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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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..............................
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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
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.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 +
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() 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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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
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II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K
II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F
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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d dt A B C = A B C d dt x = Ax, A 0 B 0 C 0 = mm 0 mm 0 mm AP = PΛ P AP = Λ P A = ΛP P d dt x = P Ax d dt (P x) = Λ(P x) d dt P x =
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No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ
y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
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