|
|
|
- こうしろう かたいわ
- 5 years ago
- Views:
Transcription
1
2 sn cn dn
3 6 44 x 6. y = b sn b = k c k c
4 4
5 . ϕ y x = a sin ϕ y = b cos ϕ. x a + y b =. s ds = +.3. x = a cos ϕ dϕ y = b sin ϕ dϕ.3 ds = a b k = a A s a cos ϕ + b sin ϕ dϕ = a a b a sin ϕ dϕ. ds = a k sin ϕ dϕ s a = ϕ k sin ϕ dϕ k ϕ Ek ϕ Ek ϕ = ϕ k sin ϕ dϕ..4 ϕ = π Ek k Ek = E π Ek = π k sin ϕ dϕ..5 s = 4aEk. y = ±b x a 5
6 ds = + = b + a z = sin ϕ s = a z = x a z x a sin ϕ k z Ek ϕ = z dz Ek = x = a a k z z dz k z z. k z z dz.6.. x = aϕ sin ϕ y = a cos ϕ. x = aϕ b sin ϕ y = a b cos ϕ. x + y ax = b x + y. x a y b =. y b = sin x a x = b a cos x a. sin x a = x cos a φ = x a dφ = dφ a ds = + = + b a b sin x a a. = a + b b sin φ dφ = a + b k sin φ dφ k = a + b a a + b = b a + b sφ = s ds = a + b φ = a + b Ek φ. x a = π k sin φ dφ.7.s = a + b Ek.8 6
7 . Ek ϕ = ϕ k sin ϕ dϕ k = k = ϕ dϕ F k ϕ = k sin ϕ k <.9.9 ϕ = π Kk Kk = F k π = π dϕ k sin ϕ.. k i ϕ π z = sin ϕ dz = cos ϕ dϕ ϕ = π z = z F k ϕ = ϕ π cos ϕ z F k ϕ = cos ϕ = dz k z ii π ϕ π z = sin π ϕ ϕ = π π z = dz k z cos ϕ. sin ϕ = z z = z dz z k z dz = cosπ ϕ dϕ = cos ϕ dϕ = cos ϕ dϕ F k ϕ = z dz k z cos ϕ. π φ π cos ϕ cos ϕ = sin ϕ = z dz F k ϕ = = z k z z z dz z k z. i ii F k ϕ = z dz z k z F k ϕ Ek 7
8 .. lemnicate r = a cos θ x = r cos θ y = r sin θ x + y = a x y x + y = r x y = r cos θ. x + y = a cos φ x y = a cos 4 φ. + cos φ = sin φ x = a cos φ sin φ y = a cos φ sin φ x = a cos φ sin φ y = a sin φ cos φ a sin φ = 3 sin φ + sin φ dφ = ds = + sin φ 3 + sin φ + ds = a dφ sin φ a sin φ dφ. sin φ sin φ = φ = r = a s = a φ dφ = a F sin φ φ..3 s = 4 a K.4 k = i x = a z + z3 z3 y = az + z4 z z 3.5 Serret 845 z x + y = a x y ds = + = a dz + z 4 x + y = a z + z 4 x y = 4a z z 4
9 . cos φ = z sin φ = z + z 4 + z 4.6 sin φ = + z + z 4 dφ = + z dz + z 4 dφ = dz. sin φ + z 4.6 cos φ = z + z 4 z dz = + z 4 F φ.5 z x y z.5 x + y = a + z 4 x y x + y = azx + y x az + y az az = az az z z.5 x y ii x = η η = sin ϕ c c = π x 4 = x 4 c φ dϕ sin ϕ c = cos φ c x 4 x F φ = c dη η η = {F π F } φ = F φ.7 = K..8 = dz x 4 z + z 4.6 c = cos φ = z + z 4 9
10 dz Gauss + z 4 x 4 fx = x iii x 4 ds = dr + rdθ =. + r r dr dθ = a sin θ = a 4 r 4. ds = + r4 a 4 r 4 dr. dθ dr dr.8.4 s = s = 4 r a dr a4 r 4..9 a 4a = K x px px px 3 4 f x px. px 3 4 x = sin ϕ F k ϕ = x ϕ x k x = dϕ. k sin ϕ px x 4 px 3 I = c x 3 x = + 3 y y + x = c c = + 3 y y + y = c + 3 c y y = c + 3 c y + 3y 3y + 3 3y + 3 = y + = y + = 3 y + I = y 3 y+ + 3 y y+ 3 = 4 3 y y { y }
11 + 3 y 3 < y y < y = z z z = y y + z = dz z I = z 4 dz z 3 z z { z } = 4 z 3 dz z z k = = sin ϕ = z cos ϕ dϕ = dz 4 4 I = 4 3 ϕ cos ϕ dϕ sin ϕ k sin ϕ = 4 3 F k ϕ px x 3 4 z = sin ϕ 3 px k z Ek ϕ = z dz = k z z dz = k sin ϕ sin cos ϕdϕ = k sin ϕdϕ ϕ πk n ϕ = dz + nz z k z = cos ϕdϕ = sin ϕ k sin ϕ dϕ + n sin ϕ k sin ϕ dz. z k z k z dz.3 z dz + nz.4 z k z Legendre-Jacobi k n. x x x 4 x = t x = dt x dt t = sin t = sin x ϕ = π Kk = π π dϕ k sin ϕ Ek = k sin ϕdϕ
12 .4.4. r Q yz Q y θ Q, r cos θ, r sin θ xy a, b, P P Q P Q = a + b r cos θ + r sin θ = a + b br cos θ + r = a + b + r 4br cos θ P U = π θ= rdθ P Q = r θ = π ϕ dθ = dϕ U = r π dθ = 4br a +b+r cos θ k = 4bra + b + r π dθ a + b + r 4br cos θ r a + b + r π dϕ 4br a +b+r cos π ϕ U = π 4r a + b + r dϕ k sin π ϕ π = 4r a + b + r dϕ k sin ϕ = k r b Kk.4. m x α β m d x dt = αx + βx3 β = β α β s = m t t y = α x x m xt = x α s m m dt = α x α s d dt = m m α x α s d x dt = α α m β y + y3 d y ds = d ds = y + y 3 ds.5 dt dt d x dt = x + x3.5 = E dt x 4 x4.6
13 E U = x + 4 x4.6 t = ± x E x x4 x a E = U a 4 + a 4E = a = + + 4E E x x4 = 4E x x 4 = {a x + a + x a x } = a x + a + x x = a cos ϕ = a sin ϕdϕ k = + a ϕ t = ± = ± + a a sin ϕdϕ = ± a a cos ϕ + a a cos ϕ + a ϕ dϕ k sin ϕ a ϕ dϕ a +a sin ϕ T T = 4 kkk a E αx βx4 = 4E αx βx 4 = {αa x + βa + x a x } = a x {α + βa + x } k = ϕ t = ± a sin ϕdϕ m a a cos ϕ{α + βa a cos ϕ} βa α + βa m t = ± α + βa ϕ = ± m α + βa ϕ dϕ βa α+βa sin ϕ dϕ.7 k sin ϕ a = α + α + 4βE T = 4 m β α + βa Kk m m β T = 4 K = π α α.4.3 Ux E m = + Ux = E.8 dt 3
14 .8 dt = ± {E Ux} x t = ± {E Ux} x = a x = a {E Ux} = x a a xv x.9 x = a x = a a a V x x x = a + a a a cos ϕ.3 x = a + a a sin ϕ.3 = a a sin ϕ = a a sin ϕ ϕ t = ± ϕ = ± a a sin ϕdϕ { a + a a } { a cos ϕ a a a + a + a a cos ϕ } V x dϕ V x.3 i Ux = α x + β 4 x4 a = a a = a a = α + α + 4βE β {E Ux} = E αx β x4 = 4E αx βx 4 = βa x a + x + α β b = a + α β {E Ux} = βa x b + x x = a cos ϕ = a cos ϕ x + aa xv x = βa x b + x V x = β a + xa xb + x x + aa x.3 k = b + a ϕ t = ± dϕ β b + a a sin ϕ a = ± βb + a ϕ = β b + a a sin ϕ dϕ = ± a b +a sin ϕ βb + a 4 ϕ dϕ k sin ϕ
