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
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- れんか ぜんじゅう
- 5 years ago
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1 4 4.1 conductor E E E 4.1: 45
2 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
3 ) ε 0 div E = ρ E =0 ρ =0 0 0 a Q Q/4πa 2 ) r E r 0 Gauss E r r) = Q 4πε 0 r 2 a<r) 0 0 r<a) 4.2 E 0 0 a r φ 0 0 a r 4.2: φp) = P S E r r)dr φs)
4 48 4 S φ ) =0 P S φr) = r Q 4πε 0 r 2 dr = Q 4πε 0 r a<r) φa) =Q/4πε 0 a) 0 φr) = Q 4πε 0 a 0 r a ) electrostatic shielding S Gauss 0 0 A B S 4.3:
5 A B 0 [ ] [ B ] A E ds =0= E ds E ds A B Q Q >0 4.4 Gauss S 1 S 1 Q S 2 Q S 2 S 1 S 1 4.4: Q S Q Q 0 Q Q 4.4 Q Q 0 Q
6 Laplace Poisson Laplace oulomb Gauss div E = ρ ε 0 4.1) E = grad φ. 4.2) φ div grad φ )= ρ ε 0. div grad φ )= φ )= 2 φ 2 Laplacian φ 2 φ = ρ ε 0 4.3) Poisson Poisson euation ρ =0 2 φ = 0 4.4) Laplace Laplace euation div grad φ ) = x i y j z k ) = 2 φ x 2 2 φ y 2 2 φ z 2 φ x i φ y j φ z k ) Laplacian 2 = 2 x 2 2 y 2 2 z )
7 φr) = 4πε 0 r. r 0 Laplace r = x 2 y 2 z 2 Laplacian x 2 φ x 2 = ) ) φ x = x x 4πε 0 x x 2 y 2 z 2 ) 3/2 = 2x 2 y 2 z 2 4πε 0 x 2 y 2 z 2 ) 5/2 2 φ y 2 = 2y 2 z 2 x 2 4πε 0 x 2 y 2 z 2 ) 5/2, 2 φ z 2 = 2z 2 x 2 y 2 4πε 0 x 2 y 2 z 2 ) 5/2 x = y = z = a Q r Q r>a) 4πε 0 r φr) = [ Q 3 4πε 0 a 2 1 ) ] r 2 r a ) 2 a 3.2 Laplace Poisson Laplace r 2 = x 2 y 2 z 2 Laplacian 2 r 2 x 2 = 2 r 2 y 2 = 2 r 2 z 2 =2 2 φ = 1 Q ε 0 4πa 3 /3 a Poisson
8 Laplace 4.4) Poisson 4.3) 1) φ i 2) Q i Laplace 1) φ i Laplace φ i 0 2) Q i Gauss n i i n i p ki k =1, 2, n n φ k = p ki i k =1, 2, n ) j j j φ k = p kj j k =1, 2, n ) n i i =1, 2, n
9 n 2 p kj k,j=1, 2, n φ k = p k1 1 p k2 2 p kn n k =1, 2, n ) n φ 1 p 11 p p 1n φ 2 p 21 p p 2n = ) φ n p n1 p n2... p nn n p kj j k p kj p kj = p jk 1 2nn 1) n φ k k =1, 2, n n j j =1, 2, n 4.6) 1 c 11 c c 1n φ 1 2 c 21 c c 2n φ 2 =.. 4.7) n c n1 c n2... c nn φ n c jk p kj c jk = jk 4.6) jk j k c jk k k 1V j p kj c jk c jj c jk j k n c jk = c kj 1 2 nn 1) 1 2 nn 1)
10 Laplace Laplace method of images 4.5 a >0 a, 0, 0) a, 0, 0) x, y, z ) φx, y, z) φx, y, z) = 4πε 0 φ x = 4πε 0 φ = y 4πε 0 φ = z 4πε 0 ) 1 [x a) 2 y 2 z 2 ] 1/2 1 [x a) 2 y 2 z 2 ] 1/2. 4.8) x a [x a) 2 y 2 z 2 ] 3/2 y [x a) 2 y 2 z 2 ] 3/2 z [x a) 2 y 2 z 2 ] 3/2 ) x a [x a) 2 y 2 z 2 ] 3/2 ) y [x a) 2 y 2 z 2 ] 3/2 ) z [x a) 2 y 2 z 2 ] 3/2
