1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =

Size: px

Start display at page:

Download "1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f ="

ぜんましまむね
5 years ago
Views:

1 1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i<j a ij x i x j = ( x, A x), f = λ i x i x x = u cos θ v sin θ y y = u sin θ + v cos θ A = {a ij } = t A. (u, v) θ, uv f = ax + bxy + cy U = ( cos θ sin θ sin θ cos θ ) f f = λ 1 u + λ v, λ i ( ) tan(θ) = b a c, λ a b i. λ i U = b c. Φ(x) = det(x A) = x (a+c)x+(ac b ) = 0. λ i = a + c ± (a c) + 4b

2 8, :, A {λ 1,, λ n }, {u 1,, u n } U = (u 1,, u n ) ( ), t U = U 1 : (x, Ax) (Ux, AUx) = (x, t UAUx) = (x, diag(λ i )x) = λ i x i λ i det(x A) = 0 n λ 1 λ n λ i = det( A) = ( 1) n det(a)) TrXY = TrY X TrU 1 AU = λ i = a ii f = i a ii x i + i<j a ij x i x j + i b i x i x i A, f a ij = a ji A = {a ij } f = i = i a ii x i + a ij x i x j + i<j i x i j a ij x j + b i x i i b i x i = i x i (Ax) i + (b, x) = (x, Ax) + bx = (x 1 1 b, A(x A A b)) 1 4 (b, A 1 b) 1 4 (b, A 1 b) 8. C 1 (Ω), C (Ω) C (Ω) Ω f x, f y f xx, f xy, f yx, f yy f = sup( f(x, y + + f yy (x, y) ) x,y f(x, y) = i g i(x)h i (y)) f xy = ( ) y f = y ( g i(x)h i (y)) = g i(x)h i(y)) = f yx

3 8.. C 1 (Ω), C (Ω) f C () f xy = f yx = f(x + h, y + k) + f(x, y) f(x + h, y) f(x.y + k) = Φ(x + h, y) Φ(x, y) = hφ x (x + θh, y) = Ψ(x, y + k) Ψ(x, y) = kψ(x, y + θ k) Φ(x, y) = f(x, y + k) f(x, y), Ψ(x, y) = f(x + h, y) f(x, y). h(f x (x + θh, y + k) f x (x + θh, y)) = k(f y (x + hh, y + θ k) f x (x, y + θ k)) hkf xy (x + θh, y + θ k) = hkf yx (x + θ h, y + θ k), hk, h, k 0, f C f xy (x + θ h, y + θ k) f xy (x, y), f yx (x + θh, y + θ k) f yx (x, y) 8..1 [ Peano ] { xy(x y ) f(x, y) = x +y (x, y) (0, 0) 0 (x, y) = (0, 0) (1) f x (0, y), () y f(x, 0), (3) f xy (0, 0), f yx (0, 0), f xy (0, 0) f yx (0, 0). f(0, y) = f(x, 0) = 0 f(h, y) f(0, y) f(h, y) f x (0, y) = lim = lim h 0 h h 0 h 1 = lim h 0 h hy(h y ) h + y f y (x, 0) = x. f xy (0, y) = 1, f yx (x, 0) = 1 = y

4 4 8, : 8.. f, (1) f C 1, (), (3), (4), (1) () (3) (4). ( )... (1) (), () (1). (1) () ( ) f(x, y) x, y, C 1 x = x(t), y = y(t) t g(t) = f(x(t), y(t)), α = x (x),β = y (x) g(t + h) = f(x(t) + αh + o(h), y(t) + βh + o(h)) = f(x(t), y(t)) + f x (x(t), y(t))αh + f y (x(t), y(t))βh + o(h) = g(t) + f x x (t)h + f x y (t)h + o(h) d dx f(x(t), y(t)) = dt dt f x + dy dt f y f(x, y) x, y, C 1 x = x(s, t), y = y(s, t) s, t C 1, s f(x(s, t), y(s, t)) = x sf x + y s f y t f(x(s, t), y(s, t)) = x tf x + y t f y 8.3. x i = a ij u j u i = a ij x j, j=1 j=1 a ij = (A 1 ) ij f ui = a ij f xj, f ui,u j = k,l a ik a jl f xk,x l A ( t AA = AA t = 1) (fxi ) = (f ui ), fxi,x i = f ui,u i.

