017 8 10 f : R R f(x) = x n + x n 1 + 1, f(x) = sin 1, log x x n m :f : R n R m z = f(x, y) R R R R, R R R n R m R n R m R n R m f : R R f (x) = lim h 0 f(x + h) f(x) h f : R n R m m n M Jacobi( ) m n Jacobian( M detm) det M = J + e x dx n J 1
f : R R f(x, y) = x + y axy f = 0, x + y axy = 0 y 直線 x+y+a=0 に漸近し 原点で交叉する美しい形をしている x +y axy=0 X+Y+a=0 o x t x = at 1 + t, y = at (a > 0) 1 + t f(x, y) = x +y axy n R n S m R m f(x) = (f 1 (x), f (x),, f m (x))(x = (x 1, x,, x n )) f k (x) S f S f : S R n R m x 1,, x n )y 1,, y m ) f k (x) S C 1 f S C 1 C 1 f(x) = (f 1 (x), f (x),, f m (x)) n m
f k x j (1 k m, 1 j n) n m (m, n) Df = x 1 f f x 1 x x n f x n x x 1 x n x f Jacobi( f Jacobi Df a S M = Df(a) = ( f k x j (a)) = (α kj ) f k f k (x) f k (a + h) f k (a) = lim η k h = 0 η = t (η 1, η,, η m ) lim h 0 η h = 0 n α kj h j + η k j=1 f(a + h) f(a) = Mh + η n = m Jacobi Df = M n detm = det(df) f x 1 1 x n detm = f n f x 1 n x n f Jacobian) [ ] = (f 1,, f n ) (x 1,, x n ) x y R P=(x, y) O P r = OP OP x θ x = r cos θ y = r sin θ r 0, 0 θ < π R [ ) [ π) = (r, θ) R : : 0 r, 0 r < π } R g Jacobi [ ] cos θ r sin θ Dg = sin θ r cos θ
(x, y) J(g) = (r, θ) = r [ ] R P = (x, y, z) r = OP OP z θ, P x, y) P OP x ϕ x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ P = (x, y, z) z 立体極座標 p θ o y x φ P r θ ϕ x y z g sin θ cos ϕ r cos θ cos ϕ r sin θ sin ϕ Dg = sin θ sin ϕ r cos θ sin ϕ r sin θ cos ϕ cos θ r sin θ 0 (x, y, z) J(g) = (r, θ, ϕ) = sin θ cos ϕ(r sin θ cos ϕ) + sin θ sin ϕ(r sin θ sin ϕ) cos θ(r sin θ cos θ cos ϕ + r sin θ cos θ sin ϕ) = r sin θ f = f(x, y) f(x) = k 1 ν=0 f (ν) (0) x ν + R (R = 1 ν! k! f (k) (θx)x k )(θ (0, 1) 4
R S C n f(x, y) a, b) S (a + h, b + k) S f(a + h, b + k) t [0, 1] Φ(t) = f(a + th, b + tk) Φ [0,1] C n Φ(0) = f(a, b), Φ(1) = f(a + h, b + k) dφ dt = f x(a + th, b + tk)h + f y (a + th, b + tk)k d Φ dt = f xx h + f xy hk + f yy k d ν Φ dt ν = (h x + k y )ν f (h x + k y )ν = ν j=0 Φ(t) = n 1 1 d ν Φ ν=1 ν! dt ν (0)tν + R t = 1 f(a + h, b + k) = [ n 1 1 ν=0 (h ν! x + k ] y )ν f (a, b) + R f(a + h, b + k) = f(a, b) + (f x (a, b)h + f y (a, b)k) ( ν )h ν j k j ν j ( x) ν j ( y) j + 1! fxx (a, b)h + f xy (a, b)hk + f yy (a, b)k } + + R grad f = (f x, f y ) Hesse H f [ ] fxx f xy H f = f yx f yy ( ) h f(x + h, y + k) = f(x, y) + (grad f) + 1 ( ) h k (h, k)h f + + R k h, k)h f ( h k ) [ ] ( ) fxx f = (h, k) xy h f yx f yy k R h n f S R n C x (x 1,, x n ) S h = (h 1,, h n ) R n f(x + h) = f(x) + (grad f, h) + 1 H f [h] + R 5
(grad f, h) H f [h] = t hh f h = n n j=1 i=1 f x j x i h j h i f 11 f 1 f 1n = [h 1 h h n ] f 1 f f n f n1 f n f nn a f h 1 h. h n grad f(a) = (f x (a, b), f y (a, b)) = 0 f C grad(a) = 0 B = H f (a) f(a + h) = f(a) + 1 B [h] + R B [h] = t hh f h = f xx h + f xy hk + f yy k a) grad(a) = 0 h 0 B [h] > 0 a f b) grad(a) = 0 h 0 B [h] < 0 a f f : R R (1)f(a, b) = 0, grad f(a, b) = (f x (a, b), f y (a, b)) (0, 0) a, b) f(x, y) = 0 ()f(a, b) = 0, grad f(a, b) = (f x (a, b), f y (a, b)) = (0, 0) a, b) f(x, y) = 0 a, b) f(x, y) = 0 f(x, y) = 0 f(x, y) = x + y axy(a > 0) (1) grad f = (x ay, y ax) 6
