Kroneher Levi-Civita 1 i = j δ i j = i j 1 if i jk is an even permutation of 1,2,3. ε i jk = 1 if i jk is an odd permutation of 1,2,3. otherwise. 3 4

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1 [2642 ] Yuji Chinone ρ t + j = V S ds ds Eq.1 ρ t + j dv = ρ t dv = t V V V ρdv = Q t Q V jdv = j ds V ds V I Q t + j ds = ; S S [ Q t ] + I = Eq.1 2 2

2 Kroneher Levi-Civita 1 i = j δ i j = i j 1 if i jk is an even permutation of 1,2,3. ε i jk = 1 if i jk is an odd permutation of 1,2,3. otherwise. 3 4 *1 ε i jk 2 = 6 ; ε ikm ε jkm = 2δ i j ; ε i jk ε nmk = δ in δ jm δ im δ jn ; e i e j = ε i jk e k ; e i e j = δ i j div rot = 2-1 f f = {e i i } { } f j e j = ei e j i f j = εi jk e k i f j = e k k ε i jk e k i f j = εi jk ik f j =, ik = ki, ε i jk = ε k ji 2-2 rot grad = 2-2 f f = e j j f = e i i e j j f = e i e j i j f = e k ε i jk i j f =, i j = ji, ε i jk = ε jik A = A 2 A 6 *1

3 3-1 A = e i i { e j j ek A k } = e i e j e k i j A k = e i ε jkl e l i j A k = ε jkl ε ilm e m i j A k = ε jkl ε iml e m i j A k = δ ji δ km δ jm δ ki em i j A k = δ jm δ ki e m i j A k δ ji δ km e m i j A k = e m mk A k ii A k e k = A 2 A 3-2 A B C = B C A = C A B A B C = A i e i { B j e j Ck e k } = A i e i B j C k ε jkl e l = Ai B j C k ε jkl δ il = A i B j C k ε i jk = A i B j C k ε ki j = A i B j C k ε kil δ jl = B j e j Ck A i ε kil e l = B j e j {Ck e k A i e i } = B C A = A i B j C k ε jki = A i B j C k ε jkl δ il = C k e k A i B j ε i jl e l = Ck e k {A i e i } B j e j = C A B 3-3 A B = B A A B A B = e i i [ A j e j Bk e k ] = e i i A j B k ε jkl e l = ε jkl δ il i A j B k = εi jk i A j B k = ε i jk B k i A j + ε i jk A j i B k = B k e k e k ε i jk i A j A j e j e j ε ik j i B k = B A A B A B C = A C B A B C 9

4 4-1 LHS = Ai B j C k { ei e j e k } = Ai B j C k { ei ε jkl e l } = Ai B j C k ε jkl ε ilm e m = A i B j C k ε jkl ε iml e m = A i B j C k δ ji δ km δ jm δ ki em = A i B j C k δ jm δ ki e m A i B j C k δ ji δ km e m = A i C i B m e m A i B i C m e m = A C B A B C = RHS A B A = A B A = A i e i { B j e j Ak e k } = A i e i B j A k ε jkl e l = Ai B j A k ε jkl δ il = A i A k B j ε i jk =, A i A k = A k A i, ε i jk = ε k ji 6 z B E = m q r = x, y, z m d2 r dt = q E + 1 dr 2 dt B 11 m d2 x dt 2 m d2 y dt 2 = q m d2 z dt 2 = q = q dy dt B z dz dt B y = qb dy 12 dt dz dt B x dx dt B z = qb dx 13 dt dx dt B y dy dt B x = 14

5 6-2 v z t = = 6-2 Eq.12,13 v x + qb 2 qb v x =, v x t = C 1 os m m t Eq.12 v x t zt Eq.14 xt = C 1 sin qb m t + C 2 yt = C 1 os qb m t + C 3 zt = C 4 C 1, C 1, C 2, C 3.C 4 = Const qb qb r = xt, yt, zt = C 1 sin m t + C 2, C 1 os m t + C 3, C 4 v = v x t, v y t, v z t qb qb = C 1 os m t, C 1 sin m t, q B 6-3 y C 3 B q> C 1 y C 3 B q< C 1 z B Bz C 2 Bz x C 2 Bz x q> q< 1 Eq.15,16 q B q B

6 7 gx 7-1 dx f xδx dx f xδx 1 = = f 1 dx f X + 1δX ; x 1 = X, dx = dx, x + X dx f xδ3x dx f xδ3x = = f 3 dx 3 f X/3δX ; 3x = X, 3dx = dx, x + X dx f xδ 3x dx f xδ 3x = = f 3 dx 3 f X/3δX ; 3x = X, 3dx = dx, x + X dx f xδx 2 1 2

7 7-4 dx f xδx 2 1 = = = 1 ɛ 1+ɛ 1 ɛ ɛ 2 +2ɛ + ɛ 2 2ɛ ɛ 2 +2ɛ ɛ 2 2ɛ 1+ɛ 1 ɛ 1+ɛ ɛ 1+ɛ 1 ɛ dx f xδx dx f X + 1 δx 2 X + 1 = 1 { f 1 + f +1} 2 +1+ɛ dx f X + 1 δx 2 X ɛ + 1+ɛ dx f xδx 2 1 ; ɛ > dx f xδx 2 1 ; x 2 1 = X, 2xdx = dx, x <, ; x 2 1 = X, 2xdx = dx, x >, x 1 ɛ 1 + ɛ X ɛ 2 + 2ɛ ɛ 2 2ɛ x 1 ɛ 1 + ɛ X ɛ 2 2ɛ ɛ 2 + 2ɛ 7-5 dx f xδ gx dy dy dx f xδ gx = g δy f gx ; gx = y, x dx = g x, = f x i g x i y = x = x i 7 δax = 1 δx a 22 δx 2 a 2 = 1 {δx + a + δx a} 23 2 a δ gx = i δx x i g x i ; gx i = 24 8 gx gx = x = x n x n N x = x n n = 1, 2,... N gx = 8-1 dx f xδ gx 25

