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

Size: px
Start display at page:

Download "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"

Transcription

1 9 7

2 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 y B B x A x B B C y C y + A x B y A x A y A C x C = C x C B y x B y B C x C y C A x A y A B B x B y B x B y B = C C C x C y C x C y C x C y C = A A x A y A x A y A B x B y B.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 x y B C = B x B y B = x B y C B C y y B x C B C x + B x C y B y C x C x C y C x y A B C = A x A y A B y C B C y B x C B C x B x C y B y C x = x {A y B x C y B y C x + A B x C B C x } y {A x B x C y B y C x A B y C B C y } + { A x B x C B C x A y B y C B C y } = x {B x A y C y + A C C x A y B y + A B } + y {B y A x C x + A C C y A x B x + A B } + {B A x C x + A y C y C A x B x + A y B y } = x {B x A x C x + A y C y + A C C x A x B x + A y B y + A B } + y {B y A x C x + A y C y + A C C y A x B x + A y B y + A B } + {B A x C x + A y C y + A C C A x B x + A y B y + A B } = A x C x + A y C y + A C B x x + B y y + B A x B x + A y B y + A B C x x + C y y + C = A CB A BC 3 x = cos, y = sin x + y = cos + sin = A = sin x + cos y = sin x + cos y = 4.34 l = = = O xy = l = π π = = π

3 5.39 θ = θ = sin θ θ = sin θ O > l sin θ = /l π l [ ] l = sin θ = π sin θ = π l l = πl 6.4 V = h = = h h π [ V = = πh ] = π h 7.43 V = sin θθ O π π [ ] V = sin θθ = π [ cos θ] π 3 = 4π3 3 3 A = x + y = x y A = B = x + y + c = x y c B = + + c. E = x y 4πε + x y c 3 4πε + + c 3 = 4πε c 3/ x + c c 3/ y c 3/ E y = E = + = P c 3/ c = P c 3/ P.. c = P = 3/ + 4 = =, ± 5 5, = 5 B. < E = l + 4πε { + } 3/ t = C. C. E = l 4πε = l 4πε t t + 3/ l 4πε t l t + 4πε tt t + 3/ t + = l 4πε, + + +

4 ' 3 3 = l l = l =, =, =, = + E = 4πε 3 = l 4πε + 3/ = L L C. E = L l 4πε L + 3/ L l 4πε L = l L πε = = l + πε + 3/ L + L L E πε 4 = l = l '= l ' ' L O L = l ' = ' ' l = l = l l = l = l E =, = + E = l 4πε + 3/ = π E = π l 4πε + 3/ = l 4πε + 3/ π x = φ ' φ ' l 4πε + 3/ l ' = φ ' = ' π y = cos x + sin y π E = = l π 4πε + cos x + sin y = 3/ π = l ε + 3/ π = E = 3 E = E = l ε E = l ε cos x + π + > 3/ = = l + 3/ ε + 5/ sin y = E/ E < / > / = / E = / E E mx = l / ε {/ + } = = 3/ l 3 3ε

5 ' 4 5 = s = s E =, = + E = s 4πε + + 3/ = =, = π φ ' y E = s π 4πε + 3/ s π + 4πε + 3/ s = 4πε π + = s sgn ε + x φ ' ' = ' = ' ' ' φ ' E s ε sgn 6 = s = s x y, =, = x x + y y, = x x y y, = + x + y E = 4πε = sx y 3 4πε s E = 4πε W W x x y y + x + y 3/ x x y y + x + y 3/ y x x, y x, y x,y x, y 4 s W y s E = πε + x + y x 3/ = πε s W x = πε + x = [ ] s W x tn πε W y x s W ' y ' + x + x + y = s πε tn W x ' W ' = x' x y' y W tn x lim x + tn x = π/ E = s W tn = s W tn sgn s π πε πε πε sgn = s sgn ε 3 E = ± s ε = kx x + y y + x = y = x =, y = xy

6 5 = = sin θ cos x + sin θ sin y + cos θ = sin θθ = = = cos θ, = sin θ cos + sin θ sin + cos θ = cos θ + Ψ Ψ = D = = 4π = π π π π π 4π 3 sin θθ cos θ sin θθ cos θ + 3/ cos θ sin θθ cos θ + 3/ x ' = O = y R = cos θ + θ RR = sin θθ + Ψ = + R RR = + + R = R 3 4 R 4 > Ψ = 4 < Ψ = 4 [ + + ] [ ] = = [ R R enclose y = π π = 4π π sin θ θ = π 3. D = D = enclose y 4π = sin θ θ enclose y = L π = πl 3.4 D = D = enclose y = πl ] +

7 6 3.7 xy 3.8 enclose y = = A + = A D = D 3.6 D = D = D sgn = enclose y sgn = A 4 O sgn < : s enclose y = 3.4, 3.3 E = ε D = enclose y 4πε = > : s enclose y = s 4π 3.4, 3.3 E = ε D = enclose y 4πε = s 4π 4πε = s ε 5 O 3.5 < : enclose y = 4 3 π3 3.4, 3.3 E = D = enclose y ε 4πε = 4 3 π3 4πε = 3ε > : enclose y enclose y = 4 3 π3 3.4, 3.3 E = D = enclose y ε 4πε = 4 3 π3 4πε = 3 3ε