15 b = a + α β k =.7 ii a a + α β + = βa a α + a β βb + a = βa + α f = αx 3 γx Ux = αx γ x3 a < x < a E Ux = γx a a xa 3 x.34 γx a a xa 3 x = x a a xv x V x = γa 3 x ϕ t = ± dϕ γ { a 3 a a a sin ϕ } = ± ϕ γa3 a θ = ϕ k = a a < k = a a < a < a 3 a 3 a a 3 a t = ± γa3 a ϕ dθ k sin θ dϕ a a a 3 a sin ϕ a a ϕ π T = 4 γa3 a Kk γ a = a = E α a 3 x = E = γa a a 3 γa 3 α T = 4 K = π α α.4.4 θ α l m l θ t l θ t = v m l dθ dt mgl cos θ E α E m l dθ + mgl cos θ = E.35 dt E = mgl cos α.36 5
16 E l dθ dt = glcos θ cos α cos θ = sin θ cos α = α sin sin α θ sin dθ g dt = l k = sin α dθ g dt = k l sin θ sin α k k sin θ = k sin ϕ.37 dθ g dt = k l sin θ g = l sin α θ sin.37 sin θ = k sin ϕ.38 k k sin ϕ = g l k sin ϕ = g k cos l ϕ sin θ = k sin ϕ dθ dϕ = k cos ϕ k sin ϕ dθ dt dϕ = dϕ = t = dθ dt ϕ l g k cos ϕ k sin ϕ g l θ = α ϕ = π T T = 4 l g π k cos ϕ = l g k sin ϕ k sin ϕ dϕ = l g F k ϕ.39 k sin ϕ dϕ = 4 l Kk.4 g K k k sin ϕ k sin ϕ = a k = X fx = = ax X ax Kk = = π π f n X = n!! n a n ax n+ fx = + ax a X a3 X k sin ϕ k4 sin 4 ϕ k k k6 sin 6 ϕ + + { n!! k} n + n! 6 n!! n a n X n n! n!! n k n sin n ϕ + dϕ n!
17 Kk = π l l T = 4 g Kk = 4 g l T 4 g Kk = T π l g π + k 4 + l = π g = π Kk = π π + 4 k k4 + = + 4 k k4 + k = sin α α k = sin α k α α { T = + α T 4 9 α } = + α 6 + α i α = π k = k = sin α = sin π =.37 tan θ 4 = x t = l θ g dθ cos θ cos θ = cos θ 4 = tan θ 4 + = x + = x + x x θ dθ = 4 x + θ = θ x = tan θ 4 t = g tan θ 4 l + x 4 g tan θ x + x = 4 l x + g = + x l log + tan θ 4 tan θ 4 tan α = tan θ 4 tan β = = tan π 4 tan α + tan β tanα + β = tan α tan β l t = g log tan θ 4 + tan π 4 l θ tan θ 4 + tan π = log tan g 4 + π l θ + π = log tan. 4 g 4 4 θ = π t = ii.36 cos α = E mgl cos α = α sin sin α = cos α sin α cos α = E = mgl k = sin α E mgl k = mgl k > E > mgl k = E <.37 t = l θ dθ = l θ dθ g k sin θ k g = k sin k l θ dθ g θ k sin θ 7
18 θ = ϕ θ = θ ϕ = ϕ dθ = dϕ t = k l ϕ dϕ g k sin ϕ = k l g F k l ϕ = k g F π l θ = π T T = k g dθ k sin θ k θ = k l g Kk 8
19 y x x = sin x sin y = x 3. x π y π π y π x = sin y SIN y x = sin yy R π + Nπ y π + Nπ sin y = N SINy + Nπ sin y u x x k x = sn x 3. Kk u Kk k u = sn x k x = sn u = snu k sn u Kk u Kk x = sn u SN u Kk + NKk u Kk + NKk N SNu NKk = sn u x = sn ux R < u < snu, K k = sin u sin u cn u = sn u 3.3 Kk u Kk cn u = sn u cn cos cnu k cn u Kk u Kk x = cn u CN u Kk + NKk u Kk + NKk N CNu NKk = cn u x = cn ux R < u < dn u = dnu k dn u = k sn u Kk u Kk dn u = k sn u dn dnu k dn u Kk u Kk x = dn u DN u Kk + NKk u Kk + NKk DNu NKk = dn u 9
20 x = dn ux R < u < k = snu = sin u cnu = cos u dnu = sn u cn u 4Kk dn u Kk sn u cn u 4Kk Kk + N + 4Kk u + 4Kk < Kk + N + 4Kk snu + 4Kk = sn u cnu + 4Kk = cn u snu + 4Kk = N+ SN{u + 4Kk N + 4Kk} = N+ SNu NKk = N SNu NKk = N SNu NKk = sn u cnu + 4Kk = N+ CN{u + 4Kk N + 4Kk} = N+ CNu NKk = N CNu NKk = N CNu NKk = cn u dn Kk Kk+N +Kk u+kk < Kk+N +Kk dnu+kk = dn u snu + Kk = N+ SN{u + Kk N + Kk} = N SN{u + Kk NKk Kk} = N SNu NKk = sn u dnu + Kk = k sn u + Kk = k sn u = k sn u = dn u k sn u cn u sin u cos u k = 3. uk = = x x = x { x + } = + x + x log x = tanh x k = sn snu = tanh u cnu = dnu = tanh u = sinh u cosh u = cosh u sinh u cosh = u cosh u cnu = dnu = sech u 3.. sn 3. x = sn u u = x cn d cn u du x k x du = x k x du = x k x = cn u dn u 3.5 d sn u du = cn u dn u = d sn u = sn u d sn u du sn u du = sn u cn u dn u = sn u dn u sn u
21 dn d dn u du = d k sn du u = k sn u d sn u k sn u du = k sn u k sn u cn u dn u = k sn u cn u sn u = cn u dn u 3.6 cn u = sn u dn u 3.7 dn u = k sn u cn u sn 3. cn dn x = cn u cn u = sn u dn u = cn u k sn u = x k x k = k u = du = x k + k x = x k + k x x x k + k x = x k + k x = cnx k 3. cn dn x = dn u dn u = k dn u sn k u = x x k k = k u = x du = x x k x x x k = x x k = dnx k x x = sin θ = cos θdθ y = θ cos θdθ θ sin θ = dθ x = sin y 3. x = sin φ = cos dφ u = φ cos φdφ = sin φ k sin φ φ dφ k sin φ = sn sin φ 3.9