5 A ) t AA = A t A = 1 i a ija ik = i a kia ji = δ jk. ( i = j δ ij = 1, i j δ ij = 0). (f ui ) = ( a ij f xj )( a ik f xk ) = ( a ij a ik )f xj f xk = δ jk f xj f xk = (f xj ) i i j k j,k i j,k j f ui = k a ikf xk,. f ui u j = l a il a jk f xk x l k x = r cos θ, y = r sin θ r = (x + y ) 1/, θ = tan 1 (y/x), f r = cos θf x + sin θf y, f θ = r sin θf x + r cos θf y f x = cos θf r sin θ r f θ, f y = sin θf r + cos θ f θ r f x = r x f r + θ x f θ = x r f y r x + y f θ = cos θf r sin θ r f θ, f y = r y f r + θ y f θ = y r f r + x x + y f θ = sin θf r + cos θ f θ, r = x + y = r + 1 r θ + 1 r θ x = (cos θ r (sin θ/r) θ ), y = (sin θ r + (cos θ/r) θ ) ( x ) = cos θ r (cos θ r )(sin θ/r) θ ((sin θ/r) θ ) cos θ r + ((sin θ/r) θ ) = cos θ r (sin θ/r) r θ + (sin θ/r) θ +(cos θ sin θ/r ) θ + (sin θ/r) r + (cos θ sin θ/r ) θ ( y ) = sin θ r + (sin θ/r) r θ + (cos θ/r) θ. (cos θ sin θ/r ) θ + (cos θ/r) r (cos θ sin θ/r ) θ

6 6 8, : 8.3.4, x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ, ( ( ) r = (x + y + z ) 1/, ϕ = tan 1 y ), θ = cos 1 z x x + y + z = 3 /x i , f(x + h, y + k) = f(x + th, y + tk) t=1 g(t) t=1 = g(0) + g (0) + 1 1! g (0) + + (n 1)! g(n 1) (0) + 1 n! g(n) (θ) g (t) = d dt f C () (D) f(x + h, y + k) = n 1 k=0 g(t) = (dx dt x + dy dt )f(x(t), y(t)) = (h y x + k )f(x, y) y 1 k! (h x + k y ) k f(x, y) + 1 n! (h x + k y ) k f(x + θh, y + θk) = f(x, y) + f x (x, y)h + f y (x, y)k + 1 (f xx(x, y)h + f xy (x, y)hk + f yy (x, y)k ) +o(( h + k ) ) f x = f y = 0 x, y f(x + h, y + k) = f(x, y) + 1 ( ) ( ) h h <, A > +O( h 3 + k 3 ) k k ( ) A f xx f xy f xy f yy λ 1, λ λ 1 + λ = f xx + f yy, λ 1 λ = f xx f yy f xy.,. λ 1 h + λ k (1) A x R nm < x, Ax >> 0 () A x < x, Ax > 0.

