1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1."

Transcription

1

2 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.

3 1.1 V 1 ev = J e = C 1.1 L M T Q 1. imensional analysis m k A T T = m α k β A γ F = kx [k] = MLT /L = M/T = T, = M α M T β L γ = M α+β T β L γ α + β = 0, β = 1, γ = 0 α = 1/, β = 1/, γ = 0 T = m k 1.3 T = π m/k A 1. g T l g m G = m 3 kg 1 s M = kg R = m M = kg R = m 3

4 1.3 rt = xt, yt, zt v a vt = rt, at = vt = rt 1.4 v = ṙ = ẋ, ẏ, ż ft + h ft ft = lim h 0 h 1.5 t n sin t f f + g = + g, f fg = g + f g cf = cf x fxt = x f 1 x = fx x y fy 1.9 y = f 1 x x = fy f = x f x, y x = 1 x y φ t φ = φt sin φ φ sin φ sin φ = φ = φ cos φ cos φ 4

5 1.3 1 ft f = f, n n ftgt = n r=0 gt ft = ġf g f f n f r tg n r t r f r t ft r n r n n! = r r!n r! e = e = lim n = n n n=0 1 n! 1.10 log t = t 1 u u log t = 1 t 1.11 logxy = log x + log y y = e x x = log y 1.1 e x+y = e x e y logxy = log X + log Y log1 + t = 1 + t, et = e t log1 + t = t t + t3 3 = 1 n 1 tn n n=1 e t = 1 + t + t! + t3 3! + = t n n! n=0 1 < t 1 sin t = cos t, cos t = sin t, tan t = sec t tan t sec t = 1/ cos t 5

6 e it = cos t + i sin t 1.13 i = 1 e ix+y = e ix e iy sin t = t t3 3! + t5 t, cos t = 1 5!! + t4 4! 1.14 e t = 1 + t + t! + t3 3! + e it = 1 + it + it! = cos t + i sin t + it3 3! + = 1 t! + t4 4! + i t t3 3! + t5 5! g m z = mg z = g 1.15 z0 = z 0, ż0 = v 0 zt = z 0 + v 0 t 1 gt m g µ v 0 t vt T L 1.6 m v m v = kv + mg v0 = v 0 lim t vt 6

7 1.7 T a T a 3 T a 10 8 km ε T a f = G Mm r M = m = a 7

8 8

9 .1 1. f = 0 a = 0. ma = f 3. f 1 = f 1 4. f = G m 1m r f a f 1 = f 1 = 0 k f = kx ω = k/m mẍ = kx ẍ = ω x.1 xt = A cosωt + B sinωt. x0 = x 0, ẋ0 = v 0 A = x 0, B = v 0 /ω..1 + iω iω xt = 0 + iω xt = 0, iω xt = 0 e at / = ae at e ±iωt sinωt cosωt..1 1 mẋ 1 kx = m ω x E = m ẋ + ω x E 9

10 V x E = m ẋ + V x ẋ mẋẍ = V xẋ mẍ = V x.3 1 mẋ + V x = 0 E = 1 mẋ + V x =.4.1 F mẍ = mω x + F x0 = 0, ẋ0 = 0 Et = m ẋ + ω x. = V x x ẋ Lx, ẋ Lagrangian Lx, ẋ = 1 mẋ V x.5.3 L L ẋ x = 0.6 mẋ V x = 0 mẍ + V x = 0 mẍ = V x.3 10

11 r = x 1, x, x 3 Lr, ṙ L L ṙ r = 0 L L = 0 j = 1,, 3.7 x j x j L = m ẋ + ẏ mω x + y L L ẋ x = 0 mẍ + mω x = 0 L ẏ L y = 0 mÿ + mω y = 0 x, y. l m θ L = m l θ mgl1 cos θ θ ml θ = mg sin θ.3 r = r f f = r V r = V rˆr ˆr = r r = r.8 L = r p, p = mv = mṙ.9 r p L x = yp z zp y, L y = zp x xp z, L z = xp y yp x.10 11

12 L = 0 L = ṙ p + r ṗ = 0.11 ṙ = p/m ṗ = f r 1 r ṙ L = r p.4 M, m G V r = GMm/r L = m ṙ + GMm r.1 xy x = r cos φ, y = r sin φ ẋ = ṙ cos φ r φ sin φ, ẏ = ṙ sin φ + r φ cos φ.13 ṙ = ẋ + ẏ = ṙ cos φ r φ sin φ + ṙ sin φ + r φ cos φ = ṙ + r φ.14 L = m ṙ + r φ + GMm r.15 L L ṙ r = 0, L φ L φ = L m r mr φ + GMm r = 0, r φ = = h mr φ = 0.17 r = h r 3 GM r.18 1

13 m r = rφ t φ = h r = h r φ.19 u = 1/r.18 GM u = u + φ h.0 φ t.1 ω = 1 u = GM l h h + A cos φ r =, l = 1 + ε cos φ GM, ε = Ah GM.1 l ε ε 0 ε < 1 ε = 1 ε > 1.3 E E = m ṙ + r φ GMm r.1 0 ε < 1 E.4 T E = GMm 1 ε l.18 ṙ t ṙ = C h r + GM r, C = = r min = r max r 1 = h 1 1 r min r r 1 r max r min = l/1 + ε, r max = l/1 ε T T = r max r min h rmax r min rr rmax rr r min r = r min cos θ + r max sin θ T = πl h1 ε 3/ = 4π a 3 a = r min + r max / = l/1 ε GM 1/ 13