22 3.9 am φ = am u = amu k x = sin φ = sinam u sn u = sinam u am amplitude 3.3 cn u = sin am u = cosam u am 3. amu = ϕ amnkk = nπ n = am u = amu amu + Kk = amu + π 3. ϕ+π dθ k sin θ = ϕ dθ ϕ+π k sin θ + = u + Kk ϕ dθ ϕ k sin θ = dθ π k sin θ + dθ k sin θ z = sin φ Ek φ = φ k sin φdφ 3. sin φ = sn u π φ π, Kk u Kk 3.3 cos φ = sin φ = sn u = cn u cos φdφ = cn u dn udu 3.4 dφ = dn udu 3. k sin φ = k sn u = dn u 3.3 Ek φ = u dn udn udu = 3.4 z u dn udu ɛu 3.5 dz + nz z k z z = sn w z, Kk w Kk dz = cn w dn wdw = w cn w dn wdw w + n sn w sn w k sn w = cn w dn wdw w + n sn w cn w dn w = dw + n sn w. 3. sn x dn u
23 3.4 a + y b = x, y = a sin φ, b cos φ φ = amu k x x = a snu k = a sn u y = b cnu k = b cn u 3.6 k = a cn u dn udu = b sn u dn udu ds = a cn u + b sn u dn udu = a = a a b a ds = + = a a cn u + b sn u dn udu sn u dn udu. cn u + b a sn u dn udu = a sn u dn u k b a sn u dn udu k = a b a 3.7 ds = a k sn u dn udu = a dn u dn udu = a dn udu s.4 s = a u dn udu = aɛu s = aek φ x = a sn u y = b cn u 3.8 k 3.7 e : a b k = = e. 3.9 a F d F P s t Hs F P : P H = e : < e < F P = s + t P H = t e = F P P H s + t e = t e = s + t t 3
24 a = e e b = e e k = s + t = et e t e t e = e e e e 3.9 e y e e + x =. e e = e e e e e e = e4 + e e e = e 3.: p P S S BS = a, OS = a b = ae, b = a e = ae e = e. P a sn u b cn u Sae, P S < e < P S = a sn u ae + b cn u = a e sn u e sn u + = a e sn u = a e sn u. P S = a + e sn u P S + P S = a e sn u + a + e sn u = a =. 3.8 P ρ = ds dθ P a sn u b cn u y = e sn u cn u x + ae cn u x = y = ae cn u y = x = ae cn u cn u e sn u = a sn u a M sn u N ae cn u a MN = sn u + a e cn u = a e sn u a sn u cn = dn u u sn u cn u = a dn u sn u cn u 4
25 π MN x θ θ < π sin θ = ON MN = sin θ = sin 8 θ ae cn u a dn u sn u cn u = ae cn u sn u cn u a dn u = e sn u dn u sin θ = e sn u dn u e θ = sin sn u π dn u θ < π dθ = e sn u dn u e sn u dn u ρ = ds dθ ds = a dn udu du = e cn u e sn u + e sn u dn u ρ = a dn udu dn u e du = a dn3 u e cn u du = e sn u dn u cn u du = e dn u du 5
26 4 k k k k snu k sin u cnu k cos u dnu k. k snu k tanh u cnu k dnu k sech u. sn sin sinu + v = sin u cos v + cos u sin v tanhu + v = sinu + v = sin u d sin v dv tanh u + tanh v + tanh u tanh v + sin v d sin u du d tanh v dv = cosh v = sech v = tanh v d tanh u = du tanh u d tanh v dv + tanh v d tanh u du cosh u = sech u = tanh u = tanh u + tanh v tanh u tanh v tanh 4. tanh u + tanh v = tanh u d tanh v dv + tanh v d tanh u du tanh u tanh v tanh u d tanh v dv + tanh v d tanh u du tanhu + v = + tanh u tanh v tanh u tanh v = tanh u d tanh v dv + tanh v d tanh u du tanh u tanh. 4.4 v fu sin u tanh u fu + v = fu dfv dv. + fv dfu du k fu fv k = 4.4 k = 4.5 k = dfu fu = sn u = cn u dn u sn du snu + v = sn u cn v dn v sn v cn u dn u k sn u sn v 6
27 sn snu + v = sn u cn v dn v + sn v cn u dn u k sn u sn v sn u cnc u dnc u + snc u cn u dn u u + v = c F u = k sn u sn c u 4.6 sn c = F u 4.7 c 4.6 u F u 4.7 sn c F = F u/ u = F u u sn u = s cn u = c dn u = d snc u = s cnc u = c dnc u = d F u u F u = N D N = s c d + s c d D = k s s s = c d c = s d d = k s c s = c d c = s d d = k S c N u N = s s s s N = s s + s s s s s s = s s s s. s = c d + c d = s d + c k s c = + k s + k s 3 s = c d c d = s d c k s c = + k s + k s 3 N = s { + k s + k s 3 } s { + k s + k s 3 } = k s s s s D = k s s + s s = k s s c d s s c d = k s s s c d s c d. N F u = = N D ND D d 7
28 ND = s c d + s c d k s s s c d s c d = k s s s s k s s = N D F u = F u u F u = F = sn c. cn dn cn u cn v sn u sn v dn u dn v cnu + v = k sn u sn 4.8 v dnu + v = dn u dn v k sn u sn v cn u cn v k sn u sn 4.9 v 4.8 s = sn u s = sn v c = cn u c = cn v d = dn u d = dn v 4. cn u + v = sn u + v = k s s c d + s c d k s s. k s s = k s s k s s = c + s d c + s d k s s s c d + s c d = c s s d d + s s c c d d = c c + s s d d cn u + v = c c + s s d d k s s cnu + v = ± c c + s s d d k s s v = cn u dn u + v = k sn u + v = k s s k s c d + s c d k s s. k s s = k s s k s s = k s d k s s k s d k s s = d + k s c d + k s c k s s k s c d + s c d = d d k s s c c d d + k 4 s s c c = d d k s s c c dn u + v = d d + k s s c c k s s dnu + v = ± d d + k s s c c k s s 8
29 v = dn u 4.9 sn snu + v = sn u cn v dn v + sn v cn u dn u k sn u sn v 4. sn u cn v dn v sn v cn u dn u sn u cn v dn v sn v cn u dn u = sn u sn v k sn v sn v sn u k sn u 4. = sn u sn vk sn 4 u sn v + k sn 4 v sn u = sn u sn v k sn u sn v snu + v = sn u sn v sn u cn v dn v sn v cn u dn u s c d s c d s s snu + v = s s s c d s c d. = s k s s + k s 4 + s k s s + k s 4 s s c c d d s 4 + s s s 4 = s c d s c d s s c c d d = s c d s c d cn u + v = sn u + v cnu + v = s c d s c d s c d s c d : θ α k = sin α.38 sin θ = k sin ϕ ϕ dϕ g g k sin ϕ = l t ϕ = am l t ω = g l sin θ g g = k sin am l t = k sn l t. sin θ = k sn ω t
