12 2 E ds = 1 ρdv ε 1 µ D D S S D B d S = 36 E d B l = S d S B d l = S ε E + J d S 4 4 div E = 1 ε ρ div B = rot E = B 1 rot µ E B = ε + J

Size: px
Start display at page:

Download "12 2 E ds = 1 ρdv ε 1 µ D D S S D B d S = 36 E d B l = S d S B d l = S ε E + J d S 4 4 div E = 1 ε ρ div B = rot E = B 1 rot µ E B = ε + J 37 3.2 3.2."

Transcription

1 nkiyono@mail.ecc.u-tokyo.ac.jp ρp, t EP, t BP, t JP, t 35 P t xyz xyz t 4 ε µ D D S S 35 D H D = ε E B = µ H E B

2 12 2 E ds = 1 ρdv ε 1 µ D D S S D B d S = 36 E d B l = S d S B d l = S ε E + J d S 4 4 div E = 1 ε ρ div B = rot E = B 1 rot µ E B = ε + J D E d S = 1 ε D ρdv 72 E E D

3 12 3 div EdV = E ds 73 D E D D D div EdV = 1 ε E D D ρdv div EP, t 1 ε ρp, t P, t div EP, t > 1 ε ρp, t E ρ 38 P D P div EP, t > 1 ε ρp, t t D div EdV > 1 ε D ρdv 74 div E = 1 ε ρ D D B d S = div B = 38 ρ ρ

4 S E d B l = S d S 75 E S rot E ds = E d l 76 S E S S S rot E ds B = S d S 77 E S 77 rot EP, t B P, t P, t t B = F rot EP, t F P, t xyz rot E F R 1 x, y, z, t F 1 x, y, z, t R 2 x, y, z, t F 2 x, y, z, t R 3 x, y, z, t F 3 x, y, z, t

5 12 5 R 3 x, y, z, t > F 3 x, y, z, t x, y, z P rot E F 39 P D P R 3 x, y, z, t > F 3 x, y, z, t x, y, z P P xy S D S xy S z t rot E ds > S S F d S 77 rot E = F = B S B d l = ε E + J ds µ S S 1 rot µ E B = ε + J

6 12 6 ρ J div E = 78 div B = 79 rot E = B rot B = ε µ E 81 t rot B = ε 2 E µ rotrot E = ε µ 2 E 2 82 xyz E E = E 1 E 2 rotrot E E E 3 E = = y z E 3 y E 2 z E 1 z E 3 = E 2 E1 y E1 + E 2 y + E 3 z E 1 + E2 y + E3 z E1 = E E + E 2 y + E 3 z 2 E 2 y 2 E 1 y 2 E 1 2 z + 2 E 3 2 z 2 E 3 z y 2 E 2 z 2 E E 1 2 y 2 E 1 z 2 E 3 2 E 3 2 y + 2 E 2 2 y z y z E y z E y z E = y z E = 1 E = E 82 E = ε µ 2 E B E B E 4 t rot rot grad div rot rot

7 t B B 2 2 B = ε µ 2 B 2 85 E B 2 f fx, y, z, t = ε µ x, y, z, t 86 2 fx, y, z, t 1 ε µ 1 ε µ = c 3. c f 2 x, t = 1 2 f v 2 x, t 87 2 fx, t 1 φy ψy fx, t = φx vt + ψx + vt 88 1 X Y X = x vt, Y = x + vt fx, t X, Y gx, Y gx, Y gx vt, x + vt = fx, t 87 X, Y 2 g X, Y = 89 X Y 2 X Y Y X 89 gx, Y X 1 φx Y 1 ψy gx, Y = φx + ψy X = x vt Y = x + vt x t 88 42

8 x x xy v > y = φx vt t y = φx v t x y φx vt x = x vt y x, t = x, t y x vt, y vt y t x y = ψx + vt v v y = ψx t v E B x 3 F x, t 1 f a k v F x, t = f k x vt a k F k v E B E x, t = f k x vt e B x, t = g l x ut b 9 k, l, e, b, v, u f g f k x vt e = ε µ v 2 f k x vt e g l x ut b = ε µ u 2 g l x ut b k l. f k x vt e ε f k x vt k x k = f k x vt = fk 1 x 1 + k 2 x 2 + k 3 x 3 vt x 1 f k 1x 1 + k 2x 2 + k 3x 3 vtk 1 k 1 k 2 k 3, x = x 1 x 2 x 3