14 3 3.1 Lx, ẋ I = t1 t 0 Lx, ẋ 3.1 xt, ẋt I functional xt ẋt xt xt + δxt, ẋt ẋt + δẋt I I I + δi δi δi = t1 t 0 Lx + δx, ẋ + δẋ Lx, ẋ = t1 t 0 L L δx + x ẋ δẋ 3. δẋ = δx δxt 0 = δxt 1 = 0 δi = t1 t 0 L x L δxt 3.3 ẋ = 0 I 3.3 δi δxt = L x L ẋ 3.4 fx f x = 0 I 3.4=0 principle of the least action 3.1 I = t1 t 0 m ẋ ω x xt 0 = x 0, xt 1 = x 1 I 14

15 3. m 1, m r 1, r V r 1 r L = 1 R r r 1, r m1 ṙ 1 + m ṙ V r1 r 3.5 R = m 1r 1 + m r m 1 + m, r = r 1 r 3.6 r 1 = R + m m 1 + m r, r = R m 1 m 1 + m r 3.7 R, r reuce mass L = m 1 + m Ṙ m 1 m + ṙ V r 3.8 m 1 + m µ = m 1m m 1 + m 1 µ = 1 m m L = L + L M, m µ = Mm/M + m V r = GMm/r L = µ ṙ + GMm r m µ m/m = µ = m µ = m/ m 1 + m R = 0, µ r = r V r 3.3. ω = g/l θ = ω sin θ

16 θ sin θ θ θ = ω θ 3.11 θ 1 θ ω cos θ = = ω cos θ θ 0 θ/ θ = ω cos θ cos θ 0 = 4ω sin θ 0 θ sin t = 0 θ0 = θ 0, θ0 = 0 0 < θ 0 < π θ θ 0 θ = ω sin θ0 sin θ t = ωt 3.14 T T 3.14 T = ω = 4 ω θ0 0 π 0 θ sin θ0 sin θ φ 1 k sin φ k = sin θ 0 sin φ = sinθ/ sinθ 0 / 4 Kk 3.15 ω Kk 0 k < 1 π φ Kk = 1 k sin φ = π k + k K0 = π/, lim k 1 Kk = T = π/ω = π l/g 60 θ 0 = π/3 k = sinπ/6 = 1/ =1/16 6 % x = x +, 1 xα = 1 αx + x < 1 16

17 4 4.1 canonical formalism x x, ẋ Lx, ẋ x x p Hx, p x, p canonical variables Lx, ẋ 1 Lx, ẋ p = L ẋ x p = mẋ L = m ẋ V x 4.1 L = m p = L = mẋ 4. ẋ ṙ + r φ V r p r = L ṙ = mṙ, p φ = L φ = mr φ 4.3 p φ Hx, p = ẋp Lx, ẋ 4.4 Hamiltonian ẋ p = L/ ẋ ẋ x, p 4. H = ẋp Lx, ẋ = p m 4.3 H = ṙp r + φp φ L = p r m + p φ mr m m pr m p V x = m p + V x 4.5 m + r p φ mr = 1 p r + p φ m r + V r

18 3 x, p ẋ = H p, ṗ = H x 4.7 ẋ ẋ = ẋx, p Hx, p = ẋx, pp Lx, ẋx, p H p H x = ẋ p p + ẋ L ẋ ẋ p = ẋ = ẋ L x p x + L ẋ ẋ x = ṗ = L x = L ẋ L/ ẋ = p x, p p/ x = H = p m + V x ẋ = H p = p m, ṗ = H x = V x H = 1 p r + p φ m r + V r ṙ = H = p r p r m, φ = H = p φ p φ mr, p r = H r = p φ mr 3 V r, 4.9 p φ = H φ = p r, p φ r, φ ṗ φ = 0 p φ = mr φ = 4. H = p m + mω x, ẋ = p m, ṗ = mω x

19 p ẍ = ω x ξ = p + imωx i = 1 ξ = ṗ + imωẋ = mω x + iωp = iω p + imωx = iωξ ξt = ξ0 e iωt 4.1 ξt = pt + imωxt, ξ0 = p0 + imωx0 e iωt = cosωt + i sinωt pt = p0 cosωt mωx0 sinωt, xt = x0 cosωt + p0 sinωt 4.13 mω x, p Hx 1, x, p 1, p = 1 p 1 + p + ω x 1 + x + γx 1 x 4.14 m = 1 ẋ 1 = H p 1 = p 1, ẋ = H p = p, ṗ 1 = H x 1 = ω x 1 γx, ṗ = H x = ω x γx 1 p 1, p ẍ 1 = ω x 1 γx, ẍ = ω x γx x = x 1 + x, y = x 1 x ẍ = ω + γx, ÿ = ω γy ω x = ω + γ, ω y = ω γ xt = x0 cosω x t + ẋ0 ω x sinω x t, yt = y0 cosω y t + ẏ0 ω y sinω y t, 4.16 x 1, x 4.3 m q E B m r = q E + ṙ B

20 4.1 E = E 0, 0, 0 B = 0, 0, B 0 m v = q E + v B v0 = φ A L = A ṙ mṙ + qa = m r + q t L r E = graφ A, B = rota 4.18 t Lr, ṙ = m ṙ + qṙ A qφ ṙ A, = q φ r + q φ ṙ A = q + q ṙ A + ṙ rota r r m r = q A t graφ + qṙ rota = q E + ṙ B A Ar, t = + ṙ A, graa b = a b + b a + a rotb + b rota 4.0 t 4. gauge transformation φ φ χ t, A A + graχ 4.18 E B p = L ṙ = mṙ + qa 4.1 0

21 p mṙ m Hr, p = ṙ p L = ṙ mṙ + qa ṙ + qṙ A qφ = m ṙ + qφ = 1 m p qa + qφ D t roth = j, ive = ρ, B + rote = 0, ivb = t j ρ D = ε 0 E, H = B/µ 0 ε 0 µ 0 c c = 1/ε 0 µ 0 E, B, φ, A L = 1 ε 0 E 1µ0 B ε 0 E φ + Ȧ 1 B rota ρφ + j A 4.4 µ 0 t1 I = mc 1 ẋ c 4.5 t 0 ẋ ṙ 1 ẋ /c 1 ẋ /c I = t1 t 0 m ẋ mc 4.5 mẋ = ẋ /c p = mẋ/ 1 ẋ /c ṗ = 0 1