8 7 6 L 3.6 < : enclose y = 3.4, 3.5 E = ε D = enclose y πε L = > : enclose y L enclose y = s πl 3.4, 3.5 E = D = enclose y ε πε L = s πl πε L = s ε 7 L 3.6 < : L enclose y enclose y = π L 3.4, 3.5 E = D = enclose y ε πε L = π L πε L = ε > : enclose y L enclose y = π L 3.4, 3.5 E = D = enclose y ε πε L = π L πε L = ε 8 xy A 3.8 < : enclose y = A 3.4, 3.7 E = ε D = enclose y ε A sgn = A ε A sgn = ε > : enclose y = A 3.4, 3.7 E = ε D = enclose y ε A sgn = A ε A sgn = sgn ε

9 8 9.4,,,,.4 A = A + A,, + A +,, A,, + A +,, { + } = + A +,, A,, A = A,, A + A, +, + A,, + A, +, = A, +, A,, A = A A,, + A,, + + A,, + A,, + = A,, + A,, A = A.4 = A = lim = A + 3. = + A = lim A + A A + A + A.5, θ,, θ,.5 A = θ + θ+ θ A + A, θ, + A +, θ, + + A, θ, sin θ θ + A +, θ, { + sin θ θ } + + = + A +,, A, θ, sin θ θ A sin θ θ = A

10 9 + θ + θ+ θ + A, θ, θ + A, θ + θ, θ θ θ+ θ A, θ, sin θ θ + A, θ +, { sinθ + θ θ } = sinθ + θa θ, θ + θ, sin θa θ, θ, θ θ sin θ θ sin θa θ sin θ θ = sin θ θ sin θa θ A, θ, + A, θ, + + A, θ, θ + A, θ, + θ = A, θ, + A, θ, θ A sin θ sin θ θ =.5 = + θ+ θ + θ sin θ + θ + θ 3. A = lim = A + 3.c A sin θ sin θθ = {cos θ cosθ + θ} 3 θ θ + θ sin sin θ θ A = lim sin θ θ sin θa θ + A + A sin θ sin θ θ sin θa θ + A sin θ 4 C = cos x + sin y π/3 C E = 4sin x + cos y, W = = 8 A B π/3 π/3 E = = = sin x + cos y sin 3 + cos 3 = 8 [ = 8 cos cos sin x + cos y sin x + cos y sin sin3 3 π/3 ] π/3 { cos sin + sin cos } = = V = 4πε x + y + + 4πε x + y + + = 4πε x + y + = x + y x + y + = x + y + + = xy

11 E = F, F = ε V = E = ε F = [ ε F = ε F F ε = ε + ε = ] ε F E = F, F = = ε V = E = ε F = [ ] ln F + ln F ε ε = } {ln ln + ln ε ε = ε ln + ε ln E = sgn F = sgn, V = = ε = ε ln ln ε F, F = = = sgn E = sgn F = [ F ε ε = V = ε = ] + ε + ε + = + ε F ε + 4 B = A V 5 V = A B = s ε E = A B sgn s ε sgn = s ε [ + ] = s ε + +

12 V = s ε + s ε + = s ε + s ε s ε V = V = s = s V =, = + V = 4πε = s 4πε + =, = π V = π = s 4πε π π s 4πε + = s 4πε [ ] + = s + ε V = V V = s ε B = A V 3 6 A A V = E = E + + = E B B > s V = E = ε = s ε = s [ ] ln ε = s ln ε < s V = E E = ε = s ε = s [ ] ln ε = s ln ε 6 B = A V 3 7 A A V = E = E + + = E B B > V = E = ε = ε = [ ] ln ε = ln ε < V = E = = ε ε E = E = ε ε [ [ ln ] ε ] ε ln + = ε

13 7 A B 3 4 > A A V = E = E + θ θ + sin θ = E B B s V = E = ε = s ε = s [ ] = s ε ε < s V = E E = ε = s ε = s [ ] = s ε ε 8 A B 3 5 > A A V = E = E + θ θ + sin θ = E B B 3 V = E = 3ε = 3 3ε = 3 [ ] = 3 3ε 3ε < V = E = 3 3ε [ ] E = 3ε [ ] 3 3ε = 3 3ε 3ε = = 3ε 6ε ε 3 3ε 9 R, R P V = + 4πε R R R P, θ, φ + θ + x + x / x + 3 x x / x O R y R = R = + cos θ = cos θ + + cos θ + 3 cos θ + cosπ θ = + cosπ θ + 3 cos π θ / = + cos θ + 3 cos θ 3 / cosπ θ + = cos θ + 3 cos θ 3

14 3 V 4πε [{ + cos θ } { + 3 cos θ + 3 cos θ } + 3 cos θ ] 3 = 3 cos θ = 3 cos θ 4πε 3 4πε [ 3 E = V = V V θ θ = 4πε 3 cos θ [ = 3 cos θ 3 4πε 4 ] 3 cos θ sin θ 4 θ = 3 4πε 4 { 3 cos θ + sin θ cos θ θ } cos θ ] θ θ V, 4.35, 4.36 V = V = = l πε [ l ln ] = πε [ ] ln = l πε [ l ln πε = l πε = ] l ln πε 5 < < E > c E V V enclose y = > c 3.4, 3.3 E = ε D = enclose y 4πε = 4πε = E enclose y = < < 3.4, 3.3 V E = ε D = enclose y 4πε = V = c c E = 4πε = E 4πε = 4πε c V c c V = E E = 4πε 4πε = 4πε + c < < c