7 < x, Ax > A = A ( < Ax, y >=< x, Ay >). B = (A + A )/, C = i(a A )/, A = B + ic. < x, Ax > R, < x, Ax >=< x, Bx > +i < x, Cx > R. < x, Ax > R, < x, Cx > R, < x, Cx >= 0 x. C = 0 A = A A z C n, < z, Az >> 0 ( ). A A 1 = a 11 > 0, A = a 11 a a 1 > 0,, A n = A > 0 ( ). (, ) a 11 a 1 a 1n a 11 a 1 a 1k a 1 a a n a 1 a a k A =, A k = a n1 a n.. ann a k1 a n.. akk 1,,, n 1, ( n. ) a = t (a 1n, a n,, a n 1,n ) E n 1 A 1 n 1, P = a t 0 1 ( ) A = t A n 1 0 P BP, B t 0 d d = a nn < a, A 1 n 1 a >, det(a) = det(a n 1) d A n 1 d > 0 z = t (x, y) R n, x R n 1, y R, < z, Bz >=< x, A n 1 x > +dy B, A ( Feshbach Krein ) ( ) ( ) ( ) h h <, A >= ah + bhk + ck = c(k + bc k k h) + ac b k c, c > 0, (ac b )/c > 0.,.., y i = k b ijx j a ii x i + a ij x i x j = i<j n α i yi, α i p, m, 0 z. (p, m, z)..

8 8 8, : 8.5, df = f x i dx i, d f = j=1 f x i x j dx i d xj =< dx, Adx >, df = 0, f xi = 0 (i = 1,,, n) U ε (x) = {z; z x < ε} x R n ε.. D, x D A, ε > 0 U ε (x) D. D intd D o. 3. D, x D, x D c. ε > 0 U ε (x) D c = R n \D = {x; x / D} 4., D c, D. 5. x D,. D D. (i),, ε > 0, U ε (x) D, U ε (x) D c. (ii) x D {x n D}, {y n D c } lim x n = x, lim y n = x A = {(x, y); x + y < 1}, A = {(x, y); x + y = 1}. A = {(x, y); x + y 1}, A = {(x, y); x + y = 1} 3. {x n R} 4. {1/n; n = 1,, }. {0} {1/n; n = 1,, }. 5. R. ϕ D = D D, intd = D\D = D (D) c (1)int(D c ) = (D) c, ()(D c ) = (intd) c ( A i ) c = A c i, ( A i) c = A c i. (1) (D) c = (D D) c = D c (D) c = D c (D c ) c = int(d c ) () (1) D D c, (D c ) c = D intd = (D c ) c, (intd) c = D c

9 (1) D, D = D D,. () D. f(x, y) D,, intd, D f = x + axy + y ( x, y 1). (i) : : f = (x + ay) + (1 a )y, a < 1 0, (1 + a ) + (1 a ) = (1 + a ). a > 1 (0, 0) df = (x + ay)dx + (ax + y)dy 0. y = 1, f = x + ax + 1, x < 1, (1 + a ), (1 a ). : π/4, f = x + y + a(x y ) = (1 + a)x + (1 a)y.. 3: f x = x + a = 0, f y = a + y = 0, a ±1 x = y = 0. a > 1, a < f = x + axy + y (x + y 1).. (i) : : x = r cos θ, y = r sin θ (0 < r < 1) r (1 + a sin θ) [r (1 a ), r (1 + a )], a < 1 a > 1 f = r (1 + a sin θ) [0, 1 + a ] f = r (1 + a sin θ) [1 a, 1 + a ] (ii), (iii) f = x 3 + y 3 3xy, 1/3 x, y / u i, f = 3 ( p i log p i + u i p i ) ( p i = 1, 0 p i 1) n [ p i u i 3 p i log p i ] p 3 = 1 p 1 p, f = ( p i log p i + u i p i ) + ( (1 p 1 p ) log(1 p 1 p ) + u 3 (1 p 1 p ) n Lagrange S = s(s a)(s b)(s c), s = (a + b + c)/ (a, b, c > 0).

