f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) B p.1/14
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1 B p.1/14
2 f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) B p.1/14
3 f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) f(x 1,...,x n ) (x 1 x 0,...,x n 0), (x 1,...,x n ) i x i f xi (x 1,...,x n ) B p.1/14
4 f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) f(x 1,...,x n ) (x 1 x 0,...,x n 0), (x 1,...,x n ) i x i f xi (x 1,...,x n ) [ ] x (x 0,y 0 ) y (x 0,y 0 ) f (x 0,y 0 ) ( ) B p.1/14
5 B p.2/14
6 [ ] f(x,y) x(t),y(t) f(x(t),y(t)) df dt = x dx dt + y dy dt B p.3/14
7 [ ] f(x,y) x(t),y(t) f(x(t),y(t)) df dt = x dx dt + y dy dt [ ] f(x(t),y(t)) = f(x(t 0 ),y(t 0 )) + f x (x(t 0 ),y(t 0 )) (x(t) x(t 0 )) +f y (x(t 0 ),y(t 0 )) (y(t) y(t 0 )) + R (x(t),y(t)) B p.3/14
8 [ ] f(x,y) x(t),y(t) f(x(t),y(t)) df dt = x dx dt + y dy dt [ ] f(x(t),y(t)) = f(x(t 0 ),y(t 0 )) + f x (x(t 0 ),y(t 0 )) (x(t) x(t 0 )) +f y (x(t 0 ),y(t 0 )) (y(t) y(t 0 )) + R (x(t),y(t)) f(x(t),y(t)) f(x(t 0 ),y(t 0 )) t t 0 =f x (x(t 0 ),y(t 0 )) x(t) x(t 0) t t 0 +f y (x(t 0 ),y(t 0 )) y(t) y(t 0) t t 0 + R(x(t),y(t)) t t 0 B p.3/14
9 [ ] f(x,y) x(t),y(t) f(x(t),y(t)) df dt = x dx dt + y dy dt [ ] f(x(t),y(t)) = f(x(t 0 ),y(t 0 )) + f x (x(t 0 ),y(t 0 )) (x(t) x(t 0 )) +f y (x(t 0 ),y(t 0 )) (y(t) y(t 0 )) + R (x(t),y(t)) f(x(t),y(t)) f(x(t 0 ),y(t 0 )) t t 0 =f x (x(t 0 ),y(t 0 )) x(t) x(t 0) t t 0 +f y (x(t 0 ),y(t 0 )) y(t) y(t 0) R(x(t),y(t)) R(x(t),y(t)) t t 0 = {x(t) x(t 2 0 )} 2 +{y(t) y(t 0 )} R(x(t),y(t)) {x(t) x(t 0 )} 2 +{y(t) y(t 0 )} 2 t t 0 + R(x(t),y(t)) t t 0 {x(t) x(t 0 )} 2 +{y(t) y(t 0 )} 2 t t 0 = {x (t 0 )(t t 0 )+R 1 (t)} 2 +{y (t 0 )(t t 0 )+R 2 (t)} 2 t t 0 B p.3/14
10 [ ] f(x 1,...,x n ) x 1 (t),...,x n (t) f(x 1 (t),...,x n (t)) f = f x1 x f xn x n B p.4/14
11 [ ] f(x 1,...,x n ) x 1 (t),...,x n (t) f(x 1 (t),...,x n (t)) f = f x1 x f xn x n [ ] f(x,y) x(u,v),y(u,v) f(x(u,v),y(u,v)) u = x x u + y y u, v = x x v + y y v B p.4/14
12 [ ] f(x 1,...,x n ) x 1 (t),...,x n (t) f(x 1 (t),...,x n (t)) f = f x1 x f xn x n [ ] f(x,y) x(u,v),y(u,v) f(x(u,v),y(u,v)) u = x x u + y y u, v = x x v + y y v f(x 1,...,x m ) x 1 (y 1,...,y n ),...,x m (y 1,...,y n ) f(x 1 (y 1,...,y n ),...,x m (y 1,...,y n )) = x x m y i x 1 y i x m y i B p.4/14
13 B p.5/14
14 [ ] f(x) x(u,v) f(x(u,v)) u = df dx x u, v = df dx x v B p.6/14
15 [ ] f(x) x(u,v) f(x(u,v)) u = df dx x u, v = df dx x v f(x) x(y 1,...,y n ) f(x(y 1,...,y n )) = df y i dx x y i B p.6/14
16 B p.7/14
17 [ ] n = α β (x,y) f(x,y) f(x + tα,y + tβ) f(x,y) D n f(x,y) = lim t 0 t B p.8/14
18 [ ] n = α β (x,y) f(x,y) f(x + tα,y + tβ) f(x,y) D n f(x,y) = lim t 0 t = α x + β y = (f x,f y ) α β B p.8/14