15 4 enclose y = > c 3.4, 3.3 E = ε D = enclose y 4πε = 4πε = E enclose y = < < E = V V = c V V = c E = E c 4πε = c E = 4πε c 4πε = 4πε c < < c 3 enclose y = > c E = enclose y = < < 3.4, 3.3 E = ε D = enclose y 4πε = V = V = c c E = 4πε = E E E E = c 4πε = 4πε O < < enclose y = 3.4, 3.3 E = ε D = enclose y 4πε = 4πε = = + θ θ + sin θ V = E = E = V = V 4πε = V = V / / V / / / / = V / / 4πε = [ ] = 4πε 4πε

16 5 3 E E = A + B cos θ C + D cos θ A CE + B DE cos θ = θ cos θ A CE = B DE = f θ = E = A C = B D A A = = cos θ / = P P V V = 4πε A P + 4πε A P = ' ' θ = A O y = A P A P x P =, P = A P = A P A P = A A P + P = cos θ + A P = A P A P = A A P + P = cos θ + = cos θ + cos θ = = + cos θ + cos θ = / P,, / = A P = A P = 3 E = V = s = D = = ε E = sin θθ t = E + F cos θ E + F cos θ 3/ = E = = A P 4πε A P 3 + = A P 4πε A P = A P 3 4πε A P + 3 { } 4πε A P 3 A P A P A P 3 4πε A P

17 6 s = ε E = = = 4π + cos θ 3/ { } 4π A P 3 i i = s = = = 4π = π π s sin θθ = 4π [ ] π π + cos θ 4 > 3.4, 3.3 π = D = enclose y 4π = 4 3 π3 4π = 3 3, E = D = 3 ε 3ε < 5.5, 5.6, 3.3 D = enclose y 4π = 4 3 π3 4π > 3 V = E = 3ε = 3 3ε < V = E E = = 3, E = ε ε D = 3 3ε 3ε ε π 3ε ε = 3ε + 6ε ε sin θθ + cos θ 3/ = D = D = εe = ε V = ε V D = ε V = ε V = ε V = ε V = /ε 6 x E x = E x E sin θ = E sin θ E D = D θ ε x ε E cos θ = ε E cos θ θ E ε E, E E sin θ = = ε cos θ E sin θ ε cos θ

18 7 tn θ tn θ = ε ε 7 P = > P = < = P,, > E > = 4πε πε + 3 < P = > ε P,, < ' ' > ε = E < = 4πε 3, = = = x x + y y = + = x + y + ' ε < E > = = x x + y y 4πε x + y + 3/ + x x + y y + 4πε x + y + 3/ E < = = x x + y y 4πε x + y + 3/ = x y ε + ε = ε P.5. = + = P.5. P.5. P.5., = ε ε ε + ε, = ε ε + ε P F = 4πε 3 = 6πε = 6πε ε ε ε + ε ε > ε F

19 8 8 ε, ε L l 3.5 D = enclose y = πl ll πl = l π 5.5, 5.6 ε ε c E = ε ε D = E = ε ε D = l πε ε l πε ε < < c c < < E,mx = E,mx E,min = E,min l πε ε = l πε ε c l πε ε c = l πε ε ε ε = c, ε ε = c c = V = c E = = l πε c l E = πε ε ε ln c + ε ln c C = l V = 4πε /ε + /ε ln/ c l πε ε = l + ln 4πε ε ε 9 E ε, ε l L D 3. ε ε E π L + ε ε E π L = l L c ε ε E = l πε ε + ε V V = l E = πε ε + ε = = l πε ε + ε ln

20 9 C = l V = πε ε + ε ln/ O E ε, ε ε = π π/6 π/6 sin θθ = π sin θθ = π [ cos θ] π/6 = 3π ε = π π π sin θθ = π sin θθ π/6 π/6 = π [ cos θ] π π/6 = + 3π 3. ε ε E + ε ε E = E = ε ε + ε = V πε { 3ε + + 3ε } V = E = πε { 3ε + + 3ε } = = πε { 3ε + + 3ε } C = V = πε { 3ε + + 3ε } / / x ε π 6 O ε Close ufce y t + σ ε = e σ/εt e σ/εt = t t t = = = e σ/εt

21 = εe = ε E, I = J = σe = σ E V = C = V = ε E + E / + E { / + } { CR = ε E E, R = V + / I = E σ + / } E σ E = ε σ E 7 7. AB P,, xy =, = x x + / y, I = Ix x, R = = x x / y +, R = R = x + / + H AB = I R = Ix x { x x / y + } 4πR 3 4π{x + / + } 3/ = I y + 4π x {x + / + } 3/ H AB = I 4π y + / / = I y + π x {x + / + } = I 3/ π x {/ + } x + / + / D I x C y + / = I y + π C. H BC H CD H DA H = H AB + H BC + H CD + H DA = I π + /4 + / x O H = I R 4πR = x x, R = x R = x O A B x {x + / + } 3/ / {/ + } / + O 7.3 = = π = H = I 4π π = I 4 O y y x

22 3 A P 7.3 A P N A N A I N A P I N B H A = N A I + 3/ B P A B H B = N B I { + } 3/ H H = H A + H B = I [ ] N A + + N B 3/ { + } 3/ = / H = H H / =/ = H = I [ ] 3/ + N A 3/ + N 5/ { + } 5/ B = = / = + / / = / 3/ 5 /4 5/ N A + 3/ 5 /4 5/ N B = N A / = N B / 4 J = J C I net though = J = π J = π J 7.8 H = H = I net though = J π C xy C < I net though = 7.9 H = I net though = π > I net though = I 7.9 H = I net though = π I π