30 θ dθ dt 4.5 cos θ = sin θ = k sn ω t = dn ω t. 4.5 θ cos θ = kω cn ω t dn ω t. θω t = ω k cn ω t dn ω t θ = ω k cn ω t θ = ω k sn ω t dn ω t θ = ω sin θ. 4.4 l α θ p O EOA BB D AD P Q N AD = OA OD = l l cos α = l cos α = l sin α = lk. 4.7 AE = l k = AD AE. 4.: P 7 θ AN = AQ sin QDN = AD sin QDN = lk sin QDN. AN = OA ON = l l cos θ = l cos θ = l sin θ. 3
31 4.3 sin θ = k sin QDN. QDN = ϕ = am u. u = ω t. 4.8 ϕ P Q... 4.: O BPAB P 4. P P P τ B BB P B BP AB. P τ P.P P τ P P. P P P P. 4.3 P P P P R P P P P R P P. 4.3: P P = vdt, P P = v dt P N N P P DOA m v v P P υ D N m g DN = mv. υ = g DN υ = g DN. P P P P P υ P υ. P P lim = P P P P DN DN 4.9 3
32 . RP P RP P P P P = P P P P RP = P RP RP P RP P P P P P 4.9 RP lim P P RP = lim P P DN = P P P P DN DA B P P BR B P P 4.4: BB PP BR B P 4 R. P R P R C 4.4 P R = P C CR = P C CB. 4. P C = P O + CO + OC ON = OB + OC + OC ON., 4. OB = DB + OD, CB = DB + CD OB CB = OD CD = OD + CDOD CD = OD + CD OC , 4.3, 4. P R = OC OD + CD + OC + ON = OC OD + ON = OC ND. 4.4 P R = OC N D P R N D P R = ND R R 4. DA. 4.5: P 5 3
33 4.5 P P P P P P R P P. BB P P P P P P R P P.BB c 4.6 R P P c 4.6: DA B P 5 P P DA. O P P τ. a OC = a. 4.7: AOP = θ AOP = θ ADQ = φ ADQ = ψ P AOP = θ AOP = θ.ad P N P N Q Q ADQ = φ ADQ = ψ. DQ = AD cos φ ND = DQ cos φ = AD cos φ 4.. k l l = OP α AD = l l cos α = l cos α = l sin α = lk k = sin α R R P R = a ND = 4alk cos φ. 33
34 P R = 4alk cos ψ. P R + P R = alkcos φ + cos ψ. P P = OP + OP OP OP cosθ θ = l + l l cos θ θ = l { cosθ θ} = 4l sin θ θ. P P = l sin θ θ = l sin θ θ sin θ θ = sin θ θ kcos φ + cos ψ a kcos φ + cos ψ l = a l =. 4.6 φ = amu ψ = amv g. ω = l 4.8 u = ω t v = ω t τ :. ω τ = u v. sin θ = k sn u sin θ = cos θ = cos θ = k sn v dn u dn v φ = am u cn u = cos φ ψ = am v cn v = cos ψ 4.8 a l = sin θ cos θ + cos θ sin θ k sn udnv + k sn v dn u sn u dn v dn u sn v = = = 4.9 kcos φ + cos ψ kcn u + cn v cn v + cn u τ v =. u w = ω τ. a l = sn w cn w + cn w = cn w = + cn w cn w + cn w 34
35 . u v = w 4.9 cn u v + cn u v = sn u dn v dn u sn v cn v + cn u = sn u dn v dn u sn v cn v + cn u sn u dn v dn u sn v sn u dn v dn u sn v = = sn u dn v dn u sn v cn v + cn usn u dn v dn u sn v = cn u k + k cn v k + k cn u cn v cn v + cn usn u dn v dn u sn v cn v cn u cn v + cn usn u dn v dn u sn v = cn v cn u sn u dn v + dn u sn v.4. cn u v + cn u v = s d s d = c c. c + c s d + s d cn u v + cn u v = c c s d s d c + c s d + s d. 4.3 A = cn u v B = c c s d s d C = c + c s d + s d. A + A = B C A = C B C + B B = c c s d s d = s c d s c d s c d + s c d C = c + c s d + s d = s c d + s c d + s c d + s c d C B = s c d + s c d C + B = s c d + s c d. A = C B C + B = s c d + s c d s c d + s c d. A = cn u v = s c d + s c d s c d + s c d. v cn u + v = s c d s c d s c d s c d
36 sn u = = x φ x k x dφ k sin φ x x = sin φ 5. x = sn u K u K. 5. u Kk Kk, u sn u sn Kk u, v Kk Kk u + v Kk u, v sn K u + v K sn sn u + v = sn u cn v dn v + sn v cn u dn u k sn u sn v u + v = {u + v K K} K u, v, u + v K 5.3 sn u + v = sn {u + v K + K} = sn u + v K cn K dn K + sn K cn u + v K dn u + v K k sn u + v K sn K K u + v K. sn K =, cn K =, dn K = k = k sn {u + v K + K} = sn u + v K k + cn u + v K dn u + v K k sn u + v K = cn u + v K dn u + v K. v = K. 8 cn u K >< u K dn u K sn u = cn u + K >: K u dn u + K sn u ± v = ± cn u dn u, 8 >< k sn u K dn u K cn u = >: k sn u + K dn u + K sn u k cn u ± v = k, dn u ± v = dn u dn u u K K u cnu + v K snu + v = dnu + v K 8 k >< dn u = dn u K dn u = k >: dn u = dn u + K u K K u
37 i u K, v K sn u cn v dn v + sn v cn u dn u k sn u sn v = cn u K dn u K k sn v K dn v K k cn v K dn v K + dn v K k k cn u K dn u K cn v K dn v K sn u K dn u K k dn u K = k sn v K cn u K dn u K k sn u K cn v K dn v K dn u K dn v K k cn u K cn v K = { k sn u K}{ k sn v K} k { sn u K}{ sn v K} = k { k sn u K sn v K} = k { k sn u K sn v K} sn v K cn u K dn u K k sn u K cn v K dn v K k sn u K sn v K = sn {v K + u K} = sn u + v K. K u + v K K i u + v K K { } cn u + v K cn u + v K snu + v K = sn{u + v K K} = = = sn u + v. dn u + v K dn u + v K ii K u + v K cnu + v K snu + v K = sn{u + v K} = = snu + v. dnu + v K i ii sn ii K u, v K iii u K, K v iv K u, K v sn K u + v K 3K u + v 3K, 4K u + v 4K K u + v K 5.., 5. K =, u = x dt t = sin x x = iy iy dt = t sin iy t = iη y dη = i. + η = iv 37 i =.