19 [ ] n = α β (x,y) f(x,y) f(x + tα,y + tβ) f(x,y) D n f(x,y) = lim t 0 t = α x + β y = (f x,f y ) α β f(x,y) n n B p.8/14
20 [ ] n = α β (x,y) f(x,y) f(x + tα,y + tβ) f(x,y) D n f(x,y) = lim t 0 t = α x + β y = (f x,f y ) α β f(x,y) n n (f x,f y ) gradf B p.8/14
21 [ ] n = α β (x,y) f(x,y) f(x + tα,y + tβ) f(x,y) D n f(x,y) = lim t 0 t = α x + β y = (f x,f y ) α β f(x,y) n n (f x,f y ) gradf D n f(x,y) f(x,y) n gradf B p.8/14
22 Dnf(x,y) (x,y) n B p.9/14
23 B p.10/14
24 [ ] f(x,y) f x (x,y),f y (x,y) f 2 f x = 2 x x, 2 f y x = y x, 2 f x y = x y, 2 f y = 2 y y B p.11/14
25 [ ] f(x,y) f x (x,y),f y (x,y) f 2 f x = 2 x x, 2 f y x = y x, 2 f x y = x y, 2 f y = 2 y f xx,f xy,f yx,f yy ( f xy,f yx x,y ) y B p.11/14
26 [ ] f(x,y) f x (x,y),f y (x,y) f 2 f x = 2 x x, 2 f y x = y x, 2 f x y = x y, 2 f y = 2 y f xx,f xy,f yx,f yy ( f xy,f yx x,y ) [ ] f xy f yx f xy = f yx y B p.11/14
27 [ ] f(x,y) f x (x,y),f y (x,y) f 2 f x = 2 x x, 2 f y x = y x, 2 f x y = x y, 2 f y = 2 y f xx,f xy,f yx,f yy ( f xy,f yx x,y ) [ ] f xy f yx f xy = f yx y B p.11/14
28 [ ] f(x,y) f x (x,y),f y (x,y) f 2 f x = 2 x x, 2 f y x = y x, 2 f x y = x y, 2 f y = 2 y f xx,f xy,f yx,f yy ( f xy,f yx x,y ) [ ] f xy f yx f xy = f yx f(x 1,...,x n ) k f (= f xi1 x x ik x ik ) ( ) i1 y B p.11/14
29 B p.12/14
30 [ ] x = r cosθ,y = r sin θ ( ) f(x,y) f = 2 f x + 2 f 2 y = 2 f 2 r r r f r 2 θ 2 B p.13/14
31 [ ] x = r cosθ,y = r sin θ ( ) f(x,y) f = 2 f x + 2 f 2 y = 2 f 2 r r r f r 2 θ 2 x = r cosθ,y = r sin θ,z = z ( ) f(x,y,z) f = 2 f x + 2 f 2 y + 2 f 2 z = 2 f 2 r r r f r 2 θ + 2 f 2 z 2 B p.13/14
32 [ ] x = r cosθ,y = r sin θ ( ) f(x,y) f = 2 f x + 2 f 2 y = 2 f 2 r r r f r 2 θ 2 x = r cosθ,y = r sin θ,z = z ( ) f(x,y,z) f = 2 f x + 2 f 2 y + 2 f 2 z = 2 f 2 r r r f r 2 θ + 2 f 2 z 2 x = r sin θ cos ϕ,y = r sin θ sin ϕ,z = r cosθ ( ) f(x,y,z) f = 1 ( r 2 ) + 1 ( sin θ ) f r 2 r r r 2 sin θ θ θ r 2 sin 2 θ ϕ 2 B p.13/14
33 [ ] x = r cosθ,y = r sin θ ( ) f(x,y) f = 2 f x + 2 f 2 y = 2 f 2 r r r f r 2 θ 2 x = r cosθ,y = r sin θ,z = z ( ) f(x,y,z) f = 2 f x + 2 f 2 y + 2 f 2 z = 2 f 2 r r r f r 2 θ + 2 f 2 z 2 x = r sin θ cos ϕ,y = r sin θ sin ϕ,z = r cosθ ( ) f(x,y,z) f = 1 ( r 2 ) + 1 ( sin θ ) f r 2 r r r 2 sin θ θ θ r 2 sin 2 θ ϕ 2 Laplacian B p.13/14
34 ( A) B p.14/14
DVIOUT
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1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim
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1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
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63 6 6.1 6.1.1 v = v 0 =v 0x,v 0y, 0) t =0 x 0,y 0, 0) t x x 0 + v 0x t v x v 0x = y = y 0 + v 0y t, v = v y = v 0y 6.1) z 0 0 v z yv z zv y zv x xv z xv y yv x = 0 0 x 0 v 0y y 0 v 0x 6.) 6.) 6.1) 6.)