23 6 w 7.7 x = J s = I/w x = J s = I/w x I x = x > H top = J s x = {I/w } x = I w y x < H top = J s x = {I/w } x = I w y O w I y x = x > H ottom = J s x = { I/w } x = I w y x < H ottom = J s x = { I/w } x = I w y < x < H = H top + H ottom = I w y + I w y = I w y 7.4,,,, C = C C = + + = + + = = = A = A + A + A + A C + + C = : = =, = = : = =, = + = : = =, + = : = =, = + + = + = + = + = = + A A + A A,, + + A, +, + = A,, + A, +, = A, +, A,, A = A

24 3 A + A A,, + A, +, + + = A,, + A,, + 7. A A = A = lim = A,, + A,, A = A C A = A A C = C C = + + = + + = = = A = A + A + C + A + + A C = : = =, = + = : = =, + = : = =, = : = =, = = = = + = = + = + A A + + A A,, + + A,, + = A,, + + A,, A + A A,, + 7. A A = A = lim = A,, + A,, A = A + A +,, + = A,, + A +,, = A +,, A,, A = A A = A C A

25 4 C = C C = + + = + + = = =.38c 7. A = A + A + A + A C + + C = : = =, = = + + = : = =, = = + = : = =, + = = + + = : = =, = = A A + A A,, + + A +,, + = A,, + A +,, { + } = + A +,, A,, A + A A, +, + 7. A A = A = lim A = A + A,, + = A, +, + A,, = A, +, A,, A = A C A = , θ,, θ, [ A A ] C = C C = + θ+ θ θ = + θ+ θ θ sin θθ sin θ θ

26 5 = = sin θθ A = A + A + A + A C θ θ+ θ + C θ = : = = sin θ, = : = θ = θ θ, = = θ θ+ θ θ + θ = : = = sin θ, θ sin θ = sin θ + θ θ = θ = θ + = : = θ = θ θ, = + + θ sinθ + θ = sinθ + θ θ θ = θ θ+ θ A A + θ A A, θ, + A, θ + θ, θ+ θ θ θ+ θ = A, θ, sin θ + A, θ + θ, { sinθ + θ } A + A A, θ, + 7. A A = A = lim = sinθ + θa, θ + θ, sin θa, θ, θ θ sin θ θ sin θa sin θ θ = sin θ θ sin θa + A, θ, + + = A, θ, θ θ + A, θ, + θ θ = A θ, θ, + A θ, θ, θ A θ sin θ sin θ θ = A sin θ C A = [ sin θ θ sin θa A ] θ C θ θ = C θ θ C θ θ = + + θ = + + sin θ sin θ θ = θ = sin θ A = A + A + A + A C θ + + C θ

27 6 = : = = sin θ, = : = =, = = + = + + = : = = sin θ, = + + = : = =, = + + sin θ = sin θ + sin θ = + sin θ + = A A + A A, θ, + A +, θ, + + = A, θ, sin θ + A +, θ, { + sin θ } A + A A, θ, + 7. A θ A θ = A θ = lim θ = + A +, θ, A, θ, sin θ A sin θ = A θ + A, θ, + + = A, θ, + A, θ, + = A, θ, + A, θ, A sin θ sin θ = A sin θ θ θ C θ A = [ sin θ A ] A C = C C = θ+ θ + θ = θ+ θ + θ θ θ = = θ.39c 7. A = A + A + C θ A + + A θ+ θ C θ = : = θ = θ θ, = θ θ = θ θ θ+ θ + θ = : = =, = = θ θ+ θ + = : = θ = θ θ, = + θ θ = + θ θ θ

28 7 θ + θ = : = =, = = θ + A A + A A, θ, + A +, θ, + + = A, θ, θ θ + A +, θ, { + θ θ } = + A θ +, θ, A θ, θ, θ A θ θ = A θ A + A A, θ, + A, θ + θ, θ+ θ θ θ θ+ θ = A, θ, + A, θ + θ, 7. A A = A = lim = A, θ + θ, A, θ, θ θ A θ θ = A θ C A = [ A θ A ] θ 7.3c 9 B = A = / / / = A B =, B = A, B = A B = A = µ I π A = µ I ln + C C π A = A = µ I π ln + C φ = f φ = f 4πR R = 7.3 A = A x x + A y y + A = µ J 4π R = µ J x x + J y y + J 4π R µ J x µ J y µ J = 4πR x + 4πR y + 4πR A x = µ J x µ J y 4πR, A y = 4πR, A = A x = µ J x, A y = µ J y, A = µ J µ J 4πR

29 8 B. A = A x x + A y y + A = µ J x x + µ J y y + µ J = µ J x x + J y y + J = µ J B.3 A = A A 7.33 A = H = µ B = µ A = µ { A A } = µ { µ J } = J 8 F = B m t = B m x t = y B B y, m y t = B x x B, m t = x B y y B x x B = B x B > m x t m y t m t = P.8. = B P.8. = y B P.8.3 t = B = B x < R R y x =, y =, =, x = y = = x > P.8. x =, x = y P.8. t P.8.3 y t + B y = P.8.4 m P.8.4 y = y t = cos ω c t y = ω c sin ω c t P.8.5 P.8.6 ω c ω c = B m P.8.7

30 9 P.8.5 P.8. = t = sin ω c t = ω c cos ω c t P.8.8 P.8.9 >, < P.8.6, P.8.9 y + ± R = R P.8. R R = ω c = m B P.8. P.8.6, P.8.9, P.8. ω c, R ω c, R P.8.6, P.8.9 >, < e e E B m t = ee + B B x E O y =, V x E = E = V m x t = P.8. m y t m t = e B = e V + e yb P.8.3 P.8.4 t = x = y = =, x = y = = P.8. x =, x = y P.8.4 t P.8.3 t + ω c = P.8.5 ω c ω c = eb m P.8.6