9 12 9 x 1 f k 1x 1 + k 2x 2 + k 3x 3 vtk 2 1 x 2 x 3 2 f k x vt = f k x vt k k k 2 3 = f k x vt k E x, t v B x, t u c v = u = 1 ε µ = c 9 u = v = c 78, 79, 8, 81 f k x ct k e = 91 g l x ct l b = 92 f k x ct k e = cg l x ct b 93 cg l x ct l b = f k x ct e , 79, 8, 81 9 u = v = c 91, 92, 93, 94 f g f x, t g x, t k e = l b = k e l b E B 93 b k e 94 e l b e b k, l e, b

10 12 1 e b k l l = k l = k 93 l f k x ct k e l = cg l x ct b l 94 1 f k x ct k e l = f k x ct e f x, t k e l = e 95 a a = k e k, e, a 95 a, l, e l, e, a l k l = k E B E B 3.4

11 F grad φ = F φ rot G = F G F F rot F = 3 2 divrot G = F = div F = F xyz F F x, y, z = F 1 x, y, z F 2 x, y, z F 3 x, y, z G 3 y G 2 z = F 1 G 1 z G 3 = F 2 G 2 G 1 y = F 3 G 1 x, y, z, G 2 x, y, z, G 3 x, y, z G 1 x, y, z = z F 2 x, y, tdt G 1 x, y 1 2 x, y, x, y, z z 1 G 1 G 2 x, y, z = G 1 z x, y, z = F 2x, y, z x F 3 t, y, z + G 1 t, y, z dt y x 1 G 2 x, y, z G 1 y x, y, z = F 3x, y, z G 3 x, y, z = y F 1 x, t, z + G 2 x, t, z dt z 43

12 12 12 y 1 div F = G 3 = = = = = y y y y y F1 G 3 y G 2 z x, y, z = F 1x, y, z F 1 + F 2 y + F 3 z = F 1 x, t, z + G 2 x, t, z z x, t, z + 2 G 2 x, t, z z F 2 dt dt y x, t, z F 3 z x, t, z + 2 G 2 z 2 G 1 y z x, t, z 2 G 2 z x, t, z + 2 G 1 dt = x, t, z dt y z x, t, z + 2 G 2 z x, t, z dt G 1 z G 3 = G 1 z = F 2 F div F = B A rot A = B A rot A 8 B rot E = rot A rot E + A = E + A grad φ = E + A

13 12 13 φ φ, A E, B 79 8 E = grad φ A B = rot A φ div A = 1 ε ρ 96 divgrad φ = φ 8 32 µ µ ε = 1 c 2 1 rotrot A µ = ε grad φ ε 2 A 2 + J rotrot A = graddiv A A 83 grad div A + 1 φ c 2 1c 2 A 2 2 = µ J div A + 1 c 2 φ = 98 ρ = J = φ A c φ = 1 c 2 2 φ 2 A = 1 c 2 2 A c A A 1 c 2 2 A 2 ac + bc = a + bc 45 46

14 ξx, t = x vt, ηx, t = x + vt fx, t = g ξx, t, ηx, t x f g x, t = X = g X ξ ξx, t, ηx, t + g η ξx, t, ηx, t Y g ξx, t, ηx, t + ξx, t, ηx, t Y x 2 f x, t = g ξx, ξ 2 t, ηx, t X X + g ξx, η t, ηx, t Y X + g ξx, ξ t, ηx, t X Y + g ξx, η t, ηx, t Y Y = 2 g 2 g 2 g ξx, t, ηx, t + 2 ξx, t, ηx, t + ξx, t, ηx, t X 2 X Y Y 2 t f g ξ x, t = ξx, t, ηx, t X = v g X ξx, t, ηx, t + v g Y + g η ξx, t, ηx, t Y ξx, t, ηx, t t 2 f x, t = v g ξx, ξ 2 t, ηx, t X X + v g ξx, η t, ηx, t Y X + v g ξx, ξ t, ηx, t X Y + v g ξx, η t, ηx, t Y Y = v 2 2 g 2 g 2 g ξx, t, ηx, t 2 ξx, t, ηx, t + ξx, t, ηx, t X 2 X Y Y g 2 g 2 g ξx, t, ηx, t + 2 ξx, t, ηx, t + ξx, t, ηx, t X 2 X Y Y 2 = 2 g 2 g 2 g ξx, t, ηx, t 2 ξx, t, ηx, t + ξx, t, ηx, t X 2 X Y Y 2 2 g ξx, t, ηx, t = X Y ξx, t ηx, t X Y X Y 1 g X X, Y = g X, Y dy = χx Y X