22 5 5.1 L = ẋp H t1 t1 I = L = t 0 x x + δx, p p + δp t 0 ẋp Hx, p 5.1 δi = = = t1 t 0 t1 t 0 t1 t 0 ẋ + δẋp + δp Hx + δx, p + δp ẋp Hx, p δp ẋ H p p δp ṗ + H x δẋp + ẋδp H H δx x δx 5. δxt 0 = δxt 1 = 0 δi = 0 ẋ = H p, ṗ = H x x, p X, P Hx, p H X, P Ẋ = H P, P = H X canonical transformation generating function 5.4 L = ẋp Hx, p = ẊP H X, P + W 5.5 W W x, P, t 5.5 W = W t + W x ẋ + W P P ẋp H = ẊP H + W t + W x ẋ + W P P

23 ẊP = XP X P p = W x, X = W P, H = H + W t 5.6 XP x, y Ω X, Y x = X cosωt + Y sinωt, y = X sinωt + Y cosωt 5.7 W x, y, P X, P Y, t W = P X x cosωt y sinωt + P Y x sinωt + y cosωt X = W P X = x cosωt y sinωt, p x = W x = P X cosωt + P Y sinωt, 5.6 W/ t Y = W P Y = x sinωt + y cosωt, 5.9 p y = W y = P X sinωt + P Y cosωt 5.10 H = H + Ω XP Y Y P X 5.11 H Z L Z ΩL Z 5.1 Ω Hx, y, p x, p y = 1 p m x + p mω y + x + y X, Y 5.3 x, p X, P W x, P, t W = xp X = W P = x, W = xp ɛ p = W x = P 5.1 W = xp + ɛgx, P 5.13 G generator G ɛ Gx, p P 3

24 5.6 X = x + ɛ G P, G X = x + ɛ G p, p = P + ɛ G x P = p ɛ G x 5.14 G P p x X = x + ɛ W = xp + ɛp X = W P = x + ɛ, p = W x = P 5.15 Ω ɛ 5.9 x X = x ɛy, G = xp Y yp X W = xp X + yp Y + ɛxp Y yp X X = W P X = x ɛy, p x = W x = P X + ɛp Y, y Y = y + ɛx Y = W P Y = y + ɛx, 5.16 p y = W y = P Y ɛp X 5.17 ɛ G = xp y yp x = L z t T = t + ɛ xt + ɛ xt = ɛẋ = ɛ H p, pt + ɛ pt = ɛṗ = ɛ H x 5.18 G G

25 6 6.1 x, p x, p Ax, p, Bx, p {A, B} = A B x Poisson bracket p A p x 1, x,, x N, p 1, p,, p N {A, B} = N j=1 B x A B A B x j p j p j x j {x, x} = 0, {x, p} = 1, {p, x} = 1, {p, p} = {x j, x k } = 0, {x j, p k } = δ jk, {p j, x k } = δ jk, {p j, p k } = G 5.14 X = x + ɛ G p, P = p ɛ G x 6.5 x = X x p = X p X + P x X + P p P = 1 + ɛ G x p P = G ɛ p X + X G ɛ x P, 1 ɛ G x p P A B x p A p B x = 1 + ɛ G x p = A X ɛ G p A X + B P A P A X G A ɛ x P 1 ɛ G x p A P ɛ X + ɛ G B p 1 + ɛ G x p B P 1 ɛ G x p B X G B ɛ x P B X + Oɛ {A, B} = {B, A}. {AB, C} = {A, C}B + A{B, C} 5

26 3. {A, {B, C}} + {B, {C, A}} + {C, {A, B}} = 0 {A, B} = A B x p B A B x p = A x p A x B = {B, A} p {x, p} = {x, p}x + x{x, p} = x, {x n, p} = nx n 1 {x, p } = {x, p}p + p{x, p} = p, {x, p n } = np n n 6.1 {x, p } 6. x, p Ax, p, t A Ax, p, t = t + A x ẋ + A p ṗ = A t + A x H p A p H x = A + {A, H} 6.7 t A explicitly A/ t = 0 A = {A, H} x, p X, P G 6.5 X = x + ɛ G p, P = p ɛ G x F X, P = F x, p + ɛ{f, G} 6. F F = H 6.8 HX, P = Hx, p + ɛ{h, G} = Hx, p ɛ G G G 6

27 6.3 m = 1 H = 1 p + ω x 6.9 a = 1 ω ωx + ip, a = 1 ω ωx ip 6.10 {a, a } = 1 1 {ωx + ip, ωx ip} = ω ω iω{x, p} + iω{p, x} = i, {a, a} = 0, {a, a } = 0 H = ωa a 6.11 a, a a = {a, H} = ω{a, a a} = iωa, a = {a, H} = ω{a, a a} = iωa 6.1 at = a0e iωt, a t = a 0e iωt 6.13 L = L x, L y, L z L x = yp z zp y, L y = zp x xp z, L z = xp y yp x {L x, L y } = {yp z zp y, zp x xp z } = y{p z, z}p x + x{z, p z }p y = xp y yp x = L z, {L y, L z } = {zp x xp z, xp y yp x } = z{p x, x}p y + y{x, p x }p z = yp z zp y = L x, 6.15 {L z, L x } = {xp y yp x, yp z zp y } = x{p y, y}p z + z{y, p y }p x = zp x xp z = L y, 6.3 L ± = L x ± il y, L = L x + L y + L z {L ±, L z } = ±il ±, {L +, L } = il z, {L, L x } = {L, L y } = {L, L z } = 0 7