10 10 8, : (1) s(s a)(s b)(s c) = 3(s/3)(s a)(s b)(s c) 3((10s/3 (a+b+c))/4) = 3(s/3) a = b = c = s/3 (). (), c = s a b f(a, b) = (s a)(s b)(a + b s). (a, b) 0 a s, 0 b s, s a + b.. a = b = s/3 : f a = (s a)(s b) (s b)(a + b s) = (s b)(s a b) f b = (s a)(s a b) 8.6, f(x + h, y + k) = f(x + th, y + tk) t=1 g(t) t=1 = g(0) + g (0) + 1 1! g (0) + + (n 1)! g(n 1) (0) + 1 n! g(n) (θ) g (t) = d dt f C () (D) f(x + h, y + k) = n 1 k=0 g(t) = (dx dt x + dy dt )f(x(t), y(t)) = (h y x + k )f(x, y) y 1 k! (h x + k y ) k f(x, y) + 1 n! (h x + k y ) k f(x + θh, y + θk) = f(x, y) + f x (x, y)h + f y (x, y)k + 1 (f xx(x, y)h + f xy (x, y)hk + f yy (x, y)k ) +o(h + k ) f x = f y = 0 x, y f(x + h, y + k) = f(x, y) + 1 ( ) ( ) h h <, A > +O( h 3 + k 3 ) k k ( ) A f xx f xy f xy f yy λ 1, λ λ 1 + λ = f xx + f yy, λ 1 λ = f xx f yy f xy.,. λ 1 h + λ k., < ( h k), A ( h k) > (h, k),, f xx > 0, f xx f yy (f xy ) > 0

11 (i) f xx = a, f yy = c, f xy = f yx = b ( ) ( ) h h <, A >= ah + bhk + ck = a(h + ba k k k) +, a > 0, (ac b )/a > 0. ac b k a (ii) A, x (a + c)x + ac b = 0 a + c > 0, ac b > 0. ac > 0 (ac > 0 ) a > 0, ac b > ( ) ( ) ( ) h h <, A >= ah + bhk + ck = c(k + bc k k h) + ac b k c, c > 0, (ac b )/c > 0.,.., y i = k b ijx j a ii x i + a ij x i x j = i<j n α i yi, α i p, m, 0 z (1)f(x, y) = x 3 + xy + 5x + y, ()f(x, y) = xy(a x y), (a 0) (1) f x = 6x + y + 10x, f y = y(x + 1). f xx = 1x + 10, f yy = (x + 1), f xy = y. f x = f y = 0 (x, y) = ( 1, ±), (0, 0), ( 5/3, 0) D = f xx f yy fxy, ( 1, ±) D < 0,, (0, 0) D > 0, f xx > 0, ( 5/3, 0) D > 0, f xx < 0, () f x = y(a x y), f y = x(a x y) f xx = y, f yy = x, f xy = a (x + y). f x = f y = 0, (x, y) = (0, 0), (0, a), (a, 0), (a/3, a/3) (x, y) = (0, 0), (0, a), (a, 0) D < 0. (a/3, a/3) D > 0, f xx = a/3 (a > 0) (a < 0) (1) f = x + (x y) + (y z) z () f = (x y) + (y z) + (z x),.

12 1 8, : : (1),() 8.7.3,, ). [014 ], f(x, y) = x 4 + y 4 x + 4xy y 8.7.4,, ), 0 < a < b. (1) f(x, y) = (ax + by ) exp[ x y ] () f(x, y) = sin x sin y sin(x + y) sin(x y) (3) f(x, y) = sin ax sin bx a, b > 0 (1) () f x = sin y sin(x + y), f y = sin x sin(x + y) f x = f y = 0, (x, y) = (mπ, nπ) (x, y) = ((n m)π/3, (m n)π/3) (m, n ). f xx = sin y cos(x + y), f yy = sin x cos(x + y) f xy = sin((x + y)). (mπ, nπ), (x, y) = ((n m)π/3, (m n)π/3),.. (3) f x = a cos ax sin by, f y = b sin ax cos by, cos ax = cos by = 0, sin by = sin ax = 0 (1) x = (π/ + mπ)/a, y = (π/ + nπ)/b, () x = mπ/a, y = nπ/b f xx = a sin ax sin bx, f yy = b sin ax sin bx, f xy = ab cos ax cos by (), (1) f xy = 0, m.n f xx < 0, f yy < 0, m.n f xx > 0, f yy > 0, m, n f xx f yy < 0.

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) 0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()