31 3 P.8.5 t = = = C sin ω c t C P.8.4 y = m eb t V B = C cos ω ct V B t = y = C = V/B = ev/ω c m y, y = y t = = t = ev ω c m cos ω ct ev ω c m sin ω ct P.8.7 P.8.8 t = y = = y = ev ω cm sin ω ct ω c t = ev ω cm cos ω ct P.8.9 P.8. mx = ev/ω cm mx = H c H c = mv µ e P.8.9, P.8. 3 P I H = I π + cos x I P = y + cos y + sin = + cos y + sin = sin y + cos O I I P, y, φ y O ' F = I µ H = I sin y + cos = µ I I π cos y + sin + cos C.7 F = µ I I π = µ I I π = µ I I π π π cos + cos y µ I I π + cos π I I > π y + µ I I π π y = µ I I µ I π + cos x sin + cos [ ln + cos ] π y

32 3 4 B I F = F + F = I y B = B y F = F = IL B = I x B y = IB O I = I = I x F = I B = I B T = F = I B = I B = I [ B B] = I B B = B x x + B y y + B cos x + sin y = B x cos + B y sin T = I B x cos + B y sin sin x + cos y = I B x cos sin B y sin x + I B x cos + B y sin cos y T = =π = T π π = I B x cos sin B y sin x + I B x cos + B y sin cos y = I π B y x + B x y = Iπ B = I B = m B n [tuns/m] B = µ ni n Λ = nφ = n B = µ n I = µ n I π L = Λ I = nφ I = µ n π B = µni π Φ Φ = B = µni π = µni π

33 3 =, = + cos, = Φ = µni π = µni π = µni π π [ π + cos + cos ] π = µni O ' ' φ ' + ' cosφ ' ] [ = µni = µni C.7 N Λ = NΦ L = Λ I = NΦ I = µn 8 l B = µ N I /l Φ = B n = µ N I n = µ N I cos θ π l l N Λ = N Φ M = Λ I = N Φ I = µ π N N cos θ l θ n 9 M = µ l 4π = µ 4π = µ 4π = µ 4π = µ 4π = µ 4π l l l l l + ln [ + + ] l {ln [ l + ] [ ]} + l ln + + ln l + l ln + { [ ] ln + + ln [ l + ] } + l { [ ln [ + ] ] l + + } = = R = + P.8. [ l ln [ l + + l ] + l ] l

34 33 = = µ π l ln + l + l /l M = µ l π ln l + + l + + l + l + l µ l π ln l P.8. Int. = ln [ l + ] + l = l ln [ l + ] + l = l ln [ l + ] + l l + l/ + l l + + l l + l = l ln [ l + + l ] + l I P B = µ I π + cos x Λ = B x = µ I π = µ I π π = µ I π + cos = µ I I O P φ O' [ ] = µ I C y C.7 M = Λ I = µ 9 Λ = NΦ = NΦ sin ωt V e V e = Λ t = NΦ ω cos ωt y B = µ H y H > t = y t n C B = µ H y ω n = cos ωt x + sin ωt y y t Φ C Φ = B n = µ H y cos ωt x + sin ωt y x ω t n

35 34 = µ H sin ωt = µ H sin ωt N Λ = NΦ = Nµ H sin ωt V e = Λ t = Nµ Hω cos ωt 3 xy O B = B OA = = ω B = B ω E = B = ω B = ωb OA O A' A l y V e = O E = ωb = ωb A l l x = ωb l 4 B E = C t P.9. C xy B E = E E P.9. E = C π E + + = π : P.9. B π t = ωb cos ωt = ωb cos ωt P.9. = ωb cos ωt π = ωb cos ωt π E π = ωb cos ωt π E = E = ωb cos ωt π E = E = E π > : P.9. > B/ t = B π t = ωb cos ωt = ωb cos ωt = ωb cos ωt π = ωb cos ωt π π π

36 35 P.9. E π = ωb cos ωt π E = E = ωb cos ωt 5 = + B = µ H = µ H W m = B H B Hπ = µ H π * 7.6 I W m = µ H π = I c I c µ π + µ π π + µ π π π c = µ I 4π µ I 4π = µ I 6π + µ I 4π ln + µ I 4π [ + µ I 4πc c c 4 c ln c 3c 4c c ] L = W m = µ I 8π + µ π ln + µ [ c 4 π c ln c ] 3c 4c 6 L L M W m = L I + L I + MI I L L F = W m = M I I y y 8 M = µ F = [ µ ] I I y = µ I I y 7 H = J + D t = J + H = D t x, y, / t D = J = t D = t * = + = π + π π

37 36 8 E = µ H t, H = ε E t, E =, H = E = E E E = E E = E = E = µ H = µ t t H = µ t E = µε E t = ±/ µε ε E = µε E t t h 9.57 E E E t w 9.57 C B t 5.5 C t n = s, = = w h s w h h B t = B t B t B t B t = w h B t + B s t 9.57 E E t w = B.4 E E t = E E n s = s { n E E } h 9.57 H H H t w J J s s w C D C t t n = s, = = w h s w h h D t = D t D t + + D t D t = w h D t + D s t 9.57 H H t w = J s s w