15 12 15 X X Y X 1 g gx, Y = X, Y dx = χxdx + ψy X ψy Y Y χx X X 1 φx gx, Y = φx + ψy 45 e, b, l x k E Ex 1, x 2, x 3, t = 78 E 1 x 1, x 2, x 3, t E 2 x 1, x 2, x 3, t E 3 x 1, x 2, x 3, t E 1 1 x 1, x 2, x 3, t + E 2 2 x 1, x 2, x 3, t + E 3 3 x 1, x 2, x 3, t = Ex 1, x 2, x 3, t = fk 1 x 1 + k 2 x 2 + k 3 x 3 ct e 1 e 2 e 3 i = 1, 2, 3 E i i x 1, x 2, x 3, t = k i f k 1 x 1 + k 2 x 2 + k 3 x 3 cte i 78 E = f k s ct e 91 f k 1 x 1 + k 2 x 2 + k 3 x 3 ctk 1 e 1 + k 2 e 2 + k 3 e 3 = f k x ct k e = E B f g k l e b 92 g l x ct l b =

16 12 16 B Bx 1, x 2, x 3, t = 8 B 1 x 1, x 2, x 3, t B 2 x 1, x 2, x 3, t B 3 x 1, x 2, x 3, t E 3 x 1, x 2, x 3, t E 2 x 1, x 2, x 3, t = B x 1, x 2, x 3, t E 1 x 1, x 2, x 3, t E 3 x 1, x 2, x 3, t = B x 1, x 2, x 3, t E 2 x 1, x 2, x 3, t E 1 x 1, x 2, x 3, t = B x 1, x 2, x 3, t Ex 1, x 2, x 3, t = fk 1 x 1 + k 2 x 2 + k 3 x 3 ct e 1 e 2 Bx 1, x 2, x 3, t = gl 1 x 1 + l 2 x 2 + l 3 x 3 ct b 1 b 2 e 3 b 3 k 2 f k 1 x 1 + k 2 x 2 + k 3 x 3 cte 3 k 3 f k 1 x 1 + k 2 x 2 + k 3 x 3 cte 2 = cg l 1 x 1 + l 2 x 2 + l 3 x 3 ctb 1 k 3 f k 1 x 1 + k 2 x 2 + k 3 x 3 cte 1 k 1 f k 1 x 1 + k 2 x 2 + k 3 x 3 cte 3 = cg l 1 x 1 + l 2 x 2 + l 3 x 3 ctb 2 k 1 f k 1 x 1 + k 2 x 2 + k 3 x 3 cte 2 k 2 f k 1 x 1 + k 2 x 2 + k 3 x 3 cte 1 = cg l 1 x 1 + l 2 x 2 + l 3 x 3 ctb 3 f k 1 x 1 + k 2 x 2 + k 3 x 3 ct k 2 e 3 k 3 e 2 k 3 e 1 k 1 e 3 = cg l 1 x 1 + l 2 x 2 + l 3 x 3 ct b 1 b 2 k 1 e 2 k 2 e 1 b 3 f k x ct k e = cg l x ct b E B ε µ = 1/c 2 f g e b k l c 2 c c cg l x ct l b = f k x ct e 94

17 t div A = 1 2 φ c ρ = φ = 1 c 2 2 φ J = A = 1 c 2 2 A rot rot A = 1 c 2 2 rot A = B grad 1 t B = 1 c 2 2 B 2 2 rot A grad φ = 1 c 2 grad 2 φ 2 A = 1 3 A c 2 3 grad φ + A = 1 2 c 2 2 grad φ + A E = grad φ A 1 E = 1 c 2 2 E 2 84

7-12.dvi

7-12.dvi 26 12 1 23. xyz ϕ f(x, y, z) Φ F (x, y, z) = F (x, y, z) G(x, y, z) rot(grad ϕ) rot(grad f) H(x, y, z) div(rot Φ) div(rot F ) (x, y, z) rot(grad f) = rot f x f y f z = (f z ) y (f y ) z (f x ) z (f z )

More information

http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n

http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 1 1 1.1 ɛ-n 1 ɛ-n lim n a n = α n a n α 2 lim a n = 1 n a k n n k=1 1.1.7 ɛ-n 1.1.1 a n α a n n α lim n a n = α ɛ N(ɛ) n > N(ɛ) a n α < ɛ

More information

168 13 Maxwell ( H ds = C S rot H = j + D j + D ) ds (13.5) (13.6) Maxwell Ampère-Maxwell (3) Gauss S B 0 B ds = 0 (13.7) S div B = 0 (13.8) (4) Farad

168 13 Maxwell ( H ds = C S rot H = j + D j + D ) ds (13.5) (13.6) Maxwell Ampère-Maxwell (3) Gauss S B 0 B ds = 0 (13.7) S div B = 0 (13.8) (4) Farad 13 Maxwell Maxwell Ampère Maxwell 13.1 Maxwell Maxwell E D H B ε 0 µ 0 (1) Gauss D = ε 0 E (13.1) B = µ 0 H. (13.2) S D = εe S S D ds = ρ(r)dr (13.3) S V div D = ρ (13.4) ρ S V Coulomb (2) Ampère C H =