grad f = 0 x, y) = (0, 0) (a, a) Hesse [ B ] fxx f xy B = f yx f yy f xx = x (x ay) = 6x f xy = y (x ay) = a f yx = a f yy = 6y [ ] 6x a B = a 6y x = [ y = 0 ] 0 a B = a 0 ( ) h h, k)b = f xx h + f xy hk + f yy k k = 6ahk h, k) B [h] 0, 0) x = [ y = a ] 6a a B = a 6a ( ) h h, k)b = 6ah + ( a)hk + 6ak k = 6a(h hk + k ) ( ) h (h, k) (0, 0) B > 0 k a, a) f f(a, a) = a f () f x = x ay f y = y ax (f x, f y ) = (0, 0) x = y = 0 x = y = a f 0, 0) = 0 0, 0) f(x, y) = 0 f(a, a) = a f(x, y) = 0 0, 0) () (1) 7
() x + y + a = 0 (1) () r = f(θ) θ = α, θ = β(α β) S β β S = 1 r dr = 1 f(θ)} dθ α α (1) x = r cos θ, y = r sin θ r cos θ + r sin θ ar cos θ sin θ = 0 a sin θ cos θ r = cos θ + sin θ S = 1 π/ a sin θ cos θ ( 0 cos θ + sin θ ) dθ π/ = a (1 + tan θ) 0 (1 + tan θ) dθ [ ] π/ = a 1 1 + tan θ 0 = a (θ π 1 1 + tan θ 0) d tan θ = sec θ = 1 dθ cos θ ) () π 1 S = 4 π ( a cos θ + sin θ ) 1 } a sin θ cos θ ( cos θ + sin θ ) dθ + a π } 9 sin θ cos θ ( 4 π cos θ + sin θ ) ( cos θ + sin θ ) dθ + a π 4 π (cos θ + sin θ) 9 sin θ cos } θ (cos θ + sin dθ + a θ) π cos θ(1 + tan θ) 9 sin θ cos } θ cos 6 θ(1 + tan dθ + a θ) 4 π t = tan θ dt dθ = 1 cos θ 8
π 4 π cos θ(1 + tan θ) 9 sin θ cos } θ cos 6 θ(1 + tan cos θdt + a θ) 0 1 (1 + tan θ) 9 } tan θ (1 + tan dt + a θ) 0 1 (1 + t) 9t } (1 + t ) dt + a [ = a 1 1 + t + ] 0 1 + t + a 1 [ ] 0 = a (t + 1)(t ) (1 + t)(1 t + t + a ) = a 1 (1) () R R R n R m S R C 1 f f(a, b) = 0 f y (a, b) 0 a R G ϕ(x) (1)ϕ(a) = b () a, b) S U x G f(x, ϕ(x)) = 0 ()ϕ(x) C 1 x G dϕ dx = f x/f y = f x (x, ϕ(x))/f y (x, ϕ(x)) [ ]f y (a, b) > 0 f y a, b) U( S) x, y) U f y (x, y) > 0 x f y f(a, b) = 0 x = a f(a, c) < 0 f(a, d) > 0 (a, c) U a, d) U f(x, c) f(x, d) x a R G = (a 1, a ) x G f(x, c) < 0, f(x, d) > 0 x G f(x, y) y f(x, y) = 0 y (c, d) y y = ϕ(x) f(x, ϕ(x)) = 0 ϕ(a) = b x x + h G ϕ(x) = y ϕ(x + h) = y + k f(x + h, y + k) f(x, y) = 0 θ (0, 1) 0 = f x (x + θh, y + θk)h + f y (x + θh, y + θk)k f y 0 h 0 k 0 1 h (ϕ(x + h) ϕ(x)) = k h = f x(x + θh, y + θk)/f y (x + θh, y + θk) () 9
y = ϕ(x) f(x, y) = 0 T r + m) R r+m T x, y) = (x 1,, x r, y 1,, y m ) T m R m C 1 f f(x, y) = (f 1 (x, y),, f m (x, y)) f j f N f N f = (x, y) T : f(x, y) = 0} f a, b) N f M = y m y m ((x, y) = (a, b)) det M 0 a R r A, b R m B, A B C 1 ϕ A B T N f (A B) = ϕ ϕ ϕ (x, ϕ(x) : x A} ϕ Jacobi Dϕ x A a, b) f 1 (x, y) = f (x, y) = = f m (a, b) = 0 Dϕ(x) = M(x, y) 1 y 1 = ϕ 1 (x) = ϕ 1 (x 1, x r ) y m = ϕ m (x) = ϕ m (x 1, x r ) a [ ] R R ϕ C 1 x, x + h A y = ϕ(x), ϕ(x + h) = y + k f j (x + h, y + k) f j (x, y) = 0 r f j m f j h α (x + θ j h, y + θ j k) + k β (x + θ j h, y + θ j k) = 0 (1 j m) x α y β α=1 β=1 10
θ j h 1. h r + M k 1. k m = 0 k 1.. k m = M 1 h 1.. h r 11