2 x x, y, z x,, z F c : x x x cos y sin z z 8 F F F F F x x x F x x F 9 F c J Fc J Fc x x x y y y cos sin 0 sin cos 0 0 0, J Fc 0 J Fc t x /x J Fc,, z
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1 The Austalian National Univesity 9 5 F n n 0,, 2, F F ě i F f F g F g 2 ij F F x. ě i F F x i i, 2, 3 2 f i ě i f f x i. 3 g ij ě i g ě j g x i ě j g j x i. 4 g ij g x g 2 x g 3 x g x 2 g 2 x 2 g 3 x 2 g x 3 g 2 x 3 g 3 x 3. 5 x i i g j j F 2 T T 3 T ijk T T ijk ě i ě j ě k. 6 tiads tiads ě i ě j ě k T ijk T ě k ě j ě i ě k ě j ě i T T ě k ě j ě k T ě j T jk. 7 x i x i x i
2 2 x x, y, z x,, z F c : x x x cos y sin z z 8 F F F F F x x x F x x F 9 F c J Fc J Fc x x x y y y cos sin 0 sin cos 0 0 0, J Fc 0 J Fc t x /x J Fc,, z, /, z e, e, e z e, e, e z z F x/x F t J F c F x xx F x F x, J F c J F c x e, F x F x, cos sin 0 sin cos x e, F x F x, 2 J F 3 c x e z. 4 e e 0 e e 0 e 0 e e e 0 5 e z 0 e z 0 e z 0. 2
3 gadient f f f f f + + z e f + e f + e f z 6 f e f 7 f e f f. 8 f e f f, 9 f z e z f f. 20 I e e + e e + e z e z e e I e e e + e e + e z e z e e + e e + e e z e e. 2 e e e, 22 e z g g e g + e g + e z g z g g e g + e g + e z g z g e + g e + g z e z + g e + g e g e + e g + e g g z e + e + e g z + e + e g z + e g z z g g g g g + g g g g + e z e z + e e g e e g g z g z g z g e 24 divegence otation/cul g g T g e i g e i g i x i 25 3
4 e T e e, 26 e 0, e z g g e g + e g + e z g z g e g + e g + e z g z + g g + g + g + g z e i e j g e k e k g e j 29 i g e g + e g + e z g z e 0, e z 0, 30 e e e e z. 3 g e + g e + g z e z + g e z g e + e g + e g g z e + e + e g z + e + e g z + e g z z e z + g e z g z g g e + g z e + g + g g + e z g e g e z x f f f f x x i f i x x i + f i x x i i 2 f x x j x j x i + f i x x i. 34 i 4
5 x i x i x i, x i x i e e +, 35 e e e e 2 0, 36 z e z f 2 f f f 2 + f f f f g g e i g i e i g g e i. 39 T f T f + T t T f 40 T f T T f T T f + T t T f 4 T T f T f T f Tjk T jk e i e j e k f l e l f k e i e j 43 x i x i T T f f k T ik x i. 44 T T ij i e j 45 g e i g e i T t g e i. 46 a b a b + b a 47 5
6 46 g e i g e i e i g. 48 g i g i e i g T t g e i. 49 g 2, 3 e g e e e g e z g 0. e g + e g + e z g z e e e g + e g + e z g z g e g 2, 50 g e g 2, 5 52 t g e g + e g + e z g z + g e e g e e 53 T t g e T e g + e g + e z g z + g e e g e e ] e e T g e e + e g e + e g z ze + e g e e g e g 2 + g 2, 54 T t g e T e g + e g + e z g z + g e e g e e ] e e T g e e + e g e + e g z ze + e g e e g e g 2 + g 2, 55 T t g e z g g 2 g g g + g 2 g z g z 2 g + g 2 2 g + g 2 g 2 g 2 g 2, 57 g + 2 g 2 g 2,
7 3 x x, y, z x,, F p : x x x sin cos y sin sin z cos 60 J Fp J F p,, x, y, z x, y, z,, sin cos sin sin cos cos cos cos sin sin sin sin cos sin 0 sin cos cos cos sin sin sin sin cos sin sin cos cos sin 0, J Fp 2 sin, 6 J F p 2 sin 62 sin cos sin sin cos, cos cos cos sin sin, sin sin cos sin 0 63 e, e, e sin 64 e 0, e e, e e 0, e e, e 0, e 0, e sin e, e cos e, sin e cos e 65 e e e + e e, e, e 0, cos e e e + sin e e, e e e e cos sin, e e, cos sin e e, e 0, e cos sin e e 66 7