8 8-1 Eq.24 dx f xδ gx = = N n=1 dx f x f x n g x n N n=1 δx x n g x n = dx N n=1 f xδx x n g x n 9 φ φ r, t = 4πρ e r, t 26 D Alambertian = t G r, t = δ 3 rδt 28 G Green Green φ r, t φ r, t = 4π G r r, t t ρ e r, t d 3 rdt = D, t Eq.28 D, Gr, t = δ 3 rδt t Fourier ˆD ik, iω Ĝ k, ω = 1 2π 4 1 ˆD ik, iω = 2π4 Ĝ k, ω 3 Eq.26 Fourier ˆφ k, ω = 4π ˆρ e k, ω ˆD ik, iω

9 Eq.3 xt, yt x yt = ˆφ k, ω = 4π2π 4 Ĝ k, ω ˆρ e k, ω 31 xtyt t dt = 2π ˆxωŷωe iωt Eq.31 Fourier LHS = F 1 [ ˆφ k, ω ] = φ r, t RHS = F 1 [ 4π2π 4 Ĝ k, ω ˆρ e k, ω ] = 4πF 1 [ 2π 4 Ĝ k, ω ˆρ e k, ω ] = 4π [ ρ e G r, t ] = 4π G r r, t t ρ e r, t d 3 rdt φ r, t Eq.29 1 G r, t = δ 3 rδt 32 Green G r, t, for t =, for t < 33 Green Green 1-a Green Fourier G r, t = δ 3 rδt = 1 2π 4 d 3 k Ĝ k, ω e iωt+ik r 34 k k d 3 k e iωt+ik r 35 Ĝ k, ω = 2 2π 4 1 ω 2 2 k a + LHS = G r, t = d 3 k Ĝ k, ω e iωt+ik r 2 t 2 Ĝ = d 3 k +ik 2 iω2 k, ω e iωt+ik r = ω2 2 k 2 2 RHS = δ 3 rδt = 1 2π 4 2 d 3 k Ĝ k, ω e iωt+ik r d 3 k e iωt+ik r

10 Ĝ k, ω = 2 2π 4 1 ω 2 2 k 2 1-b a G r, t = 2 2π 4 d 3 1 k +ik r 37 ω 2 2 k 2 ω Green ω = k ω = +k Green 1-b Eq.37 G r, t = 2 2π 4 d 3 k 1 ω 2 2 k 2 +ik r = 2 2π 4 d 3 k e ik r ω 2 2 k 2 ω 38 ω 2 k 2 2 ω-plane C 1 -k R= ω o +k C 2 2 ω plane Green t < exp [ iωt] = exp [+iω t ] = exp +i t R[ω] exp t I[ω]

11 I[ω] > R = ω + Eq.38 C 1 ω 2 k 2 = 2 e+iω t ω 2 k 2 2 e+iω t ω 2 k 2 2 = G r, t < = 39 t exp iωt = exp iω t = exp i t R[ω] exp + t I[ω] Eq.38 C 2 ω 2 k 2 = 2 ω 2 k 2 2 ω = ±k ω 2 k 2 = 2 e ikt = 2πi [R k + R+k] = 2πi ω 2 k 2 2 2k + e ikt = 2π 2k k sinkt ω k k 3 θ k k 1 k ψ 2 3 k = k 1, k 2, k 3 = k, θ, ψ

12 G r, t = 2 2π 4 = 1 2π 2 ir = 2π 2 1 2r = 2π 2 1 2r = 1 2π 2 2r = 1 δ 4π t r r [{ [ { d 3 k 2π sinktdk } k sinkt π θ π 2π e ik r = 2π 3 e ikr os θ dθ = 1 2π 2 ir dk [ e +ikt r/ e +ikt+r/ + e ikt r/ e ikt+r/] dk e +ikt r/ + dk e +ikt r/ ; t Green } { dk e +ik t r/ dk e +ikt+r/ ] G ret r, t = 1 4π G ret r, t < = = 1 4πr δ t r r [ δ k 2 sin θ dkdθdψ [sinkt 1k ] eikr os θ dk e ikr e +ikr sinkt dk e +ikt+r/ + t r δ t + r ] }] dk e +ik t+r/ ; k = k Green ω = ±k t <, t Green t > t Green G adv r, t = 1 4π G adv r, t > = δ t + r Eq.32 r = t = r, t t t < G r, t r 11 t dt u n 11-1 dt dt = [1 1 n u ] dt 4

13 dt n 11-1 u t=t Q n P P t=t +dt O 4 P O t 1 t 1 = t + OP P O t 2 t 2 = t + dt + OP = t + dt + OP QP = t + dt + dt dt = t 2 t 1 = [1 1 ] n u dt OP n udt Eq.4

Radiation from moving charges#1 Liénard-Wiechert potential Yuji Chinone 1 Maxwell Maxwell MKS E (x, t) + B (x, t) t = 0 (1) B (x, t) = 0 (2) B (x, t)

Radiation from moving charges#1 Liénard-Wiechert potential Yuji Chinone 1 Maxwell Maxwell MKS E (x, t) + B (x, t) t = 0 (1) B (x, t) = 0 (2) B (x, t) Radiation from moving harges# Liénard-Wiehert potential Yuji Chinone Maxwell Maxwell MKS E x, t + B x, t = B x, t = B x, t E x, t = µ j x, t 3 E x, t = ε ρ x, t 4 ε µ ε µ = E B ρ j A x, t φ x, t A x, t