More information

F S S S S S S S 32 S S S 32: S S rot F ds = F d l (63) S S S 0 F rot F ds = 0 S (63) S rot F S S S S S rot F F (63)

F S S S S S S S 32 S S S 32: S S rot F ds = F d l (63) S S S 0 F rot F ds = 0 S (63) S rot F S S S S S rot F F (63) 211 12 1 19 2.9 F 32 32: rot F d = F d l (63) F rot F d = 2.9.1 (63) rot F rot F F (63) 12 2 F F F (63) 33 33: (63) rot 2.9.2 (63) I = [, 1] [, 1] 12 3 34: = 1 2 1 2 1 1 = C 1 + C C 2 2 2 = C 2 + ( C )

More information

. p.1/14

. p.1/14 . p.1/14 F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y). p.2/14 F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y) (x,y) h. p.2/14 F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y) (x,y) h h { F 2 (x+ h,y) F 2 2(x h,y) F 2 1(x,y+ h)+f 2 1(x,y

More information

II 2 II

II 2 II II 2 II 2005 yugami@cc.utsunomiya-u.ac.jp 2005 4 1 1 2 5 2.1.................................... 5 2.2................................. 6 2.3............................. 6 2.4.................................

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

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

0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,,

0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,, 2012 10 13 1,,,.,,.,.,,. 2?.,,. 1,, 1. (θ, φ), θ, φ (0, π),, (0, 2π). 1 0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ).

More information

i 0 1 0.1 I................................................ 1 0.2.................................................. 2 0.2.1...........................

i 0 1 0.1 I................................................ 1 0.2.................................................. 2 0.2.1........................... 2008 II 21 1 31 i 0 1 0.1 I................................................ 1 0.2.................................................. 2 0.2.1............................................. 2 0.2.2.............................................

More information

(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)

(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) 2017 12 9 4 1 30 4 10 3 1 30 3 30 2 1 30 2 50 1 1 30 2 10 (1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) (1) i 23 c 23 0 1 2 3 4 5 6 7 8 9 a b d e f g h i (2) 23 23 (3) 23 ( 23 ) 23 x 1 x 2 23 x

More information

( 12 ( ( ( ( Levi-Civita grad div rot ( ( = 4 : 6 3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n

More information

untitled

untitled 20010916 22;1017;23;20020108;15;20; 1 N = {1, 2, } Z + = {0, 1, 2, } Z = {0, ±1, ±2, } Q = { p p Z, q N} R = { lim a q n n a n Q, n N; sup a n < } R + = {x R x 0} n = {a + b 1 a, b R} u, v 1 R 2 2 R 3

More information

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { ( 5 5.1 [ ] ) d f(t) + a d f(t) + bf(t) : f(t) 1 dt dt ) u(x, t) c u(x, t) : u(x, t) t x : ( ) ) 1 : y + ay, : y + ay + by : ( ) 1 ) : y + ay, : yy + ay 3 ( ): ( ) ) : y + ay, : y + ay b [],,, [ ] au xx

More information

24.15章.微分方程式

24.15章.微分方程式 m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt

More information

body.dvi

body.dvi ..1 f(x) n = 1 b n = 1 f f(x) cos nx dx, n =, 1,,... f(x) sin nx dx, n =1,, 3,... f(x) = + ( n cos nx + b n sin nx) n=1 1 1 5 1.1........................... 5 1.......................... 14 1.3...........................

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

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9 1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),

More information

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx

More information

Untitled

Untitled II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j

More information

. p.1/11

. p.1/11 . p.1/11 [ ] F(x,y,z) = (F 1 (x,y,z),f 2 (x,y,z),f 3 (x,y,z)) = F 1 i+f 2 j+f 3 k div F = F 1 x + F 2 y + F 3 z F (divergence). p.1/11 [ ] F(x,y,z) = (F 1 (x,y,z),f 2 (x,y,z),f 3 (x,y,z)) = F 1 i+f 2 j+f

More information

III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2 lim. (x,y) (1,0) x 2 + y 2 lim (x,y) (0,0) lim (x,y) (0,0) lim (x,y) (0,0) 5x 2 y x 2 + y 2. xy x2 + y

III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2 lim. (x,y) (1,0) x 2 + y 2 lim (x,y) (0,0) lim (x,y) (0,0) lim (x,y) (0,0) 5x 2 y x 2 + y 2. xy x2 + y III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2. (x,y) (1,0) x 2 + y 2 5x 2 y x 2 + y 2. xy x2 + y 2. 2x + y 3 x 2 + y 2 + 5. sin(x 2 + y 2 ). x 2 + y 2 sin(x 2 y + xy 2 ). xy (i) (ii) (iii) 2xy x 2 +