28 7 7.1 x 1, x,, x N p 1, p,, p N x 1,, x N, p 1,, p N N phase space N N H = p m + mω x 7.1 p m + mω x = E 7. x, p N N N 1 7. x, p x, p X, P XP = xp X, P x, p = G 6.5 X, P x, p X x = ɛ G p, = X x P x = 1 + ɛ P p = ɛ G x X p P = p G x p G p x 1 + ɛ G x p ɛ G x + Oɛ ɛ G p 1 ɛ G p x 7.4 = 1 + Oɛ 7.5 x, p G = H T xt + T pt + T = xtpt 7.6 [ ] 8

29 7.3 action variable J = px 7.7 = Hx, p = E 7.8 p p x E 7.7 p m + mω x = E J = π me E/mω = πe/ω J h +1/ E n = n + 1 hω, h h/π g m l A ω = g/l l l + δl ω A δe 0 δω/ω δa/a δe/e δl δe/ω = 0 E/ω aiabatic invariant J = πe/ω 9

30 1.1 1 T 1 Hz s 1 L 1 MT Pa Nm 3 L MT 1 Q Ω VA 1 4 L 3 M 1 T Q F/m / CV 1 m g = GM/R g = 9.8m/s T = l α g β m γ α = 1/, β = 1/, γ = 0 T = l/g 3.15 = 3 g = GM/R G g /g = M /M R /R = 0.17 T /T = g /g = gt = 1/ft ftgt = 1 fg + fġ = 0 g = f 1 f 1 / = f/f g/f = g f 1 n [ n ftgt = + n ftgt ] = t=t n r=0 t = t [ n r f n r ] g r r n r = t=t n r=0 n f r g n r r 1.4 u = 1 + t u log1 + t = u log u = 1 1 u = t u = e t t = log u /u = log u/u = 1/u u/ = u e t / = e t m v = µmg vt = v 0 µgt 3 T = v 0 /µg, L = v 0 T/ = v 0/µg v = k m v mg k v mg k = k m v mg k vt mg/k = v 0 mg/ke kt/m v = mg/k x F mω = ω x F mω xt = F 1 cosωt/mω E/ = mẍ + ω xẋ = F ẋ Et = F xt 30

31 . v = l θ K = ml θ / V = mgl1 cos θ L = K V L θ L θ = 0 ml θ = mg sin θ.3 φ = 0 r min = l/1 + ε ṙ = 0, φ = h/r min E = m h 1 + ε l h = GMl GMm 1 + ε l = GMm 1 ε l ṙ ṙ r = h r 3 ṙ GM r ṙ ṙ = ṙ r, 1 r = ṙ r 3, 1 r = ṙ r r min, r max ṙ = 0 r = r min cos θ + r max sin θ r = r max r min sin θ cos θθ, r r min = r max r min sin θ, r max r = r max r min cos θ T = 4 r max r min h π/ = πr max + r min r max r min h 0 rmin cos θ + r max sin θ θ = πl h1 ε 3/ a 3.1 ẍ = ω x xt 0 = x 0, xt 1 = x 1 xt = A sinωt t 0 +B sinωt 1 t A = x 1 / sinωt, B = x 0 / sinωt T t 1 t 0 I = = t1 mω x sin 0 cosωt 1 t + x 1 cosωt t 0 x 0 x 1 cosωt 1 + t 0 t ωt t 0 mω x sinωt 0 + x 1 cosωt x 0 x 1 31

32 L L Ṙ R = 0 m 1 + m R = 0 L L = 0 µ r = ṙ r r V r 3.3 1/ 1 x = 1 + x/ + π/ Kk = φ k sin φ + = π 1 + π/ 0 φ sin φ = π/4 0 1 k m v x = qe 0 + v y B 0, m v y = qv x B 0, m v z = 0 v0 = 0 ω = qb 0 /m v x t = E 0 B 0 sinωt, v y t = E 0 B 0 1 cosωt, v z t = 0 4. E = graφ χ t A A + graχ = graφ t t = E, B = rota + graχ = rota = B rotgraχ a b = a x b x + a y b y + a z b z a = a x x + a y y + a z z 4.4 ṙ = H p = 1 p qa, m ṗ = H r = q graφ + 1m p qa A + p qa rota 4.0 p = mṙ + qa m r = q graφ A t ṙ rota = qe + ṙ B H = 1 P m X + PY mω + X + Y + ΩXP Y Y P X 3

33 Ẋ = P X m ΩY, P X, P Y P X = mω X ΩP Y, Ẏ = P Y m + ΩX, P Y = mω Y + ΩP X Ẍ = ω Ω X ΩẎ, Ÿ = ω Ω Y + ΩẊ ξt = Xt + iy t ξ = ω Ω ξ + iωξ ξt = ξ0e iωt cosωt Xt, Y t ω Ω 6.1 {AB, C} = AB x C p AB C p x = = {A, C}B + A{B, C} A C A C x B + A B x p p B + A B p x {x, p } = {x, p }x + x{x, p } = p x + x p = 4xp 6. X = x + ɛ G/ p, Y = y ɛ G/ x F X, Y = F x + ɛ G p, p ɛ G = F x, y + ɛ x F G x p F p G = F x, p + ɛ{f, G} x ɛ 6.3 {L +, L z } = {L x + il y, L z } = L y + il x = il x + il y = il +, {L, L z } = {L x il y, L z } = L y il x = il x il y = il, {L +, L } = {L x + il y, L x il y } = il z, {L, L x } = {L x + L y + L z, L x } = L y {L y, L x } + L z {L z, L x } = L y L z + L z L y = ml θ = mg sin θ mgθ 33