11 : 2 φ x 2 = 2 φ y 2 = 2 φ z 2 = 2x a) 2 y 2 z 2 4πε 0 [x a) 2 y 2 z 2 ] 5/2 2x ) a)2 y 2 z 2 [x a) 2 y 2 z 2 ] 5/2 x a) 2 2y 2 z 2 4πε 0 [x a) 2 y 2 z 2 ] 5/2 x ) a)2 2y 2 z 2 [x a) 2 y 2 z 2 ] 5/2 x a) 2 y 2 2z 2 4πε 0 [x a) 2 y 2 z 2 ] 5/2 x ) a)2 y 2 2z 2 [x a) 2 y 2 z 2 ] 5/2 4.8) Laplace 2 φ x 2 2 φ y 2 2 φ z 2 =0. x =0 φ0,y,z)=0 4.8)
12 56 4 Laplace E = grad φ ) x a E x x, y, z) = 4πε 0 [x a) 2 y 2 z 2 ] 3/2 x a [x a) 2 y 2 z 2 ] 3/2 ) y E y x, y, z) = 4πε 0 [x a) 2 y 2 z 2 ] 3/2 y [x a) 2 y 2 z 2 ] 3/2 4.9) ) z E z x, y, z) = 4πε 0 [x a) 2 y 2 z 2 ] 3/2 z [x a) 2 y 2 z 2 ] 3/2 x =0 E x 0,y,z)= 2a 4πε 0 [a 2 y 2 z 2 ] 3/2, E y 0,y,z)=E z 0,y,z)=0 E y = E z =0 E x < a R R >a > O A P D a O B A 4.6: B P 0 φ = 1 ) 4πε 0 PA PA =0 PB PB = AB : D O B=R RR = a 2 R = a2 R
13 OA φ = 1 ) 4πε 0 R a a R =0 = a R R a = a R P OP r OA OP θ AP = r 2 R 2 2Rr cos θ, BP = r 2 R 2 2R r cos θ P ) 1 φr, θ, ϕ) = 4πε 0 r 2 R 2 2Rr cos θ r 2 R 2 2R r cos θ = 1 4πε 0 r 2 R 2 2Rr cos θ ) a R 2 r 2 a 4 2a 2 Rr cos θ 4.10) P r = a φ =0 E = grad φ grad φ = φ r e r 1 φ r θ e θ 1 φ r sin θ ϕ e ϕ 4.10) E r r, θ, ϕ) = [ r R cos θ 4πε 0 [r 2 R 2 2Rr cos θ] 3/2 ar 2 r a 3 ] R cos θ [R 2 r 2 a 4 2a 2 Rr cos θ] 3/2 [ R sin θ 1 E θ r, θ, ϕ) = 4πε 0 [r 2 R 2 2Rr cos θ] 3/2 a 3 ] [R 2 r 2 a 4 2a 2 Rr cos θ] 3/2 E ϕ r, θ, ϕ) =0 r = a E r a, θ, ϕ) = 4πε 0 a R 2 a 2 [R 2 a 2 2aR cos θ] 3/2, E θ a, θ, ϕ) =E ϕ a, θ, ϕ) =0 E θ = E ϕ =0 E r < 0 4.7
14 58 4 O B 4.7: 4.4 a 0 R R >a 0 O R A OA O R R = a 2 /R = a/r = ) /4πε 0 r) E θ E ϕ E r /4πε 0 r 2 ) φ = /4πε 0 a)
15 φ φ = ) capacitance F= V 1 a φ = /4πε 0 a) φ = 1 4πε 0 a =4πε 0 a φ φ = f) 1 φ 1 = f 1 ) 2 φ 2 = f 2 ) 1 2 φ 1 φ 2 = f 1 2 ) f 1 )f 2 )=f 1 2 ) f) k f) =k k =1/ F µf =10 6 F pf = F 4.5 R = km =4πε 0 R = F = 708 µf
16 condenser A >0 B φ A φ B φ A φ B = ) spherical condenser 4.8 a b b >a Q Q >0 r a<r<b r b Q a Q 4.8: Gauss ε 0 Er) 4πr 2 = Q Er) = Q 4πε 0 r 2 φ = b a Er)dr = Q 1 4πε 0 a 1 ) b = Q b a 4πε 0 ab = Q φ =4πε ab 0 b a 4.13)
17 b 4.13) b ab =4πε 0 b a 4πε 0 a a = b b a b a b 4.6 parallel plate condenser 4.9 S d Q d Q 4.9: Q Q σ = ±Q/S Gauss 0 A ε 0 EA= σa E = σ ε 0 = Q ε 0 S φ φ = B A E ds = Ed.