38 = i sinh y iy = sin iv = i sinh v sin sn w = x dt t k t = sn x, k. 5.5 t = sin θ, x = sin φ dt = cos θ dθ = t dθ w = 5.5 x = iy i t = iη φ v dη = dθ k sin θ = sn sin φ, k dt y t k t = sn iy, k 5.6 y dη = i η k η. 5.7 = iv 5.8 η = tan ψ, 5.9 y = tan ϕ 5. dψ + tan cos ψ = ψ + η dψ = dψ 5. cos ψ cos ψ + k η = + k tan ψ = sin ψ + k sin ψ cos ψ = k sin ψ cos. 5. ψ k = k 5., v = ϕ cos ψ + η k sin ψ + η ϕ cos ψ dψ = dψ k sin ψ = sn sin ϕ, k sin ϕ = sn v, k. 38
39 cos ϕ = cn v, k y = tan ϕ = sin ϕ cos ϕ = sn v, k cn v, k. 5.3 iy = sn iv sn iv, k = i sn v, k cn v, k sn 5.4 y < y < +, ϕ π < ϕ < π, v K v K K = π dψ k sin ψ = Kk 5.4 sn iv, k v < v < 5.4 sn iv, k cn iv, k = sn iv, k = + sn v, k cn v, k = cn v, k = cn v, k. 5.5 k sn iv, k dn iv, k = k sn iv, k = + k sn v, k cn v, k = k sn v, k cn v, k = dn v, k cn v, k 5.6 v 5. u v snu + iv = cnu + iv = snu, k cniv, k dniv, k + sniv, k cnu, k dnu, k k sn u, k sn, iv, k 5.7 cnu, k cniv, k sniv, k cnu, k dnu, k + dniv, k k sn u, k sn, iv, k 5.8 dnu + iv = dnu, k dniv, k k sniv, k snu, k cnu, k + cniv, k k sn u, k sn, 5.9 iv, k sniv, k = i snv, k cnv, k, cniv, k = cnv, k, dniv, k = dnv, k cnv, k snu + iv = snu, k dnv, k + snu, k dnu, k snv, k cnv, k k dn, u, k sn v, k 5. cnu + iv = cnu, k cnv, k i cnu, k dnu, k snv, k dnv, k k dnu, k snv, k, 5. dnu + iv = dnu, k cnv, K dniv, k ik snu, k cnu, k snv, k k dn, 5. u, k sn v, k 39
40 sn z z sn snz + z = sn z cn z dn z + sn z cn z dn z k sn z sn z k k z = x + iy, z = x + iy snu + v = sn{x + x + iy + y } x, x, y, y = snx + x cn{iy + y } dn{iy + y } + sn{iy + y } cnx + x dnx + x k sn x + x sn. {iy + y } snz + z = sn z cn z dn z + sn z cn z dn z k sn z sn z cnz + z = cn z cn z sn z cn z dn z + dn z k sn z sn z, dnz + z = dn z dn z k sn z sn z cn z + cn z k sn z sn z sn sn cnu + K snu + K = sn {u + K + K} = dnu + K = k sn u dn u = sn u, 5.3 k snu + 4K = sn{u + K + K} = snu + K = sn u 5.4 sn 4K sn =, sn K =, sn K =, sn 3K =, sn 4k = u. sn 4K sn dn u sniv, k = i snv, k cnv, k 5.6 v sn v K sniv + ik, k = i snv + K, k cnv + K, k = i snv, k cnv, k = sniv, k sniv + ik, k = sniv, k. 5.7 k sn ik 5. v = K cnk, k = snik = 5.8 4
41 5. v = K snk, k =, cnk, k =, dnk, k = k = k 5.9 snu + ik = snu, k cnik, k dnik, k + snik, k cnu, k dnu, k k sn u, k sn ik, k = snu, k dnv, k + snu, k dnu, k snv, k cnv, k k dn u, k sn v, k k snu, k = dnu, k = k snu, k. 5. v = K snu, ik = k snu, k s = snk, k =, c = cnk, k =, d = dnk, k = 5.3 snu + ik = snu, k cnik, k dnik, k + snik, k cnu, k dnu, k k sn u, k sn ik, k = d snu, k dnk, k + snu, k dnu, k snk, k cnk, k k dn u, k sn K, k k snu, k = dnu, k = = sn u. k snu, k 5. snu, ik = snu, k 5.3 u sn ik u m n snu + 4K + ik = sn u 5.33 snu + 4mK + nik = sn u K ik 4 ik sn u sn u = ik 5.3 sn sniv + K = sniv = i snv, k cnv, k 5.35 sniv + K = dnv, k 5.36 u = K sn u = K. 4
42 5.3. cn dn sn cn dn cn cnu + K = k sn u dn u snu + K cnu + K = cn{u + K + K} = k dnu + K = k cn 4K cn u dn u k dn u 5.37 = cn u 5.38 cnu + 4K = cn{u + K + K} = cnu + K = cn u 5.39 cnu + K + ik cnu + K = dn = cnu + K = cn u 5.4 u + K cn K + ik cnu + 4K + K + ik = cn u 5.4 cn 4K K + ik dn dnu + K = k dn u dnu + K = dn{u + K + K} = dn K k dnu + K = k k dn u dnu + 4iK = dn u ik sn u cn u dn u dn 4iK = dn u 5.43 = dn u 5.44 dnu + K + 4iK = dn u 5.45 dn K 4iK m m Ω Ω sn w cn w dn w fw + mω + m Ω = fw 5.46 snk + iv = dnv, k, cnk + iv = ik snv, k dnv, K k snv, K = ik snv, k dnv, k dn = ik snv, k v, k dnv, k, dnk + iv = k cnv, k dnv, k k sn v, k = k cnv, k dnv, k dn = k cnv, k v, k dnv, k, snu + ik = k sn u, cnu + ik ik sn u dn u ik sn u dn u ik sn u dn u = dn = u k sn = u k sn = i dn u u k sn u, dnu + ik = ik sn u cn u dn = ik sn u cn u u k sn u = ik sn u cn u k sn = i cn u u sn u, snu + ik = sn u, cnu + ik cn u i sn u dn u = dn = cn u, u dnu + ik dn u i sn u cn u = dn = dn u, u
43 snu + K + ik = k snu + K = k cn u = dn u dn u k cn u, cnu + K + ik i dnu + K = k snu + K = i k dn u k cn u = ik dn u k cn u, dnu + K + ik i cnu + K = = i k sn u dn u cn u = ik sn u snu + K dn u cn u, snu + K + ik = snu + K = sn u, cnu + K + ik = cn u, dnu + K + ik = dnu + K = dn u Ω Ω fw = sn w Ω Ω 4K ik K ik K + ik fw = cn w Ω Ω 4K + ik K K ik K + ik fw = dn w Ω Ω K 4iK K + ik K ik ik ik 43
44 6 x 6. y = b sn b = k c k c sn.. sn x y = b sn b c = 6. c x s ds = + ds = + = b x x c cn dn. c c b c cn x c dn x. c cn u = sn u = + k dn u ds = + = + b [ + k { c dn x c b dn x c c k + } ] dn x b = + dn x c c c b x b k c k dn4 = c c k b c x k dn c b + c dn x c + b c x k dn4 c x k dn4 c. b = k k c 6. ds k k = k k x k dn > c x k dn c dn x k x + c k k dn4 = 4 x c k dn + c k = dn x c + dn x c k = dn x c x k = dn c + k cn x c k > + k sn x c + k k dn 4 x c ds { x } = k dn c 44
45 . s x O ds = x k dn c { ds = s = k x k dn c x dn x c x }.. k = k. 6.3 sin φ = sn u 6.4 k sin φ = dn u 6.4 cos φ dφ = cn u dn u du cos φ = sin φ = sn u = cn u 6.6 dφ = dn u du. εu u dn u du = Ek φ = φ k sin φ dφ s = c x k ε x 6.7 c x = x x = a sn u = u u = Kk. a K = Kk. = K c c = Kk a. 6.8 x = u = x = a u = K sin φ = sn u φ π. E = Ek = π k sin φ dφ = k dn u du = εk 6.9.x = x = a l x = a 6.7 l = c a k ε a = s = c εk a. c k x = K εk = εk, 6.8 c = a K 6.9 E = εk. l = c 4cE E a = k k a = 4aE a Kk 45
46 sn ω x y f = ω y 6. u = ω y f = du ρ s z l U = ρω y ds 6. y = ys U ρω ρ ω 6.. x x = x = a T T x T cos ψ = T ds x d ds T cos ψ = d T = ds ds T cos ψ = T ds = = T 6. y ds ρds y ω f f = ρω y ds