34 θt = A sinωt ω = g/l E = 1 mgla l l + δl δe δe = W + W δl < 0 W = mgδl W = = ml θ +mg cos θ T = π/ω W = 1 T T 0 ml θ + mg 1 θ = mg mga cos θ 1 θ / θt = A cosωt δe = mg mga δl + mgδl = 1 4 mga δl E = 1 mgla δe E = 1 δl l, δω ω = 1 δl l ω = g/l δ E ωδe Eδω = ω ω = E 1 δl ω l + 1 δa E = 1 mgla δl = 0 l δe E = δl l + δa A δa A = 3 δl 4 l 34

35 [ ] mω x mγẋ mẍ + γmẋ + mω x = 0 i 0 < γ < ω ii ω < γ x0 = x 0, ẋ0 = 0 xt [ ] Ω mẍ + mω x = F cosωt = Re F e iωt Re xt = ReAe iωt A = 0 mẍ + mγẋ + mω x = F cosωt = Re F e iωt A A = A e iδ A δ Ω [ ] V x m 1 E = m ẋ + V x T T = x x 1 m E V x x x 1 < x V x = E a V x = K x b V x = V 0 1 e ax [ ] L = r p ṗ = α r 3 r A = ṙ L α r r A/ = 0 A 35

36 [ ] NMR M = M x, M y, M z M x M y M z = γm B x M x T = γm B y M y T = γm B z M z M eq T 1 γ T 1, T M eq M z M eq = M z M B B = 0, 0, B 0 M z t M z 0 Mt = M x t + im y t Mt Mt M0 = M x 0 + im y 0 M x t, M y t [ ] 4.1 vt rt r0 = 0 E B y E B [ ] m = 1 H = 1 p α x x 0 α > 0, x 0 > 0 xt = = x 0, xt = + = +x 0 E = 0 [ ] r, p Hr, p rp ρr, p, t ρ = {H, ρ} t ρ 0 = Cexp H/k B T C ρ 0 rp = 1 ρ 0 0 Ar, p < A >= Ar, pρ 0 r, prp H = p /m C < H > 36

37 [ ] m mg kv m v = mg kv v0 = 0 [ ] m Ze µ z x = r cos φ, y = r sin φ cgs-gauss L = m ṙ + r φ + Zeµ φ c r r, φ p r, p φ p φ φ/ p φ H = p r ṙ + p φ φ L H = 1 p r + 1r p φ Zeµ m cr m r + 1 mr p φ Zeµ = E cr r/ p φ = 0 t r φ = ± c Zeµ r3 m E Zeµ mc r 4 φ < 0 φ > 0 + r φ = ±Ar r 4 r0 4 A, r 0 φ = 0 r = r 0 r 0/r = u r = r 0 φ x 0 u 1 u = sin 1 x 37

38 [ ] x = e αt α + γα + ω = 0 i 0 < γ < ω α = γ ± γ ω xt = e γt A cos ω γ t + B sin ω γ t ii ω < γ xt = e γt Ae γ ω t + Be γ ω t x0 = x 0, ẋ0 = 0 i 0 < γ < ω A = x 0, B = γ ω γ x 0 ii ω < γ A = x γ, B = x 0 γ 1 γ ω γ ω [ ] xt = ReAe iωt Re m Ω + ω Ae iωt = Re F e iωt A = F mω Ω A xt = a cosωt + b sinωt + F mω Ω cosωt xt = ReAe iωt Re m Ω + iγω + ω Ae iωt = Re F e iωt A = A A A δ F A = m γω, tan δ = ω Ω + 4γ Ω Ω ω Ω ω ω Ω + 4γ Ω 4ω Ω ω + γ A F 1 mω Ω ω + γ F mω Ω + iγω 38

39 Ω Lorentzian [ ] m T x + V x = E x = ± E V x m x T = x 1 x = m E V x x x 1 < x E = V x x 1 m x E V x x 1 x a V x = K x x 0 = E/K x0 m m T = x x 0 E K x = K m = K x 0 = 4 me K b V x = V 0 1 e ax y T = x x 1 x0 m x E V 0 1 e ax y = e ax y = ayx y1 y m T = ay E V 0 1 y = 1 m a x 1 < x y < y 1 V 0 y1 y 0 x x0 x y y y y y 1 y y 1 = 1 + E V 0, y = 1 E V 0 y = y cos θ + y 1 sin θ y1 y y π/ y y y y 1 y = 0 θ y cos θ + y 1 sin θ = π y1 y T T = π m a V 0 E 39

40 [ ] A = ṙ L α r r t Ȧ = r L + ṙ L α r ṙ + α r ṙr L = 0 r = α mr 3 r L = r p = mr ṙ Ȧ = α r 3 r r ṙ α r ṙ + α r ṙr r r ṙ = r ṙr r rṙ, r r = r, r ṙ = 1 r r = 1 r = rṙ r r [ ] B = 0, 0, B 0 M z M x, M y M B x = M y B z M z B y = B 0 M y, M B y = M z B x M x B z = B 0 M x, M B z = M x B y M y B x = 0, M z = M z M eq T 1 M z t M eq = M z 0 M eq e t/t1 M x = γb 0 M y M x T, M y = γb 0 M x M y T M = M x + im y M = M iγb 0 + 1T M Mt = M0exp iγb t T M x t = ReMt = e t/t M x 0 cosω 0 t + M y 0 sinω 0 t M y t = ImMt = e t/t M x 0 sinω 0 t + M y 0 cosω 0 t ω 0 = γb 0 [ ] - ẋ v x = E 0 B 0 sinωt, ẏ v y = E 0 B 0 1 cosωt t x0 = y0 = 0 xt = E 0 ωb 0 1 cosωt, yt = E 0 ωb 0 ωt sinωt 40