8 f f e + e f + e f sin 67 g g e + e g + e g e sin g + e + e g + e g e sin g + e + e g + e g e sin +g e e + e e + g e e + cos sin e e + g e e cos sin e e g g g g g g g sin sin + g g g cos sin g sin g g + g + g cos sin 68 g 2 2 g + sin g sin + g sin g e sin g sin g e sin g e g g sin + g f 2 2 cos, 2, 0 7 sin 2 f + 2 sin sin f + f 2 sin
9 e g e g 2 sin 2 sin g sin + g ] g sin + cos sin ] g e g 2 sin g sin + g cos T t g e 2 2g sin + sin g sin + g ] T t g e 2 g + cos sin g + sin 2 g + cos ] g sin 2 T t g e g 2 sin + cos sin 2 g sin 2 g ] g g 2g sin g 2 sin 2 sin g g + 2 g 2 2 cos g 2 sin 2 g 2 sin 2 g g + 2 sin 2 g 2 sin g + 2 cos g g ] x x, y, z x,, F t : x x x R + cos sin y R + cos cos z sin 82 R, J F t J Ft cos sin cos cos sin sin sin sin cos cos cos R+ cos sin R+ cos 0 cos sin sin sin R + cos cos cos cos sin cos R + cos sin sin cos 0, J Ft, R + cos 83 J F R + cos 84 t 9
10 cos sin cos cos sin, sin sin sin cos cos, cos R+ cos sin R+ cos 0 85 e, e, e R + cos 86 e 0, e e, e e 0, e e, e 0, e 0, e cos e, e sin e, cos e + sin e 87 e x, x cos 88 e e cos e + R + cos e e, 89 e e sin e R + cos e e, 90 e cos R + cos e sin e + R + cos e e 9 e + cos R+ cos, e 0, R+ cos e sin e e, e 0, e sin R+ cos e cos R+ cos e 92 e R + x x e 93 f f e + e f + e f R + cos 94 0
11 g g e + e g + e R + cos g + e + e g + e R + cos g + e + e g + e R + cos +g e cos e + R + cos e e +g cos R + cos e e + R+ cos g g e e g e + g e e sin R + cos e e sin R + cos e e g g g g g g + g g g g cos g R+ cos + g g sin R+ cos + g cos g sin 95 g R + cos R + cos g + R + cos g + g ] g g e g R + cos + g sin + g e R + cos g cos g + e g g cos R + cos, sin, 0 98 R + cos f R + cos R + cos f + 2 R + cos R + cos f 2 f + R + cos
12 e g 2 g e g 2 g + e g R + cos 2 g sin R + cos + cos g R + cos 2 sin g sin R + cos g R + cos 2 cos g + sin g ] T t g e g 2 + ] g 2 + cos g R + cos 2 + g cos g sin T t g e g 2 ] g 2 sin g R + cos 2 + g cos g sin T t g e R + cos 2 cos g + sin g ] + g g g g 2 2 g 2 2 cos g R + cos 2 g cos 2 R + cos 2 + g cos sin R + cos 2 + g sin R + cos g g g g sin g R + cos 2 g g + g sin cos R + cos 2 g sin R + cos 2 06 R + cos g sin 2 R + cos 2 07 g 2 cos g + 2 sin g ] 08 ],,. 2
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24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),
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1424 10 11 X 39,000 100% 1425 10 11 L 41,000 100% 1426 10 11 26,000 100% 1427 10 11 L 27,000 100% 1428 10 11 32,000 100% 1429 10 11 L 34,000 100% 1430 10 11 29,000 100% 1431 10 11 L 26,000 100% 1432 P
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DVIOUT
A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
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1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),
数学演習：微分方程式