D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y
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20010916 22;1017;23;20020108;15;20; 1 N = {1, 2, } Z + = {0, 1, 2, } Z = {0, ±1, ±2, } Q = { p p Z, q N} R = { lim a q n n a n Q, n N; sup a n < } R + = {x R x 0} n = {a + b 1 a, b R} u, v 1 R 2 2 R 3
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(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e 0 1 15 ) e OE z 1 1 e E xy 5 1 1 5 e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0 Q y P y k 2 M N M( 1 0 0) N(1 0 0) 4 P Q M N C EP
(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)
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2016 5 2 ) Binary Cubic Forms 2016 5 2 1 / 25 1 2 2 2 3 2 3 ) Binary Cubic Forms 2016 5 2 2 / 25 1.1 ( ) 4 2 12 = 5+7, 16 = 5+11, 36 = 7+29, 1.2 ( ) p p+2 3 5 5 7 11 13 17 19, 29 31 41 43 ) Binary Cubic
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9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x
2009 9 6 16 7 1 7.1 1 1 1 9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x(cos y y sin y) y dy 1 sin
lim lim lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d
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1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
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II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
1 3 1.1.......................... 3 1............................... 3 1.3....................... 5 1.4.......................... 6 1.5........................ 7 8.1......................... 8..............................
1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1
sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V
1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2
1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2
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II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................
5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
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2017 12 9 4 1 30 4 10 3 1 30 3 30 2 1 30 2 50 1 1 30 2 10 (1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) (1) i 23 c 23 0 1 2 3 4 5 6 7 8 9 a b d e f g h i (2) 23 23 (3) 23 ( 23 ) 23 x 1 x 2 23 x
f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f
22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )
.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(
06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,
1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (
1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +
() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (
3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc
( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x +
(.. C. ( d 5 5 + C ( d d + C + C d ( d + C ( ( + d ( + + + d + + + + C (5 9 + d + d tan + C cos (sin (6 sin d d log sin + C sin + (7 + + d ( + + + + d log( + + + C ( (8 d 7 6 d + 6 + C ( (9 ( d 6 + 8 d
() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.
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() 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >
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( ) Loewner SLE 13 February
( ) Loewner SLE 3 February 00 G. F. Lawler, Conformally Invariant Processes in the Plane, (American Mathematical Society, 005)., Summer School 009 (009 8 7-9 ) . d- (BES d ) d B t = (Bt, B t,, Bd t ) (d
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, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,,
6,,3,4,, 3 4 8 6 6................................. 6.................................. , 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p,
II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R
![II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R](/thumbs/95/125540821.jpg "II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R")
II Karel Švadlenka 2018 5 26 * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* 5 23 1 u = au + bv v = cu + dv v u a, b, c, d R 1.3 14 14 60% 1.4 5 23 a, b R a 2 4b < 0 λ 2 + aλ + b = 0 λ =
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
重力方向に基づくコントローラの向き決定方法
( ) 2/Sep 09 1 ( ) ( ) 3 2 X w, Y w, Z w +X w = +Y w = +Z w = 1 X c, Y c, Z c X c, Y c, Z c X w, Y w, Z w Y c Z c X c 1: X c, Y c, Z c Kentaro Yamaguchi@bandainamcogames.co.jp 1 M M v 0, v 1, v 2 v 0 v
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
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