41 [ ] 1 + x α x x 0 = 0 x = ± αx 0 x x 0 x x 0 x = α 1 x 0 x 0 t 0 xt = x 0 tanh αx 0 t 1 x 0 x + 1 = 1 x0 + x log x 0 + x x 0 x 0 x xt t t t 0 xt = x 0 tanh αx 0 t [ ] 0 = ρ = ρ t + ρ r ṙ + ρ pṗ ρ t = H ρ r p ρ r H p = {H, ρ} ρ 0 = C exp H/k B T H r, p H {H, ρ 0 } = 0 ρ 0 t = 0 H = p /m ρ 0 rp = 1 C r e p /mk B T p = CV πmk B T 3/ = 1 C = 1 V πmk B T 3/ C p H = Hρ 0 rp = C p r m e p /mk B T p = 1 m mk BT 3 = 3 k BT [ ] v = g k m v = k v m v mg, v k 41

42 v v v v = k m [ ] 1 v + v log = k v v v m t vt = v kv t tanh = m 0 t 0 mg kg k tanh m t [ ] p r = L ṙ = mṙ, p φ = L φ = Zeµ mr φ + cr L ṙ L φ L r = 0 p r L φ = 0 p φ = 0 mr φ Zeµ φ cr = 0 r = r φ Zeµ φ mcr, p φ = m H = p r ṙ + p φ φ L = ṙ + r φ 1 = p r + 1r p φ Zeµ m cr φ = 1 mr p φ Zeµ cr m r + 1 mr p φ Zeµ = E cr r = E 1 m mr p φ Zeµ cr p φ = 0 r = ± E Zeµ m mc r 4 p φ = 0 φ = Zeµ mcr 3 4

43 t r φ = r/ φ/ = ± cr3 Zeµ m E Zeµ mc r 4 ±Ar r 4 r 4 0 r r 0 r 4 0 = Zeµ mc E r ± c me Zeµ r3 = ±Ar 3 A = c me Zeµ = 1 r0 + φ = 0 r r 0 r r r 4 r 4 0 = A φ 0 φ Aφ u = r0/r u = ur/r = 1 u u r0 = u r0 sin 1 u π u = r 0 π r = sin r 0Aφ = cosφ r = r 0 cosφ A = 1/r0 φ = π/4 φ = 0 r = r 0 φ = π/4 φ = π/ 43

(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

(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)

More information

23 7 28 i i 1 1 1.1................................... 2 1.2............................... 3 1.2.1.................................... 3 1.2.2............................... 4 1.2.3 SI..............................

More information

m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d

m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m

More information

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..........................

More information

pdf

pdf 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

More information

I 1

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

More information

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,

More information

chap1.dvi

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

More information

2011de.dvi

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

More information

dynamics-solution2.dvi

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

More information

) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)

) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) 4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7

More information

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. 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..............

More information

B ver B

B ver B B ver. 2017.02.24 B Contents 1 11 1.1....................... 11 1.1.1............. 11 1.1.2.......................... 12 1.2............................. 14 1.2.1................ 14 1.2.2.......................

More information

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. 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....

More information

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 ( ) 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

More information

2 1 x 1.1: v mg x (t) = v(t) mv (t) = mg 0 x(0) = x 0 v(0) = v 0 x(t) = x 0 + v 0 t 1 2 gt2 v(t) = v 0 gt t x = x 0 + v2 0 2g v2 2g 1.1 (x, v) θ

2 1 x 1.1: v mg x (t) = v(t) mv (t) = mg 0 x(0) = x 0 v(0) = v 0 x(t) = x 0 + v 0 t 1 2 gt2 v(t) = v 0 gt t x = x 0 + v2 0 2g v2 2g 1.1 (x, v) θ 1 1 1.1 (Isaac Newton, 1642 1727) 1. : 2. ( ) F = ma 3. ; F a 2 t x(t) v(t) = x (t) v (t) = x (t) F 3 3 3 3 3 3 6 1 2 6 12 1 3 1 2 m 2 1 x 1.1: v mg x (t) = v(t) mv (t) = mg 0 x(0) = x 0 v(0) = v 0 x(t)

More information

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

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

More information

C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0

C : q i (t) C : q i (t) q i (t) q i(t) q i(t) q i (t)+δq i (t) (2) δq i (t) δq i (t) C, C δq i (t 0 )0, δq i (t 1 ) 0 (3) δs S[C ] S[C] t1 t 0 t1 t 0 1 2003 4 24 ( ) 1 1.1 q i (i 1,,N) N [ ] t t 0 q i (t 0 )q 0 i t 1 q i (t 1 )q 1 i t 0 t t 1 t t 0 q 0 i t 1 q 1 i S[q(t)] t1 t 0 L(q(t), q(t),t)dt (1) S[q(t)] L(q(t), q(t),t) q 1.,q N q 1,, q N t C :

More information

() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y

() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y 5. [. ] z = f(, y) () z = 3 4 y + y + 3y () z = y (3) z = sin( y) (4) z = cos y (5) z = 4y (6) z = tan y (7) z = log( + y ) (8) z = tan y + + y ( ) () z = 3 8y + y z y = 4 + + 6y () z = y z y = (3) z =

More information

.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(

.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,

More information

Gmech08.dvi

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.)

More information

006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................

More information

http://www.ike-dyn.ritsumei.ac.jp/ hyoo/wave.html 1 1, 5 3 1.1 1..................................... 3 1.2 5.1................................... 4 1.3.......................... 5 1.4 5.2, 5.3....................

More information

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................................

More information

x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin

x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin 2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1 x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ

More information

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 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

More information

K E N Z U 01 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.................................... 4 1..1..................................... 4 1...................................... 5................................

More information

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10% 1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n

More information

K E N Z OU

K E N Z OU K E N Z OU 11 1 1 1.1..................................... 1.1.1............................ 1.1..................................................................................... 4 1.........................................