38 37 W e = T = T = 8T = T T T W m = P s = T = T = 4T = T T [ ] ε E E t = T [ ε T Re [ Ẽe jωt] Re [ ] Ẽe jωt] t ε Ẽe jωt + Ẽ e jωt Ẽejωt + Ẽ e jωt t [ ] εẽ Ẽ + Ẽ Ẽe jωt + Ẽ Ẽ e jωt t [ ε T ] Ẽ Ẽ + Ẽ Ẽe jωt + Ẽ Ẽ e jωt ε t = 8 T 4 Ẽ Ẽ T 4 µ 4 H H, P l = [ E J Ẽe jωt + Ẽ e jωt [ [ T T ] t = T T σ Ẽ Ẽ [ Re [ Ẽe jωt] Re [ J ] e jωt] t J e jωt + J e jωt t Ẽ J + Ẽ J + Ẽ J e jωt + Ẽ J e jωt ] Ẽ J + Ẽ J + Ẽ J e jωt + Ẽ J e jωt t = ] t Re [ Ẽ J ] Ẽ = jωµ H H = J + σẽ + jωεẽ P.. P.. p = Ẽ H / p = Ẽ H = [ H Ẽ Ẽ H ] P..3 P.. H = J + σẽ jωεẽ P..4 P..3 P.. P..4 p = [ H jωµ H Ẽ J + σẽ jωεẽ ] [ = jω 4 µ H H 4 εẽ Ẽ ] σẽ Ẽ Ẽ J [ p = jω 4 µ H H ] 4 εẽ Ẽ σẽ Ẽ Ẽ J

39 38 3 Ẽ = Ẽ Ẽ, k = k k e jk = e jk jk = jke jk Ẽ = E e jk = E e jk = E jke jk = jk E e jk Ẽ = { jk E e jk } = jk E e jk = jk E jke jk = k E ke jk Ẽ = E e jk = e jk E = jke jk E = je ke jk Ẽ = {je ke jk } = j e jk E k = j jke jk E k = k E ke jk = {k ke k E k}e jk = {k E k E k}e jk Ẽ + k Ẽ = k E ke jk {k E k E k}e jk + k E e jk = = Ẽ = jk E e jk = jk Ẽ k Ẽ = H = jωµ Ẽ = jωµ je ke jk = ωµ k E e jk = ωµ k Ẽ 4 Ẽ = E e γ x H = E Z w e γ y p = Ẽ H = E e γ E x e γ y = E e γ Z w Z w e γ = e α e jβ e γ = e α p = E Z e α 5 x y Ẽ = / x / y / = Ẽ x y = jk E e jk Γe jk y Ẽ x = jωµ H = jωµ H e jk Γe jk y k E = ωµ H P..5 x y H = / x / y / H y = H y x = jk H e jk + Γe jk x = jωε Ẽ = jωε E e jk + Γe jk x k H = ωε E P..6

40 39 P..5 P..6 k = ω µ ε P..5 / P..6 E µ = H ε [ E H = Re Ẽ H ] [ ] = Re E e jk + Γe jk x H e jk Γ e jk y [ E H = Re Γ + Γe jk Γ e ] jk = E H Γ

1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x

1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x . P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +

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

all.dvi

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

More information

29

29 9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n

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

(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x

(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x Compton Scattering Beaming exp [i k x ωt] k λ k π/λ ω πν k ω/c k x ωt ω k α c, k k x ωt η αβ k α x β diag + ++ x β ct, x O O x O O v k α k α β, γ k γ k βk, k γ k + βk k γ k k, k γ k + βk 3 k k 4 k 3 k

More information

(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

(1.2) 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

More information

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

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

More information

lim lim lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d

lim lim lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d lim 5. 0 A B 5-5- A B lim 0 A B A 5. 5- 0 5-5- 0 0 lim lim 0 0 0 lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d 0 0 5- 5-3 0 5-3 5-3b 5-3c lim lim d 0 0 5-3b 5-3c lim lim lim d 0 0 0 3 3 3 3 3 3

More information

c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l

c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l c 28. 2, y 2, θ = cos θ y = sin θ 2 3, y, 3, θ, ϕ = sin θ cos ϕ 3 y = sin θ sin ϕ 4 = cos θ 5.2 2 e, e y 2 e, e θ e = cos θ e sin θ e θ 6 e y = sin θ e + cos θ e θ 7.3 sgn sgn = = { = + > 2 < 8.4 a b 2

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. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π

1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π . 4cm 6 cm 4cm cm 8 cm λ()=a [kg/m] A 4cm A 4cm cm h h Y a G.38h a b () y = h.38h G b h X () S() = π() a,b, h,π V = ρ M = ρv G = M h S() 3 d a,b, h 4 G = 5 h a b a b = 6 ω() s v m θ() m v () θ() ω() dθ()

More information

( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =

( ; ) 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 =

More information

B line of mgnetic induction AB MN ds df (7.1) (7.3) (8.1) df = µ 0 ds, df = ds B = B ds 2π A B P P O s s Q PQ R QP AB θ 0 <θ<π

B line of mgnetic induction AB MN ds df (7.1) (7.3) (8.1) df = µ 0 ds, df = ds B = B ds 2π A B P P O s s Q PQ R QP AB θ 0 <θ<π 8 Biot-Svt Ampèe Biot-Svt 8.1 Biot-Svt 8.1.1 Ampèe B B B = µ 0 2π. (8.1) B N df B ds A M 8.1: Ampèe 107 108 8 0 B line of mgnetic induction 8.1 8.1 AB MN ds df (7.1) (7.3) (8.1) df = µ 0 ds, df = ds B

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

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

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

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

More information

9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint (

9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint ( 9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) 2. 2.1 Ĥ ψ n (r) ω n Schrödinger Ĥ ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ + Ĥint (t)] ψ (r, t), (2) Ĥ int (t) = eˆxe cos ωt ˆdE cos ωt, (3)