( ) 1 / 21 1 2 3 4 ( ) 2 / 21 x(t)? ẋ + 5x = 0 ( ) 3 / 21 x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 ( ) 3 / 21 x(t)? ẋ + 5x = 0 x(t) = t 2? ẋ = 2t, ẋ + 5x = 2t + 5t 2 0 x(t) = sin 5t? ẋ
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#A A A. F, F d F P + F P = d P F, F P F F A. α, 0, α, 0 α > 0, + α +, α + d + α + + α + = d d F, F 0 < α < d + α + = d α + + α + = d d α + + α + d α + = d 4 4d α + = d 4 8d + 6 http://mth.cs.kitmi-it.c.jp/
2018_センター試験速報_数2B.indd
08 年度大学入試センター試験解説 数学 Ⅱ B 第 問 60 80 ア = 80 = 80 44 = 80 44 = 4 5 イ, ウ = 80 = 45 エオカ sin + 5 - cos + 0 = = + 5 = - 5 sin - cos { - 5 + 0 } = sin - cos - = キ 6 cos - 6 = cos cos 6 + sin sin 6 = cos + sin
[] x < T f(x), x < T f(x), < x < f(x) f(x) f(x) f(x + nt ) = f(x) x < T, n =, 1,, 1, (1.3) f(x) T x 2 f(x) T 2T x 3 f(x), f() = f(t ), f(x), f() f(t )
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difgeo1.dvi
1 http://matlab0.hwe.oita-u.ac.jp/ matsuo/difgeo.pdf ver.1 8//001 1 1.1 a A. O 1 e 1 ; e ; e e 1 ; e ; e x 1 ;x ;x e 1 ; e ; e X x x x 1 ;x ;x X (x 1 ;x ;x ) 1 1 x x X e e 1 O e x x 1 x x = x 1 e 1 + x
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応力とひずみ.ppt
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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
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DT-870/5100 &DT-5042RFB ...3 1-1...3 1-1...6 1-3...16 2....17...21 3-1...21 3-2...21 3-2...22 3-3...23 3-4...24...25 4-1....25 4-2...27 4-3...28 4-4...33 4-5...36...37 5-1....39 5-2...40 5-3...43...49
6 6.1 L r p hl = r p (6.1) 1, 2, 3 r =(x, y, z )=(r 1,r 2,r 3 ), p =(p x,p y,p z )=(p 1,p 2,p 3 ) (6.2) hl i = jk ɛ ijk r j p k (6.3) ɛ ijk Levi Civit
6 6.1 L r p hl = r p (6.1) 1, 2, 3 r =(x, y, z )=(r 1,r 2,r 3 ), p =(p x,p y,p z )=(p 1,p 2,p 3 ) (6.2) hl i = jk ɛ ijk r j p k (6.3) ɛ ijk Levi Civita ɛ 123 =1 0 r p = 2 2 = (6.4) Planck h L p = h ( h
genron-7
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1 1. x 1 (1) x 2 + 2x + 5 dx d dx (x2 + 2x + 5) = 2(x + 1) x 1 x 2 + 2x + 5 = x + 1 x 2 + 2x x 2 + 2x + 5 y = x 2 + 2x + 5 dy = 2(x + 1)dx x + 1
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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
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chap1.dvi
1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
知能科学：ニューラルネットワーク
2 3 4 (Neural Network) (Deep Learning) (Deep Learning) ( x x = ax + b x x x ? x x x w σ b = σ(wx + b) x w b w b .2.8.6 σ(x) = + e x.4.2 -.2 - -5 5 x w x2 w2 σ x3 w3 b = σ(w x + w 2 x 2 + w 3 x 3 + b) x,
知能科学：ニューラルネットワーク
2 3 4 (Neural Network) (Deep Learning) (Deep Learning) ( x x = ax + b x x x ? x x x w σ b = σ(wx + b) x w b w b .2.8.6 σ(x) = + e x.4.2 -.2 - -5 5 x w x2 w2 σ x3 w3 b = σ(w x + w 2 x 2 + w 3 x 3 + b) x,
A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π
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t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ
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