18 62 4 = Q φ = ε 0 S d. 4.14) S d 4.7 cylindrical condenser 4.10 a b a b 4.10: L QL r a<r<b r Gauss ε 0 E 2πrL = QL E = Q 2πε 0 r φ = b a E dr = Q 2πε 0 log b a = Q φ =2πε 1 0 logb/a) 4.15)
19 = ) 1 2 = ) A 1 A 3 A A 1 A : A 1 A 2 A 3 A 1 A 3 A 2 φ 1 φ 2 A 1 A 3 φ φ = φ 1 φ 2 = 1 2 φ = ) A 1 A 2
20 64 4 φ 1 φ 2 φ = φ 1 = φ 2 A 1 A 2 φ 1 = φ = φ = 1 2 φ = 1 2 = 1 φ 2 φ = 1 φ 1 2 φ 2 = 1 2 n n 1 = n k=1 1 k, = n k. k= P 1 P 2 P 1 A 1 P 1 A 1 P 2 A 2 P 2 A : A 1 A 2 1 = /3 A 1 A 2 =4/ /11
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
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](/thumbs/93/114079295.jpg "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)
1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
![1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1](/thumbs/94/121802164.jpg "1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1")
1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
2.5 (Gauss) (flux) v(r)( ) S n S v n v n (1) v n S = v n S = v S, n S S. n n S v S v Minoru TANAKA (Osaka Univ.) I(2012), Sec p. 1/30
2.5 (Gauss) 2.5.1 (flux) v(r)( ) n v n v n (1) v n = v n = v, n. n n v v I(2012), ec. 2. 5 p. 1/30 i (2) lim v(r i ) i = v(r) d. i 0 i (flux) I(2012), ec. 2. 5 p. 2/30 2.5.2 ( ) ( ) q 1 r 2 E 2 q r 1 E
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
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](/thumbs/94/118770263.jpg "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
2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
1. 1 213 1 6 1 3 1: ( ) 2: 3: SF 1 2 3 1: 3 2 A m 2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP
1 B () Ver 2014 0 2014/10 2015/1 http://www-cr.scphys.kyoto-u.ac.jp/member/tsuru/lecture/... 1. ( ) 2. 3. 3 1 7 1.1..................................................... 7 1.2.............................................