More information

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 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

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

1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3

1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A 2 1 2 1 2 3 α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 4 P, Q R n = {(x 1, x 2,, x n ) ; x 1, x 2,, x n R}

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

Gmech08.dvi

Gmech08.dvi 145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2

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

H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [

H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [ 3 3. 3.. H H = H + V (t), V (t) = gµ B α B e e iωt i t Ψ(t) = [H + V (t)]ψ(t) Φ(t) Ψ(t) = e iht Φ(t) H e iht Φ(t) + ie iht t Φ(t) = [H + V (t)]e iht Φ(t) Φ(t) i t Φ(t) = V H(t)Φ(t), V H (t) = e iht V (t)e

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

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

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

18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb

18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb r 1 r 2 r 1 r 2 2 Coulomb Gauss Coulomb 2.1 Coulomb 1 2 r 1 r 2 1 2 F 12 2 1 F 21 F 12 = F 21 = 1 4πε 0 1 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) Coulomb ε 0 = 107 4πc 2 =8.854 187 817 10 12 C 2 N 1 m 2 (2.2)

More information

c 2009 i

c 2009 i I 2009 c 2009 i 0 1 0.0................................... 1 0.1.............................. 3 0.2.............................. 5 1 7 1.1................................. 7 1.2..............................

More information

all.dvi

all.dvi 5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0

More information

( ) Note (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e

( ) Note (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e ( ) Note 3 19 12 13 8 8.1 (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e µ τ, e R, µ R, τ R (1a) L ( ) ) * 3) W Z 1/2 ( - )

More information

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.   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

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

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7

More information

第3章

第3章 5 5.. Maxwell Maxwell-Ampere E H D P J D roth = J+ = J+ E+ P ( ε P = σe+ εe + (5. ( NL P= ε χe+ P NL, J = σe (5. Faraday rot = µ H E (5. (5. (5. ( E ( roth rot rot = µ NL µσ E µε µ P E (5.4 = ( = grad

More information

Note.tex 2008/09/19( )

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

More information

r d 2r d l d (a) (b) (c) 1: I(x,t) I(x+ x,t) I(0,t) I(l,t) V in V(x,t) V(x+ x,t) V(0,t) l V(l,t) 2: 0 x x+ x 3: V in 3 V in x V (x, t) I(x, t

r d 2r d l d (a) (b) (c) 1: I(x,t) I(x+ x,t) I(0,t) I(l,t) V in V(x,t) V(x+ x,t) V(0,t) l V(l,t) 2: 0 x x+ x 3: V in 3 V in x V (x, t) I(x, t 1 1 2 2 2r d 2r d l d (a) (b) (c) 1: I(x,t) I(x+ x,t) I(0,t) I(l,t) V in V(x,t) V(x+ x,t) V(0,t) l V(l,t) 2: 0 x x+ x 3: V in 3 V in x V (x, t) I(x, t) V (x, t) I(x, t) V in x t 3 4 1 L R 2 C G L 0 R 0

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

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

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................

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

N cos s s cos ψ e e e e 3 3 e e 3 e 3 e

N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >

More information

66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3) d 1 NN K K 8.1 d σd σd M = σd = E 2 d (8.4) ρ 2 d = I M = EI ρ 1 ρ = M EI ρ EI

66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3) d 1 NN K K 8.1 d σd σd M = σd = E 2 d (8.4) ρ 2 d = I M = EI ρ 1 ρ = M EI ρ EI 65 8. K 8 8 7 8 K 6 7 8 K 6 M Q σ (6.4) M O ρ dθ D N d N 1 P Q B C (1 + ε)d M N N h 2 h 1 ( ) B (+) M 8.1: σ = E ρ (E, 1/ρ ) (8.1) 66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3)

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

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

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i 1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [

More information

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.

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. 1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 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

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

(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

量子力学A

量子力学A c 1 1 1.1....................................... 1 1............................................ 4 1.3.............................. 6 10.1.................................. 10......................................

More information

A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π

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

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

7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa, f a g b g f a g f a g bf a

7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa, f a g b g f a g f a g bf a 9 203 6 7 WWW http://www.math.meiji.ac.jp/~mk/lectue/tahensuu-203/ 2 8 8 7. 7 7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa,

More information

( ) e + e ( ) ( ) e + e () ( ) e e Τ ( ) e e ( ) ( ) () () ( ) ( ) ( ) ( )

( ) e + e ( ) ( ) e + e () ( ) e e Τ ( ) e e ( ) ( ) () () ( ) ( ) ( ) ( ) n n (n) (n) (n) (n) n n ( n) n n n n n en1, en ( n) nen1 + nen nen1, nen ( ) e + e ( ) ( ) e + e () ( ) e e Τ ( ) e e ( ) ( ) () () ( ) ( ) ( ) ( ) ( n) Τ n n n ( n) n + n ( n) (n) n + n n n n n n n n

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

II R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k

II R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k II 231017 1 1.1. R n k +1 v 0,, v k k v 1 v 0,, v k v 0 1.2. v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4.