More information

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11

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

3 0 4 3 5 6 6 7 7 8 4 9 6 0 30 33 34 3 36 4 4 5 44 6 47 7 54 8 56 9 60 0 6 64 67 3 70 4 7 5 75 6 80

3 0 4 3 5 6 6 7 7 8 4 9 6 0 30 33 34 3 36 4 4 5 44 6 47 7 54 8 56 9 60 0 6 64 67 3 70 4 7 5 75 6 80 3 0 4 3 5 6 6 7 7 8 4 9 6 0 30 33 34 3 36 4 4 5 44 6 47 7 54 8 56 9 60 0 6 64 67 3 70 4 7 5 75 6 80 7 8 3 elemet, set A, A A, A A, A A, b, c, {, b, c, }, x P x, P x x {x P x}, A x, P x {x A P x} 3 { {,,

More information

1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2

1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2 θ i ) AB θ ) A = B = sin θ = sin θ A B sin θ) ) < = θ < = Ax Bx = θ = sin θ ) abc θ sin 5θ = sin θ fsin θ) fx) = ax bx c ) cos 5 i sin 5 ) 5 ) αβ α iβ) 5 α 4 β α β β 5 ) a = b = c = ) fx) = 0 x x = x =

More information

(, ) (, ) S = 2 = [, ] ( ) 2 ( ) 2 2 ( ) 3 2 ( ) 4 2 ( ) k 2,,, k =, 2, 3, 4 S 4 S 4 = ( ) 2 + ( ) ( ) (

(, ) (, ) S = 2 = [, ] ( ) 2 ( ) 2 2 ( ) 3 2 ( ) 4 2 ( ) k 2,,, k =, 2, 3, 4 S 4 S 4 = ( ) 2 + ( ) ( ) ( B 4 4 4 52 4/ 9/ 3/3 6 9.. y = x 2 x x = (, ) (, ) S = 2 = 2 4 4 [, ] 4 4 4 ( ) 2 ( ) 2 2 ( ) 3 2 ( ) 4 2 ( ) k 2,,, 4 4 4 4 4 k =, 2, 3, 4 S 4 S 4 = ( ) 2 + ( ) 2 2 + ( ) 3 2 + ( 4 4 4 4 4 4 4 4 4 ( (

More information

橡点検記録(集約).PDF

橡点検記録(集約).PDF 942.8.8.8.7 671 86 11 1 9 9 9 1 1,792 7,23 2,483 1,324 2,198 7,23 82 7,23 6,327 9,22 9,713 8,525 8,554 9,22. 8,554. 1,79 9,713 95 947 8,525.. 944 671 81 7 17 1,29 1,225 1,241 1,25 1,375 9.3 23,264 25,

More information

A

A A05-132 2010 2 11 1 1 3 1.1.......................................... 3 1.2..................................... 3 1.3..................................... 3 2 4 2.1............................... 4 2.2

More information

ii 4 5 RLC 2 LC LC OTA, FDNR 6

ii 4 5 RLC 2 LC LC OTA, FDNR 6 There is nothing more practical than a good theory. James Clerk Maxwell ( 1831 1879) 1.1 1.1 2 3 MOSFET 3 MOSFET MOS CMOS CMOS CMOS ii 4 5 RLC 2 LC LC OTA, FDNR 6 iii 1 (90 30 ) 6 5 1 1 2013 3 1. 1.1...

More information

II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1

II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1 II 2014 1 1 I 1.1 72 r 2 72 8 72/8 = 9 9 2 a 0 1 a 1 a 1 = a 0 (1+r/100) 2 a 2 a 2 = a 1 (1 + r/100) = a 0 (1 + r/100) 2 n a n = a 0 (1 + r/100) n a n a 0 2 n a 0 (1 + r/100) n = 2a 0 (1 + r/100) n = 2

More information

曲面のパラメタ表示と接線ベクトル

曲面のパラメタ表示と接線ベクトル L11(2011-07-06 Wed) :Time-stamp: 2011-07-06 Wed 13:08 JST hig 1,,. 2. http://hig3.net () (L11) 2011-07-06 Wed 1 / 18 ( ) 1 V = (xy2 ) x + (2y) y = y 2 + 2. 2 V = 4y., D V ds = 2 2 ( ) 4 x 2 4y dy dx =

More information

untitled

untitled yoshi@image.med.osaka-u.ac.jp http://www.image.med.osaka-u.ac.jp/member/yoshi/ II Excel, Mathematica Mathematica Osaka Electro-Communication University (2007 Apr) 09849-31503-64015-30704-18799-390 http://www.image.med.osaka-u.ac.jp/member/yoshi/

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

II 2 ( )

II 2 ( ) II 2 ( 26 1 1 1 3 1.1....................................... 3 1.1.1.............................. 3 1.1.2.............................. 4 1.1.3..................... 5 1.2 : R 3...............................