More information

i

i 009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3

More information

2 Chapter 4 (f4a). 2. (f4cone) ( θ) () g M. 2. (f4b) T M L P a θ (f4eki) ρ H A a g. v ( ) 2. H(t) ( )

2 Chapter 4 (f4a). 2. (f4cone) ( θ) () g M. 2. (f4b) T M L P a θ (f4eki) ρ H A a g. v ( ) 2. H(t) ( ) http://astr-www.kj.yamagata-u.ac.jp/~shibata f4a f4b 2 f4cone f4eki f4end 4 f5meanfp f6coin () f6a f7a f7b f7d f8a f8b f9a f9b f9c f9kep f0a f0bt version feqmo fvec4 fvec fvec6 fvec2 fvec3 f3a (-D) f3b

More information

構造と連続体の力学基礎

構造と連続体の力学基礎 II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton

More information

x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)

x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b) 2011 I 2 II III 17, 18, 19 7 7 1 2 2 2 1 2 1 1 1.1.............................. 2 1.2 : 1.................... 4 1.2.1 2............................... 5 1.3 : 2.................... 5 1.3.1 2.....................................

More information

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.   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)

More information

DVIOUT

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)

More information

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

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

More information

DE-resume

DE-resume - 2011, http://c-faculty.chuo-u.ac.jp/ nishioka/ 2 11 21131 : 4 1 x y(x, y (x,y (x,,y (n, (1.1 F (x, y, y,y,,y (n =0. (1.1 n. (1.1 y(x. y(x (1.1. 1 1 1 1.1... 2 1.2... 9 1.3 1... 26 2 2 34 2.1,... 35 2.2

More information

24 I ( ) 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

24 I ( ) 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 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),

More information

r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B

r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B 1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n

More information

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5. A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c

More information

/Volumes/NO NAME/gakujututosho/chap1.tex i

/Volumes/NO NAME/gakujututosho/chap1.tex i 2012 4 10 /Volumes/NO NAME/gakujututosho/chap1.tex i iii 1 7 1.1............................... 7 2 11 2.1........................................... 11 2.2................................... 18 2.3...................................

More information

( )

( ) 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...............................

More information

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

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)

More information

() 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 (

() 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

More information

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy

More information

. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n

. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n 003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........

More information

215 11 13 1 2 1.1....................... 2 1.2.................... 2 1.3..................... 2 1.4...................... 3 1.5............... 3 1.6........................... 4 1.7.................. 4

More information

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

More information

II K116 : January 14, ,. A = (a ij ) ij m n. ( ). B m n, C n l. A = max{ a ij }. ij A + B A + B, AC n A C (1) 1. m n (A k ) k=1,... m n A, A k k

II K116 : January 14, ,. A = (a ij ) ij m n. ( ). B m n, C n l. A = max{ a ij }. ij A + B A + B, AC n A C (1) 1. m n (A k ) k=1,... m n A, A k k : January 14, 28..,. A = (a ij ) ij m n. ( ). B m n, C n l. A = max{ a ij }. ij A + B A + B, AC n A C (1) 1. m n (A k ) k=1,... m n A, A k k, A. lim k A k = A. A k = (a (k) ij ) ij, A k = (a ij ) ij, i,

More information

keisoku01.dvi

keisoku01.dvi 2.,, Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 5 Mon, 2006, 401, SAGA, JAPAN Dept.

More information

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 (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) = (, )

More information

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 (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

More information

ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx

ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x

More information

ẍ = kx, (k > ) (.) x x(t) = A cos(ωt + α) (.). d/ = D. d dt x + k ( x = D + k ) ( ) ( ) k k x = D + i D i x =... ( ) k D + i x = or ( ) k D i x =.. k.

ẍ = kx, (k > ) (.) x x(t) = A cos(ωt + α) (.). d/ = D. d dt x + k ( x = D + k ) ( ) ( ) k k x = D + i D i x =... ( ) k D + i x = or ( ) k D i x =.. k. K E N Z OU 8 9 8. F = kx x 3 678 ẍ = kx, (k > ) (.) x x(t) = A cos(ωt + α) (.). d/ = D. d dt x + k ( x = D + k ) ( ) ( ) k k x = D + i D i x =... ( ) k D + i x = or ( ) k D i x =.. k. D = ±i dt = ±iωx,

More information

振動と波動

振動と波動 Report JS0.5 J Simplicity February 4, 2012 1 J Simplicity HOME http://www.jsimplicity.com/ Preface 2 Report 2 Contents I 5 1 6 1.1..................................... 6 1.2 1 1:................ 7 1.3

More information

64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () m/s : : a) b) kg/m kg/m k

64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () m/s : : a) b) kg/m kg/m k 63 3 Section 3.1 g 3.1 3.1: : 64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () 3 9.8 m/s 2 3.2 3.2: : a) b) 5 15 4 1 1. 1 3 14. 1 3 kg/m 3 2 3.3 1 3 5.8 1 3 kg/m 3 3 2.65 1 3 kg/m 3 4 6 m 3.1. 65 5

More information

Z: Q: R: C: sin 6 5 ζ a, b

Z: Q: R: C: sin 6 5 ζ a, b Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,

More information

201711grade1ouyou.pdf

201711grade1ouyou.pdf 2017 11 26 1 2 52 3 12 13 22 23 32 33 42 3 5 3 4 90 5 6 A 1 2 Web Web 3 4 1 2... 5 6 7 7 44 8 9 1 2 3 1 p p >2 2 A 1 2 0.6 0.4 0.52... (a) 0.6 0.4...... B 1 2 0.8-0.2 0.52..... (b) 0.6 0.52.... 1 A B 2

More information

2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( (

2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( ( (. x y y x f y = f(x y x y = y(x y x y dx = d dx y(x = y (x = f (x y = y(x x ( (differential equation ( + y 2 dx + xy = 0 dx = xy + y 2 2 2 x y 2 F (x, y = xy + y 2 y = y(x x x xy(x = F (x, y(x + y(x 2

More information

() 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)

() 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 ()

More information

2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p

2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p 2012 IA 8 I 1 10 10 29 1. [0, 1] n x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 2. 1 x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 1 0 f(x)dx 3. < b < c [, c] b [, c] 4. [, b] f(x) 1 f(x) 1 f(x) [, b] 5.