47 y F y F y s f = F y s s d ds f = d ds F ys 6.4 s = ds ds 6.4 ds = df : ds = d ds F ys ds 6.5 :ds y T y F y = T sin ψ = T 6.5 ds = d T ds = d T sin ψ ds ds ds ds 6.3 ρω y ds = d T ds = d T sin ψ ds 6.6 ds ds ds y ds = ds = p ds ρω y + d ds d ds T = ds T. ds T = ds + ρω y = 6.7 T p = d ds p = dp ds = p dp ds ds = ds = ds = + p 6.9 p dp = ρω T y + p 6. 47
48 6. p p + p p dp = ρω y + p T dp = ρω T y b y + p = t p dp = dt y : b y p : p t : + p +p dt y = ρω y t T b [ t] +p = ρω T [y ] y b + p = ρω y b T + p = + ρω b y 6. T b p = = y y 6. p = = ρω b y { + ρω b y } 6. T 4T < k < 6. = ρω b T = = ρω b T = ρω b T = ρω b T = b ρω b T ρω T ρω b 4T + ρω b 4T = k } { y b + ρω b y 4T b y 4T + ρω b b ρω b y 4T 4T b 4T + ρω b { y 4T b + ρω b { y 4T b + ρω b y 4T b ρω b 4T 4T + ρω b ρω b 4T + ρω b 4T ρω b y b 4T y b 4T + ρω b 4T + ρω b y 4T b k y b } y b } y b = η = b ρω T + ρω b y 4T b k y b dη = b
49 c = ρω + ρω b = k T 4T k b k = k 6.4 b dη = b η k c η dη η k η = c sn η = η η dη η k η dη η k η x η = sn c = = x c x c x y = b sn c b = k c 6.7 k x x = x = a 6.8 c = Kk a 6.8 Kx y = b sn a l 6.9 l = 4a E k K a. l x = a 9 k b ρω 4T ω π T x ψ tan tan ψ = = b c 6.3 x y = x 6. T cos ψ = T T = T b k = T cos T c + = T k + 49
50 Kx y = b sn a Kx y = b sn 3a x x ω T ρ = s = ρds y z ρdsω y ρdsω z x y z d T = ds ds d T + ρω y = 6.9 ds ds d T dz + ρω z = 6.3 ds ds T ds = T y 6.9 z y d T dz ds z d T ds = y T dz d T dz ds z T dz d + T ds ds ds ds ds ds ds ds ds ds ds ds ds = T y dz ds z = H 6.3 ds dz 6.9 ds ds ds ds d T d ds + ρω y ds + T dz dz ds + ρω z ds ds ds ds ds ds ds = T T d ds + ds ds ds ds ds ρω y + T dz dz T dz d dz ds + ds ds ds ds ds ρω z = T ds T + dz ds ρω y + T dz ds T + ds ρω z { = } dz + + ds ds ρω y + z = T + ρω y + z = λ T H λ y + z = r y ds + z dz ds = dr
51 ds T T ds = T T = ds T 6.35 T y dz ds z ds y dz ds z ds T y dz z = H = H = H 6.34 y + yz y dz z = H T dz y + z dz = { dr } 4 dz + z = { dr } y yz dz y z dz = H T dz + z = H. T y + z + y + z y + z { + dz = { dr } + H 4 T } = { dr } + H. 4 dz T r = y + z { r + } dz = 4 { dr } + H T ds = + + dz 6.37 { r + } dz = r { + dz } { ds = r } { ds r } = 4 { dr } + H T { dr } { ds = 4r } H T. 5
52 ds 6.3 = T T 6.33 { dr } 4T = r T 4 H T. T + ρω y + z = λ T + ρω r = λ T = λ ρω r { dr } = 4r T λ ρω r 4r 4H T = 4r T 4λ 4λρω r + ρ ω 4 r 4 4r 4H T = 4ρ ω 4 { T r 6 λ } T ρω r4 ρ ω 4 4λ r H ρ ω 4 = 4ρ ω 4 T r b r c r d b c d r b c d r b > c r b c r b r c < 6.38 d > b d > b > r > c dr r = c + b c sin φ = b sin +c sin φ = b sin φ + c cos φ k = b c d c = b sin φ cos φ c cos φ sin φ = b c sin φ cos φ dφ. 6.4 b r = b b sin φ c cos φ = b sin φ c cos φ = b c cos φ r c = b sin φ + c cos φ c = b sin φ c cos φ = b sin φ c sin φ = b c sin φ d r = d b sin φ c cos φ = d c b c d c sin φ = d c k sin φ b c cos φ sin φ φ dφ = 4ρ ω 4 T r b r c r d = 4ρ ω 4 b c cos φb c sin φd c k sin φ T dφ = ρ ω 4 T d c k sin φ dφ = ρ ω 4 T d c k sin φ dφ = ρω d c T k sin φ dφ = ρω d c T k sin φ dφ k sin φ = ρω d c T dφ k sin φ = ρω T d c x. 5
53 a ρω d c T = K a φ = am φ dφ k sin φ = sn sin φ K a Kx a = = φ φ dφ k sin φ x dφ k sin φ = sn sin φ. Kx 6.39 a { } Kx Kx Kx Kx r = b sn + c sn = b sn + c cn. a a a a 6.4 y F y, x I = Iy x Iy = F y, 6.4 x I y = yx F x, y, yx = y, yx = y 6.4 y I yx. x x = x x = x y x y. I y = yx yx δy. δy y.δy x. δ Y = yx + δyx 6.43 δyx = δyx = 6.44 Y Y x = yx = y, Y x = yx = y y = yx I y = yx IY = x x Iy = { F y + δy, x x F + δ y, }
54 I δi x { } x δi = Y x yx = F y + δy, + δ F y, x x x { F = x y δy + F δ }. δ = dδy δ = d δy. δi = x x { F y δy + x F δi = δy + F δy x y 6.44 δi = x δyx F x { F y d d F F d } δy. x x x x { d } F δy. F } δy =. F y = x ρy ds. ds = + = 6.48 y ds = y α 6.49 I = a y α = δi = { } d y α y + = + 54
55 . d y α y + + { y α d { d + } = 6.5 d y + { + y α + y α + y α + + } 3 } d y s «+ = d y + y { { d { d + + y. + + log + + d y =. β + + { + d y = + + } } y y + + «+ + s «+ = = } d y = { + } = y α d y + y = y α d y + = y + y y α d y = y y α + y y α d y = y y α = 6.5 } 3 d y = log α y + A A 6.5 = βα y
56 . = y b β = α b 6.53 = α y α b α b = α αy + y 4 α + αb b 4 α b = αy + b y + y 4 + aαb b y b 4 α b = y α b y + b α b y α b = b y α b y α b = α b b y α b + b y α b = α b b y { + b y } α b α b = ρω T } = ρω b y { + ρω b y T 4T y y + α = βy + α. + = 6.55 = βy + α 6.56 = βy + α = βy + α = βy + α = βy + α. βy + α = t β = dt. x = t dt β = β βx = log t + t + C βx x = log t + t dt t = log t + t β e βx x e βx x βy + α = βy α. 56 = t + t = βy α + βy + α
57 e βx x e βx x βy α = e βx x βy α = e βx x + y + α = e βx x + e βx x β y = = e βx x + β = e βx x β cosh βx x α. α = e βx x + e βx x α β x x y = a cosh α a. a = β α x. 57
58 7 [] 4 [] 97 [3] 975 [4] 4 [5] 3 [6] 993 [7] 987 [8] 986 [9] 996 [] 4 [] IB 997 []
t (x(t), y(t)), a t b (x(a), y(a)) t ( ) ( ) dy s + dt dt dt [a, b] a a t < t 1 < < t n b {(x(t i ), y(t i ))} n i ( s(t) ds ) ( ) dy dt + dt dt ( ) d
![t (x(t), y(t)), a t b (x(a), y(a)) t ( ) ( ) dy s + dt dt dt [a, b] a a t < t 1 < < t n b {(x(t i ), y(t i ))} n i ( s(t) ds ) ( ) dy dt + dt dt ( ) d](/thumbs/80/80954781.jpg "t (x(t), y(t)), a t b (x(a), y(a)) t ( ) ( ) dy s + dt dt dt [a, b] a a t < t 1 < < t n b {(x(t i ), y(t i ))} n i ( s(t) ds ) ( ) dy dt + dt dt ( ) d")
1 13 Fall Semester N. Yamada Version:13.9.3 Chapter. Preliminalies (1 3) Chapter 1. (4 16) Chapter. (17 9) Chapter 3. (3 49) Chapter 4. (5 63) Chapter 5. (64 7) Chapter 6. (71 8) 11, ISBN 978-4-535-618-4.