More information

2 p T, Q

2 p T, Q 270 C, 6000 C, 2 p T, Q p: : p = N/ m 2 N/ m 2 Pa : pdv p S F Q 1 g 1 1 g 1 14.5 C 15.5 1 1 cal = 4.1855 J du = Q pdv U ( ) Q pdv 2 : z = f(x, y). z = f(x, y) (x 0, y 0 ) y y = y 0 z = f(x, y 0 ) x x =

More information

2010 4 3 0 5 0.1......................................... 5 0.2...................................... 6 1 9 2 15 3 23 4 29 4.1............................................. 29 4.2..............................

More information

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................ 5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)

More information

II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re

II ( ) (7/31) II (  [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier

More information

sec13.dvi

sec13.dvi 13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:

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

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

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

入試の軌跡

入試の軌跡 4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf

More information

Micro-D 小型高密度角型コネクタ

Micro-D 小型高密度角型コネクタ Micro- 1 2 0.64 1.27 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 1.09 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 3 4 J J

More information

EP7000取扱説明書

EP7000取扱説明書 EP7000 S0109-3012 3 47 811 1213 1419 2021 53 54 5560 61 6263 66 2223 2427 2830 3133 3436 3740 4142 4344 45 46 4750 5152 2 4 5 6 7 1 3 4 5 6 7 8 9 15 16 17 18 13 EP7000 2 10 11 12 13 14 19 20 21 22 23 24

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

Chap11.dvi

Chap11.dvi . () x 3 + dx () (x )(x ) dx + sin x sin x( + cos x) dx () x 3 3 x + + 3 x + 3 x x + x 3 + dx 3 x + dx 6 x x x + dx + 3 log x + 6 log x x + + 3 rctn ( ) dx x + 3 4 ( x 3 ) + C x () t x t tn x dx x. t x

More information

b3e2003.dvi

b3e2003.dvi 15 II 5 5.1 (1) p, q p = (x + 2y, xy, 1), q = (x 2 + 3y 2, xyz, ) (i) p rotq (ii) p gradq D (2) a, b rot(a b) div [11, p.75] (3) (i) f f grad f = 1 2 grad( f 2) (ii) f f gradf 1 2 grad ( f 2) rotf 5.2

More information

i

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

More information

ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4

ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4 20 20.0 ( ) 8 y = ax 2 + bx + c 443 ax 2 + bx + c = 0 20.1 20.1.1 n 8 (n ) a n x n + a n 1 x n 1 + + a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 444 ( a, b, c, d

More information

Microsoft Word - 触ってみよう、Maximaに2.doc

Microsoft Word - 触ってみよう、Maximaに2.doc i i e! ( x +1) 2 3 ( 2x + 3)! ( x + 1) 3 ( a + b) 5 2 2 2 2! 3! 5! 7 2 x! 3x! 1 = 0 ",! " >!!! # 2x + 4y = 30 "! x + y = 12 sin x lim x!0 x x n! # $ & 1 lim 1 + ('% " n 1 1 lim lim x!+0 x x"!0 x log x

More information

FX ) 2

FX ) 2 (FX) 1 1 2009 12 12 13 2009 1 FX ) 2 1 (FX) 2 1 2 1 2 3 2010 8 FX 1998 1 FX FX 4 1 1 (FX) () () 1998 4 1 100 120 1 100 120 120 100 20 FX 100 100 100 1 100 100 100 1 100 1 100 100 1 100 101 101 100 100

More information

difgeo1.dvi

difgeo1.dvi 1 http://matlab0.hwe.oita-u.ac.jp/ matsuo/difgeo.pdf ver.1 8//001 1 1.1 a A. O 1 e 1 ; e ; e e 1 ; e ; e x 1 ;x ;x e 1 ; e ; e X x x x 1 ;x ;x X (x 1 ;x ;x ) 1 1 x x X e e 1 O e x x 1 x x = x 1 e 1 + x

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

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

untitled

untitled 20 7 1 22 7 1 1 2 3 7 8 9 10 11 13 14 15 17 18 19 21 22 - 1 - - 2 - - 3 - - 4 - 50 200 50 200-5 - 50 200 50 200 50 200 - 6 - - 7 - () - 8 - (XY) - 9 - 112-10 - - 11 - - 12 - - 13 - - 14 - - 15 - - 16 -

