t<1} 5 1 1+x α + y dxdy = 1 ( α α Γ 1 2 ) ( ) 2 1 Γ 2 α α 1 R n + 1+x α 1 + x α 2 + + x α n dx 1 dx 2 dx n = 1 ( α Γ 1 n ) ( ) n 1 Γ n α α R n + = {(x 1,...,x n ) x 1 >,...,x n > }, α > n