(yx4) 1887-1945 741936 50 1995 1 31 http://kenboushoten.web.fc.com/ OCR TeX 50 yx4 e-mail: yx4.aydx5@gmail.com i Jacobi 1751 1 3 Euler Fagnano 187 9 0 Abel iii 1 1...................................
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
Z: Q: R: C: 3. Green Cauchy
7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................
熊本県数学問題正解
00 y O x Typed by L A TEX ε ( ) (00 ) 5 4 4 ( ) http://www.ocn.ne.jp/ oboetene/plan/. ( ) (009 ) ( ).. http://www.ocn.ne.jp/ oboetene/plan/eng.html 8 i i..................................... ( )0... (
21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........
(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt
S I. x yx y y, y,. F x, y, y, y,, y n http://ayapin.film.s.dendai.ac.jp/~matuda n /TeX/lecture.html PDF PS yx.................................... 3.3.................... 9.4................5..............
i 18 2H 2 + O 2 2H 2 + ( ) 3K
i 18 2H 2 + O 2 2H 2 + ( ) 3K ii 1 1 1.1.................................. 1 1.2........................................ 3 1.3......................................... 3 1.4....................................
1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 +
( )5 ( ( ) ) 4 6 7 9 M M 5 + 4 + M + M M + ( + ) () + + M () M () 4 + + M a b y = a + b a > () a b () y V a () V a b V n f() = n k= k k () < f() = log( ) t dt log () n+ (i) dt t (n + ) (ii) < t dt n+ n
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
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 ),
i
i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,
S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d
S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....
高校生の就職への数学II
II O Tped b L A TEX ε . II. 3. 4. 5. http://www.ocn.ne.jp/ oboetene/plan/ 7 9 i .......................................................................................... 3..3...............................
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)
[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s
![[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s](/thumbs/94/118579915.jpg "[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s")
[ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =
4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx
4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan
ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
2011de.dvi
211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37
1. z dr er r sinθ dϕ eϕ r dθ eθ dr θ dr dθ r x 0 ϕ r sinθ dϕ r sinθ dϕ y dr dr er r dθ eθ r sinθ dϕ eϕ 2. (r, θ, φ) 2 dr 1 h r dr 1 e r h θ dθ 1 e θ h
IB IIA 1 1 r, θ, φ 1 (r, θ, φ)., r, θ, φ 0 r
II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (
II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )
2 4 202 9 202 9 6 3................................................... 3.2................................................ 4.3......................................... 6.4.......................................
I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
Z: Q: R: C:
0 Z: Q: R: C: 3 4 4 4................................ 4 4.................................. 7 5 3 5...................... 3 5......................... 40 5.3 snz) z)........................... 4 6 46 x
I 1
I 1 1 1.1 1. 3 m = 3 1 7 µm. cm = 1 4 km 3. 1 m = 1 1 5 cm 4. 5 cm 3 = 5 1 15 km 3 5. 1 = 36 6. 1 = 8.64 1 4 7. 1 = 3.15 1 7 1 =3 1 7 1 3 π 1. 1. 1 m + 1 cm = 1.1 m. 1 hr + 64 sec = 1 4 sec 3. 3. 1 5 kg
入試の軌跡
4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf
( ) ( )
20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))
Note.tex 2008/09/19( )
1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
Part () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
http://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg
i I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................
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
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
Gmech08.dvi
51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r
t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z
I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)
II 2 II
II 2 II 2005 yugami@cc.utsunomiya-u.ac.jp 2005 4 1 1 2 5 2.1.................................... 5 2.2................................. 6 2.3............................. 6 2.4.................................
( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =
1 9 8 1 1 1 ; 1 11 16 C. H. Scholz, The Mechanics of Earthquakes and Faulting 1. 1.1 1.1.1 : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ =
X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
i
009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3
untitled
0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.
17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1,
17 ( ) 17 5 1 4 II III A B C(1 ) 1,, 6, 7 II A B (1 ), 5, 6 II A B (8 ) 8 1 I II III A B C(8 ) 1 a 1 1 a n+1 a n + n + 1 (n 1,,, ) {a n+1 n } (1) a 4 () a n OA OB AOB 6 OAB AB : 1 P OB Q OP AQ R (1) PQ
r III... IV.. grad, div, rot. grad, div, rot 3., B grad, div, rot I, II ɛ-δ web page (
r 8.4.8. 3-3 phone: 9-76-4774, e-mail: hara@math.kyushu-u.ac.jp http://www.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html Office hours: 4/8 I.. ɛ-n. ɛ-δ 3. 4. II... 3. 4. 5.. r III... IV.. grad, div,
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
2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................
Gmech08.dvi
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.)
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 =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
() 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 ()
K E N Z U 2012 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.2................................... 4 1.2.1..................................... 4 1.2.2.................................... 5................................
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 =
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
K E N Z OU
K E N Z OU 11 1 1 1.1..................................... 1.1.1............................ 1.1..................................................................................... 4 1.........................................
meiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
OABC OA OC 4, OB, AOB BOC COA 60 OA a OB b OC c () AB AC () ABC D OD ABC OD OA + p AB + q AC p q () OABC 4 f(x) + x ( ), () y f(x) P l 4 () y f(x) l P
4 ( ) ( ) ( ) ( ) 4 5 5 II III A B (0 ) 4, 6, 7 II III A B (0 ) ( ),, 6, 8, 9 II III A B (0 ) ( [ ] ) 5, 0, II A B (90 ) log x x () (a) y x + x (b) y sin (x + ) () (a) (b) (c) (d) 0 e π 0 x x x + dx e
n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1)
D d dx 1 1.1 n d n y a 0 dx n + a d n 1 y 1 dx n 1 +... + a dy n 1 dx + a ny = f(x)...(1) dk y dx k = y (k) a 0 y (n) + a 1 y (n 1) +... + a n 1 y + a n y = f(x)...(2) (2) (2) f(x) 0 a 0 y (n) + a 1 y
( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) (
6 20 ( ) sin, cos, tan sin, cos, tan, arcsin, arccos, arctan. π 2 sin π 2, 0 cos π, π 2 < tan < π 2 () ( 2 2 lim 2 ( 2 ) ) 2 = 3 sin (2) lim 5 0 = 2 2 0 0 2 2 3 3 4 5 5 2 5 6 3 5 7 4 5 8 4 9 3 4 a 3 b
(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)
1. 1.1...,. 1.1.1 V, V x, y, x y x + y x + y V,, V x α, αx αx V,, (i) (viii) : x, y, z V, α, β C, (i) x + y = y + x. (ii) (x + y) + z = x + (y + z). 1 (iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y
(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)
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
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 +
( )
7..-8..8.......................................................................... 4.................................... 3...................................... 3..3.................................. 4.3....................................
sikepuri.dvi
2009 2 2 2. 2.. F(s) G(s) H(s) G(s) F(s) H(s) F(s),G(s) H(s) : V (s) Z(s)I(s) I(s) Y (s)v (s) Z(s): Y (s): 2: ( ( V V 2 I I 2 ) ( ) ( Z Z 2 Z 2 Z 22 ) ( ) ( Y Y 2 Y 2 Y 22 ( ) ( ) Z Z 2 Y Y 2 : : Z 2 Z
.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,
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t
2000年度『数学展望 I』講義録
2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53
医系の統計入門第 2 版 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, 第 2 版 1 刷発行時のものです.