More information

untitled

untitled 19 1 19 19 3 8 1 19 1 61 2 479 1965 64 1237 148 1272 58 183 X 1 X 2 12 2 15 A B 5 18 B 29 X 1 12 10 31 A 1 58 Y B 14 1 25 3 31 1 5 5 15 Y B 1 232 Y B 1 4235 14 11 8 5350 2409 X 1 15 10 10 B Y Y 2 X 1 X

More information

n ( (

n ( ( 1 2 27 6 1 1 m-mat@mathscihiroshima-uacjp 2 http://wwwmathscihiroshima-uacjp/~m-mat/teach/teachhtml 2 1 3 11 3 111 3 112 4 113 n 4 114 5 115 5 12 7 121 7 122 9 123 11 124 11 125 12 126 2 2 13 127 15 128

More information

Fubini

Fubini 3............................... 3................................ 5.3 Fubini........................... 7.4.............................5..........................6.............................. 3.7..............................

More information

2010 II / y = e x y = log x = log e x 2. ( e x ) = e x 3. ( ) log x = 1 x 1.2 Warming Up 1 u = log a M a u = M a 0

2010 II / y = e x y = log x = log e x 2. ( e x ) = e x 3. ( ) log x = 1 x 1.2 Warming Up 1 u = log a M a u = M a 0 2010 II 6 10.11.15/ 10.11.11 1 1 5.6 1.1 1. y = e x y = log x = log e x 2. e x ) = e x 3. ) log x = 1 x 1.2 Warming Up 1 u = log a M a u = M a 0 log a 1 a 1 log a a a r+s log a M + log a N 1 0 a 1 a r

More information

0.6 A = ( 0 ),. () A. () x n+ = x n+ + x n (n ) {x n }, x, x., (x, x ) = (0, ) e, (x, x ) = (, 0) e, {x n }, T, e, e T A. (3) A n {x n }, (x, x ) = (,

0.6 A = ( 0 ),. () A. () x n+ = x n+ + x n (n ) {x n }, x, x., (x, x ) = (0, ) e, (x, x ) = (, 0) e, {x n }, T, e, e T A. (3) A n {x n }, (x, x ) = (, [ ], IC 0. A, B, C (, 0, 0), (0,, 0), (,, ) () CA CB ACBD D () ACB θ cos θ (3) ABC (4) ABC ( 9) ( s090304) 0. 3, O(0, 0, 0), A(,, 3), B( 3,, ),. () AOB () AOB ( 8) ( s8066) 0.3 O xyz, P x Q, OP = P Q =

More information

Netcommunity SYSTEM X7000 IPコードレス電話機 取扱説明書

Netcommunity SYSTEM X7000 IPコードレス電話機 取扱説明書 4 5 6 7 8 9 . 4 DS 0 4 5 4 4 4 5 5 6 7 8 9 0 4 5 6 7 8 9 4 5 6 4 0 4 4 4 4 5 6 7 8 9 40 4 4 4 4 44 45 4 6 7 5 46 47 4 5 6 48 49 50 5 4 5 4 5 6 5 5 6 4 54 4 5 6 7 55 5 6 4 56 4 5 6 57 4 5 6 7 58 4

More information

.A. D.S

.A. D.S 1999-1- .A. D.S 1996 2001 1999-2- -3- 1 p.16 17 18 19 2-4- 1-5- 1~2 1~2 2 5 1 34 2 10 3 2.6 2.85 3.05 2.9 2.9 3.16 4 7 9 9 17 9 25 10 3 10 8 10 17 10 18 10 22 11 29-6- 1 p.1-7- p.5-8- p.9 10 12 13-9- 2

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

7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6

7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6 26 11 5 1 ( 2 2 2 3 5 3.1...................................... 5 3.2....................................... 5 3.3....................................... 6 3.4....................................... 7

More information

熊本県数学問題正解

熊本県数学問題正解 00 y O x Typed by L A TEX ε ( ) (00 ) 5 4 4 ( ) http://www.ocn.ne.jp/ oboetene/plan/. ( ) (009 ) ( ).. http://www.ocn.ne.jp/ oboetene/plan/eng.html 8 i i..................................... ( )0... (

More information

Microsoft Word - 計算力学2007有限要素法.doc

Microsoft Word - 計算力学2007有限要素法.doc 95 2 x y yz = zx = yz = zx = { } T = { x y z xy } () {} T { } T = { x y z xy } = u u x y u z u x x y z y + u y (2) x u x u y x y x y z xy E( ) = ( + )( 2) 2 2( ) x y z xy (3) E x y z z = z = (3) z x y