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](/thumbs/85/91372642.jpg "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
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)
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009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3
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
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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
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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
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)
( ) 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
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
(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {
![(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {](/thumbs/94/118399854.jpg "(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {")
7 4.., ], ], ydy, ], 3], y + y dy 3, ], ], + y + ydy 4, ], ], y ydy ydy y y ] 3 3 ] 3 y + y dy y + 3 y3 5 + 9 3 ] 3 + y + ydy 5 6 3 + 9 ] 3 73 6 y + y + y ] 3 + 3 + 3 3 + 3 + 3 ] 4 y y dy y ] 3 y3 83 3
5. F(, 0) = = 4 = 4 O = 4 =. ( = = 4 ) = 4 ( 4 ), 0 = 4 4 O 4 = 4. () = 8 () = 4
... A F F l F l F(p, 0) = p p > 0 l p 0 P(, ) H P(, ) P l PH F PF = PH PF = PH p O p ( p) + = { ( p)} = 4p l = 4p (p 0) F(p, 0) = p O 3 5 5. F(, 0) = = 4 = 4 O = 4 =. ( = = 4 ) = 4 ( 4 ), 0 = 4 4 O 4 =
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)
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)
(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
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
2.4 ( ) ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) Minoru TANAKA (Osaka Univ.) I(2011), Sec p. 1/30
2.4 ( ) 2.4.1 ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) I(2011), Sec. 2. 4 p. 1/30 (2) Γ f dr lim f i r i. r i 0 i f i i f r i i i+1 (1) n i r i (3) F dr = lim F i n i r i. Γ r i 0 i n i
さくらの個別指導 ( さくら教育研究所 ) A a 1 a 2 a 3 a n {a n } a 1 a n n n 1 n n 0 a n = 1 n 1 n n O n {a n } n a n α {a n } α {a
... A a a a 3 a n {a n } a a n n 3 n n n 0 a n = n n n O 3 4 5 6 n {a n } n a n α {a n } α {a n } α α {a n } a n n a n α a n = α n n 0 n = 0 3 4. ()..0.00 + (0.) n () 0. 0.0 0.00 ( 0.) n 0 0 c c c c c
δ ij δ ij ˆx ˆx ŷ ŷ ẑ ẑ 0, ˆx ŷ ŷ ˆx ẑ, ŷ ẑ ẑ ŷ ẑ, ẑ ˆx ˆx ẑ ŷ, a b a x ˆx + a y ŷ + a z ẑ b x ˆx + b
23 2 2.1 n n r x, y, z ˆx ŷ ẑ 1 a a x ˆx + a y ŷ + a z ẑ 2.1.1 3 a iˆx i. 2.1.2 i1 i j k e x e y e z 3 a b a i b i i 1, 2, 3 x y z ˆx i ˆx j δ ij, 2.1.3 n a b a i b i a i b i a x b x + a y b y + a z b
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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
R = Ar l B r l. A, B A, B.. r 2 R r = r2 [lar r l B r l2 ]=larl l B r l.2 r 2 R = [lar l l Br ] r r r = ll Ar l ll B = ll R rl.3 sin θ Θ = ll.4 Θsinθ
![R = Ar l B r l. A, B A, B.. r 2 R r = r2 [lar r l B r l2 ]=larl l B r l.2 r 2 R = [lar l l Br ] r r r = ll Ar l ll B = ll R rl.3 sin θ Θ = ll.4 Θsinθ](/thumbs/80/82073891.jpg "R = Ar l B r l. A, B A, B.. r 2 R r = r2 [lar r l B r l2 ]=larl l B r l.2 r 2 R = [lar l l Br ] r r r = ll Ar l ll B = ll R rl.3 sin θ Θ = ll.4 Θsinθ")
.3.2 3.3.2 Spherical Coorinates.5: Laplace 2 V = r 2 r 2 x = r cos φ sin θ, y = r sin φ sin θ, z = r cos θ.93 r 2 sin θ sin θ θ θ r 2 sin 2 θ 2 V =.94 2.94 z V φ Laplace r 2 r 2 r 2 sin θ.96.95 V r 2 R
1. z dr er r sinθ dϕ eϕ r dθ eθ dr θ dr dθ r x 0 ϕ r sinθ dϕ r sinθ dϕ y dr dr er r dθ eθ r sinθ dϕ eϕ 2. (r, θ, φ) 2 dr 1 h r dr 1 e r h θ dθ 1 e θ h
IB IIA 1 1 r, θ, φ 1 (r, θ, φ)., r, θ, φ 0 r
.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 =,
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知能科学：ニューラルネットワーク
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