More information

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B 9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A

More information

( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )

( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes ) ( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)

More information

meiji_resume_1.PDF

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

More information

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. (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 +

More information

IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................

More information

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

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 ),

More information

* 1 2014 7 8 *1 iii 1. Newton 1 1.1 Newton........................... 1 1.2............................. 4 1.3................................. 5 2. 9 2.1......................... 9 2.2........................

More information

Part () () Γ Part ,

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

More information

/Volumes/NO NAME/gakujututosho/chap1.tex i

/Volumes/NO NAME/gakujututosho/chap1.tex i 2010 4 8 /Volumes/NO NAME/gakujututosho/chap1.tex i iii 1 5 1.1............................... 5 2 9 2.1........................................... 9 2.2................................... 16 2.3...................................

More information

2000年度『数学展望 I』講義録

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

More information

I, II 1, 2 ɛ-δ 100 A = A 4 : 6 = max{ A, } A A 10

I, II 1, 2 ɛ-δ 100 A = A 4 : 6 = max{ A, } A A 10 1 2007.4.13. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 0. 1. 1. 2. 3. 2. ɛ-δ 1. ɛ-n

More information

,. 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,,.

,. 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,

More information

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C 8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,

More information

) ] [ 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 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

More information

I

I I 6 4 10 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............

More information

4................................. 4................................. 4 6................................. 6................................. 9.................................................... 3..3..........................

More information

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 =

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 =

More information

6.1 (P (P (P (P (P (P (, P (, P.101

6.1 (P (P (P (P (P (P (, P (, P.101 (009 1 4 3 ( ( ( 00 1 (P.3 1 1.1 (P.3.................. 1 1. (P.4............... 1 (P.15.1 (P.15................. (P.18............3 (P.17......... 3.4 (P................ 4 3 (P.7 4 3.1 ( P.7...........

More information

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..........

More information

6.1 (P (P (P (P (P (P (, P (, P.

6.1 (P (P (P (P (P (P (, P (, P. (011 30 7 0 ( ( 3 ( 010 1 (P.3 1 1.1 (P.4.................. 1 1. (P.4............... 1 (P.15.1 (P.16................. (P.0............3 (P.18 3.4 (P.3............... 4 3 (P.9 4 3.1 (P.30........... 4 3.

More information

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou (Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fourier) (Fourier Bessel).. V ρ(x, y, z) V = 4πGρ G :.

More information

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 = 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

More information

LCR e ix LC AM m k x m x x > 0 x < 0 F x > 0 x < 0 F = k x (k > 0) k x = x(t)

LCR e ix LC AM m k x m x x > 0 x < 0 F x > 0 x < 0 F = k x (k > 0) k x = x(t) 338 7 7.3 LCR 2.4.3 e ix LC AM 7.3.1 7.3.1.1 m k x m x x > 0 x < 0 F x > 0 x < 0 F = k x k > 0 k 5.3.1.1 x = xt 7.3 339 m 2 x t 2 = k x 2 x t 2 = ω 2 0 x ω0 = k m ω 0 1.4.4.3 2 +α 14.9.3.1 5.3.2.1 2 x

More information

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 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..........................................

More information

[ ] 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 [ ]. 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 =

More information

128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds

128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds 127 3 II 3.1 3.1.1 Φ(t) ϕ em = dφ dt (3.1) B( r) Φ = { B( r) n( r)}ds (3.2) S S n( r) Φ 128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds

More information

: , 2.0, 3.0, 2.0, (%) ( 2.

: , 2.0, 3.0, 2.0, (%) ( 2. 2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................

More information

i

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,

More information

II 1 II 2012 II Gauss-Bonnet II

II 1 II 2012 II Gauss-Bonnet II II 1 II 212 II Gauss-Bonnet II 1 1 1.1......................................... 1 1.2............................................ 2 1.3.................................. 3 1.4.............................................

More information

(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)

(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

More information

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 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

More information

simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a =

simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a = II 6 ishimori@phys.titech.ac.jp 6.. 5.4.. f Rx = f Lx = fx fx + lim = lim x x + x x f c = f x + x < c < x x x + lim x x fx fx x x = lim x x f c = f x x < c < x cosmx cosxdx = {cosm x + cosm + x} dx = [

More information

数学の基礎訓練I

数学の基礎訓練I I 9 6 13 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 3 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............

More information

量子力学 問題

量子力学 問題 3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,

More information

δ ij δ ij ˆx ˆx ŷ ŷ ẑ ẑ 0, ˆx ŷ ŷ ˆx ẑ, ŷ ẑ ẑ ŷ ẑ, ẑ ˆx ˆx ẑ ŷ, a b a x ˆx + a y ŷ + a z ẑ b x ˆx + b

δ ij δ ij ˆx ˆx ŷ ŷ ẑ ẑ 0, ˆx ŷ ŷ ˆx ẑ, ŷ ẑ ẑ ŷ ẑ, ẑ ˆx ˆx ẑ ŷ, a b a x ˆx + a y ŷ + a z ẑ b x ˆx + b 23 2 2.1 n n r x, y, z ˆx ŷ ẑ 1 a a x ˆx + a y ŷ + a z ẑ 2.1.1 3 a iˆx i. 2.1.2 i1 i j k e x e y e z 3 a b a i b i i 1, 2, 3 x y z ˆx i ˆx j δ ij, 2.1.3 n a b a i b i a i b i a x b x + a y b y + a z b

More information

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

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

More information

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc 013 6 30 BCS 1 1.1........................ 1................................ 3 1.3............................ 3 1.4............................... 5 1.5.................................... 5 6 3 7 4 8

More information

W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)

W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2) 3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)

More information