More information

f (x) f (x) f (x) f (x) f (x) 2 f (x) f (x) f (x) f (x) 2 n f (x) n f (n) (x) dn f f (x) dx n dn dx n D n f (x) n C n C f (x) x = a 1 f (x) x = a x >

f (x) f (x) f (x) f (x) f (x) 2 f (x) f (x) f (x) f (x) 2 n f (x) n f (n) (x) dn f f (x) dx n dn dx n D n f (x) n C n C f (x) x = a 1 f (x) x = a x > 5.1 1. x = a f (x) a x h f (a + h) f (a) h (5.1) h 0 f (x) x = a f +(a) f (a + h) f (a) = lim h +0 h (5.2) x h h 0 f (a) f (a + h) f (a) f (a h) f (a) = lim = lim h 0 h h 0 h (5.3) f (x) x = a f (a) =

More information

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

6.1 (P (P (P (P (P (P (, P (, P.101 (008 0 3 7 ( ( ( 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

mugensho.dvi

mugensho.dvi 1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim

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

1 1 2 1 2.1................................. 1 2.2............................... 2 2.3 3............................ 3 2.4...........................

1 1 2 1 2.1................................. 1 2.2............................... 2 2.3 3............................ 3 2.4........................... 11 2 5 1 1 2 1 2.1................................. 1 2.2............................... 2 2.3 3............................ 3 2.4................................. 3 2.5...............................

More information

Quiz x y i, j, k 3 A A i A j A k x y z A x A y A z x y z A A A A A A x y z P (x, y,z) r x i y j zk P r r r r r r x y z P ( x 1, y 1, z 1 )

Quiz x y i, j, k 3 A A i A j A k x y z A x A y A z x y z A A A A A A x y z P (x, y,z) r x i y j zk P r r r r r r x y z P ( x 1, y 1, z 1 ) Quiz x y i, j, k 3 A A i A j A k x y z A x A y A z x y z A A A A A A x y z P (x, y,z) x i y j zk P x y z P ( x 1, y 1, z 1 ) Q ( x, y, z ) 1 OP x1i y1 j z1k OQ x i y j z k 1 P Q PQ 1 PQ x x y y z z 1 1

More information

grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )

grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( ) 2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))

More information

(u(x)v(x)) = u (x)v(x) + u(x)v (x) ( ) u(x) = u (x)v(x) u(x)v (x) v(x) v(x) 2 y = g(t), t = f(x) y = g(f(x)) dy dx dy dx = dy dt dt dx., y, f, g y = f (g(x))g (x). ( (f(g(x)). ). [ ] y = e ax+b (a, b )

More information

untitled

untitled F(r)=QE(r) Q ρ( r ') 3 E= ke 3 ( r r ') d r ' V r r' () Er () = Fr Q E E x y Ez ρ = x y z ε E E z y Ex E E z y Ex =, =, = y z z x x y A A x y Az diva = x y z A A z y A A x A z y Ax rot A =,, y z z x x

More information

.1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0,

.1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0, .1.1 Y K L Y = K 1 3 L 3 L K K (K + ) 1 1 3 L 3 K 3 L 3 K 0 (K + K) 1 3 L 3 K 1 3 L 3 lim K 0 K = L (K + K) 1 3 K 1 3 3 lim K 0 K = 1 3 K 3 L 3 z = f(x, y) x y z x-y-z.1 z = e x +xy y 3 x-y ( ) z 0 f(x,

More information

1 B64653 1 1 3.1....................................... 3.......................... 3..1.............................. 4................................ 4..3.............................. 5..4..............................

More information

A 2008 10 (2010 4 ) 1 1 1.1................................. 1 1.2..................................... 1 1.3............................ 3 1.3.1............................. 3 1.3.2..................................

More information

( : December 27, 2015) CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x) f (x) y = f(x) x ϕ(r) (gradient) ϕ(r) (gradϕ(r) ) ( ) ϕ(r)

( : December 27, 2015) CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x) f (x) y = f(x) x ϕ(r) (gradient) ϕ(r) (gradϕ(r) ) ( ) ϕ(r) ( : December 27, 215 CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x f (x y f(x x ϕ(r (gradient ϕ(r (gradϕ(r ( ϕ(r r ϕ r xi + yj + zk ϕ(r ϕ(r x i + ϕ(r y j + ϕ(r z k (1.1 ϕ(r ϕ(r i

More information

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

More information

( ) ( ) ( ) i (i = 1, 2,, n) x( ) log(a i x + 1) a i > 0 t i (> 0) T i x i z n z = log(a i x i + 1) i=1 i t i ( ) x i t i (i = 1, 2, n) T n x i T i=1 z = n log(a i x i + 1) i=1 x i t i (i = 1, 2,, n) n

More information