医系の統計入門第 2 版 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. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +
chap03.dvi
99 3 (Coriolis) cm m (free surface wave) 3.1 Φ 2.5 (2.25) Φ 100 3 r =(x, y, z) x y z F (x, y, z, t) =0 ( DF ) Dt = t + Φ F =0 onf =0. (3.1) n = F/ F (3.1) F n Φ = Φ n = 1 F F t Vn on F = 0 (3.2) Φ (3.1)
M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -
M3............................................................................................ 3.3................................................... 3 6........................................... 6..........................................
1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C
0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,
e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,
01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1 e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1,
IMO 1 n, 21n n (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a
1 40 (1959 1999 ) (IMO) 41 (2000 ) WEB 1 1959 1 IMO 1 n, 21n + 4 13n + 3 2 (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a = 4, b =
( 12 ( ( ( ( Levi-Civita grad div rot ( ( = 4 : 6 3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n
5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (
![5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (](/thumbs/92/107866626.jpg "5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (")
5 5.1 [ ] ) d f(t) + a d f(t) + bf(t) : f(t) 1 dt dt ) u(x, t) c u(x, t) : u(x, t) t x : ( ) ) 1 : y + ay, : y + ay + by : ( ) 1 ) : y + ay, : yy + ay 3 ( ): ( ) ) : y + ay, : y + ay b [],,, [ ] au xx
dynamics-solution2.dvi
1 1. (1) a + b = i +3i + k () a b =5i 5j +3k (3) a b =1 (4) a b = 7i j +1k. a = 14 l =/ 14, m=1/ 14, n=3/ 14 3. 4. 5. df (t) d [a(t)e(t)] =ti +9t j +4k, = d a(t) d[a(t)e(t)] e(t)+ da(t) d f (t) =i +18tj
1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
![1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1](/thumbs/94/121802164.jpg "1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1")
1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,
[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
1 3 1.1.......................... 3 1............................... 3 1.3....................... 5 1.4.......................... 6 1.5........................ 7 8.1......................... 8..............................
Δ =,, 3, 4, 5, L n = n
九州大学学術情報リポジトリ Kyushu University Institutional Repository 物理工科のための数学入門 : 数学の深い理解をめざして 御手洗, 修九州大学応用力学研究所 QUEST : 推進委員 藤本, 邦昭東海大学基盤工学部電気電子情報工学科 : 教授 http://hdl.handle.net/34/500390 出版情報 : バージョン :accepted
D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j
![D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j](/thumbs/103/158001152.jpg "D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j")
6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..
( )
18 10 01 ( ) 1 2018 4 1.1 2018............................... 4 1.2 2018......................... 5 2 2017 7 2.1 2017............................... 7 2.2 2017......................... 8 3 2016 9 3.1 2016...............................
120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2
9 E B 9.1 9.1.1 Ampère Ampère Ampère s law B S µ 0 B ds = µ 0 j ds (9.1) S rot B = µ 0 j (9.2) S Ampère Biot-Savart oulomb Gauss Ampère rot B 0 Ampère µ 0 9.1 (a) (b) I B ds = µ 0 I. I 1 I 2 B ds = µ 0
Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e
7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z
A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π
4 4.1 4.1.1 A = f() = f() = a f (a) = f() (a, f(a)) = f() (a, f(a)) f(a) = f 0 (a)( a) 4.1 (4, ) = f() = f () = 1 = f (4) = 1 4 4 (4, ) = 1 ( 4) 4 = 1 4 + 1 17 18 4 4.1 A (1) = 4 A( 1, 4) 1 A 4 () = tan
all.dvi
29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F 30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan
ORIGINAL TEXT I II A B 1 4 13 21 27 44 54 64 84 98 113 126 138 146 165 175 181 188 198 213 225 234 244 261 268 273 2 281 I II A B 292 3 I II A B c 1 1 (1) x 2 + 4xy + 4y 2 x 2y 2 (2) 8x 2 + 16xy + 6y 2
30 I .............................................2........................................3................................................4.......................................... 2.5..........................................
I-2 (100 ) (1) y(x) y dy dx y d2 y dx 2 (a) y + 2y 3y = 9e 2x (b) x 2 y 6y = 5x 4 (2) Bernoulli B n (n = 0, 1, 2,...) x e x 1 = n=0 B 0 B 1 B 2 (3) co
16 I ( ) (1) I-1 I-2 I-3 (2) I-1 ( ) (100 ) 2l x x = 0 y t y(x, t) y(±l, t) = 0 m T g y(x, t) l y(x, t) c = 2 y(x, t) c 2 2 y(x, t) = g (A) t 2 x 2 T/m (1) y 0 (x) y 0 (x) = g c 2 (l2 x 2 ) (B) (2) (1)
1 1 3 ABCD ABD AC BD E E BD 1 : 2 (1) AB = AD =, AB AD = (2) AE = AB + (3) A F AD AE 2 = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD 1 1
ABCD ABD AC BD E E BD : () AB = AD =, AB AD = () AE = AB + () A F AD AE = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD AB + AD AB + 7 9 AD AB + AD AB + 9 7 4 9 AD () AB sin π = AB = ABD AD
(4) P θ P 3 P O O = θ OP = a n P n OP n = a n {a n } a = θ, a n = a n (n ) {a n } θ a n = ( ) n θ P n O = a a + a 3 + ( ) n a n a a + a 3 + ( ) n a n
3 () 3,,C = a, C = a, C = b, C = θ(0 < θ < π) cos θ = a + (a) b (a) = 5a b 4a b = 5a 4a cos θ b = a 5 4 cos θ a ( b > 0) C C l = a + a + a 5 4 cos θ = a(3 + 5 4 cos θ) C a l = 3 + 5 4 cos θ < cos θ < 4
all.dvi
38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t
notekiso1_09.dvi
39 3 3.1 2 Ax 1,y 1 Bx 2,y 2 x y fx, y z fx, y x 1,y 1, 0 x 1,y 1,fx 1,y 1 x 2,y 2, 0 x 2,y 2,fx 2,y 2 A s I fx, yds lim fx i,y i Δs. 3.1.1 Δs 0 x i,y i N Δs 1 I lim Δx 2 +Δy 2 0 x 1 fx i,y i Δx i 2 +Δy
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
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
,.,. 2, R 2, ( )., I R. c : I R 2, : (1) c C -, (2) t I, c (t) (0, 0). c(i). c (t)., c(t) = (x(t), y(t)) c (t) = (x (t), y (t)) : (1)
( ) 1., : ;, ;, ; =. ( ).,.,,,., 2.,.,,.,.,,., y = f(x), f ( ).,,.,.,., U R m, F : U R n, M, f : M R p M, p,, R m,,, R m. 2009 A tamaru math.sci.hiroshima-u.ac.jp 1 ,.,. 2, R 2, ( ).,. 2.1 2.1. I R. c