More information

, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,,

, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,, 6,,3,4,, 3 4 8 6 6................................. 6.................................. , 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p,

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

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,,

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,

More information

9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =

9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 = 5 5. 5.. A II f() f() F () f() F () = f() C (F () + C) = F () = f() F () + C f() F () G() f() G () = F () 39 G() = F () + C C f() F () f() F () + C C f() f() d f() f() C f() f() F () = f() f() f() d =

More information

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

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

More information

i

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

More information

5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6 cos π 6.7 MP 4 P P N i i i i N i j F j ii N i i ii F j i i N ii li i F j i ij li i i i

5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6 cos π 6.7 MP 4 P P N i i i i N i j F j ii N i i ii F j i i N ii li i F j i ij li i i i i j ij i j ii,, i j ij ij ij (, P P P P θ N θ P P cosθ N F N P cosθ F Psinθ P P F P P θ N P cos θ cos θ cosθ F P sinθ cosθ sinθ cosθ sinθ 5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6

More information

A 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2.

A 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2. A A 1 A 5 A 6 1 2 3 4 5 6 7 1 1.1 1.1 (). Hausdorff M R m M M {U α } U α R m E α ϕ α : U α E α U α U β = ϕ α (ϕ β ϕβ (U α U β )) 1 : ϕ β (U α U β ) ϕ α (U α U β ) C M a m dim M a U α ϕ α {x i, 1 i m} {U,

More information

B 1 B.1.......................... 1 B.1.1................. 1 B.1.2................. 2 B.2........................... 5 B.2.1.......................... 5 B.2.2.................. 6 B.2.3..................

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

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

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

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,

More information

LLG-R8.Nisus.pdf

LLG-R8.Nisus.pdf d M d t = γ M H + α M d M d t M γ [ 1/ ( Oe sec) ] α γ γ = gµ B h g g µ B h / π γ g = γ = 1.76 10 [ 7 1/ ( Oe sec) ] α α = λ γ λ λ λ α γ α α H α = γ H ω ω H α α H K K H K / M 1 1 > 0 α 1 M > 0 γ α γ =

More information

( ) 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) (

( ) 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

More information

.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 =,

.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

More information

untitled

untitled 1 17 () BAC9ABC6ACB3 1 tan 6 = 3, cos 6 = AB=1 BC=2, AC= 3 2 A BC D 2 BDBD=BA 1 2 ABD BADBDA ABC6 BAD = (18 6 ) / 2 = 6 θ = 18 BAD = 12 () AD AD=BADCAD9 ABD ACD A 1 1 1 1 dsinαsinα = d 3 sin β 3 sin β

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

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

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

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)

More information

x E E E e i ω = t + ikx 0 k λ λ 2π k 2π/λ k ω/v v n v c/n k = nω c c ω/2π λ k 2πn/λ 2π/(λ/n) κ n n κ N n iκ k = Nω c iωt + inωx c iωt + i( n+ iκ ) ωx

x E E E e i ω = t + ikx 0 k λ λ 2π k 2π/λ k ω/v v n v c/n k = nω c c ω/2π λ k 2πn/λ 2π/(λ/n) κ n n κ N n iκ k = Nω c iωt + inωx c iωt + i( n+ iκ ) ωx x E E E e i ω t + ikx k λ λ π k π/λ k ω/v v n v c/n k nω c c ω/π λ k πn/λ π/(λ/n) κ n n κ N n iκ k Nω c iωt + inωx c iωt + i( n+ iκ ) ωx c κω x c iω ( t nx c) E E e E e E e e κ e ωκx/c e iω(t nx/c) I I

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

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

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

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

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy

z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy f f x, y, u, v, r, θ r > = x + iy, f = u + iv C γ D f f D f f, Rm,. = x + iy = re iθ = r cos θ + i sin θ = x iy = re iθ = r cos θ i sin θ x = + = Re, y = = Im i r = = = x + y θ = arg = arctan y x e i =

More information

Z: Q: R: C:

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

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

II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R

II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R II Karel Švadlenka 2018 5 26 * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* 5 23 1 u = au + bv v = cu + dv v u a, b, c, d R 1.3 14 14 60% 1.4 5 23 a, b R a 2 4b < 0 λ 2 + aλ + b = 0 λ =

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

4‐E ) キュリー温度を利用した消磁:熱消磁

4‐E ) キュリー温度を利用した消磁:熱消磁 ( ) () x C x = T T c T T c 4D ) ) Fe Ni Fe Fe Ni (Fe Fe Fe Fe Fe 462 Fe76 Ni36 4E ) ) (Fe) 463 4F ) ) ( ) Fe HeNe 17 Fe Fe Fe HeNe 464 Ni Ni Ni HeNe 465 466 (2) Al PtO 2 (liq) 467 4G ) Al 468 Al ( 468

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

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

85 4

85 4 85 4 86 Copright c 005 Kumanekosha 4.1 ( ) ( t ) t, t 4.1.1 t Step! (Step 1) (, 0) (Step ) ±V t (, t) I Check! P P V t π 54 t = 0 + V (, t) π θ : = θ : π ) θ = π ± sin ± cos t = 0 (, 0) = sin π V + t +V

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

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

,. 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

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

46 4 E E E E E 0 0 E E = E E E = ) E =0 2) φ = 3) ρ =0 1) 0 2) E φ E = grad φ E =0 P P φ = E ds 0

46 4 E E E E E 0 0 E E = E E E = ) E =0 2) φ = 3) ρ =0 1) 0 2) E φ E = grad φ E =0 P P φ = E ds 0 4 4.1 conductor E E E 4.1: 45 46 4 E E E E E 0 0 E E = E E E =0 4.1.1 1) E =0 2) φ = 3) ρ =0 1) 0 2) E φ E = grad φ E =0 P P φ = E ds 0 4.1 47 0 0 3) ε 0 div E = ρ E =0 ρ =0 0 0 a Q Q/4πa 2 ) r E r 0 Gauss

More information