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1 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 ) x (f y ) x (f x ) y = f zy f yz f xz f zx f yx f xy f xy = f yx 1 f zy f yz f xz f zx f yx f xy = ϕ rot(grad ϕ) div(rot F ) = div H y G z F z H x G x F y =(H y G z ) x +(F z H x ) y +(G x F y ) z = H yx G zx + F zy H xy + G xz F yz = 2 Φ div(rot Φ) 1 f xy f 2 f xy = f yx f 2 f xy = f yx 2

2 12 2 grad rot rot grad rot div div rot grad ϕ = Φ ϕ Φ rot(grad ϕ) = Φ = rot Φ =? rot Φ = Φ?. xyz grad f = f, rot F = F, div F = F, = x y z ( f) =, ( F ) = (1) f F F f a (ra) =, a (a b) = a (ra) =r(a a) a b a a (b c) =(a b) c a (a b) =(a a) b

3 12 3 (1) a a a a a = a a = D = x x x y x z D x xf x (xf z ) y (xf y ) z xf zy xf yz D (Df) =D xf y = (xf x ) z (xf z ) x = xf xz f z xf zx xf z (xf y ) x (xf x ) y f y + xf yx xf xy = f z f y (xf y ) x (xf y ) x = ( x f ) = x f x y x y + x f x y = f y + x 2 f x y = f y + xf yx x y D D x x y Φ C C

4 12 4 C rot Φ d = Φ dl C Φ rot Φd = rot Φ rot Φ Φ Φ C Φ C C Φ

5 12 5 rot C C C C I =[, 1] [, 1] = C 1 + C C 2 2 C 2 C C

6 12 6 C C 1 + C rot Φ d = rot Φ d + rot Φ d 1 2 Φ dl = Φ dl + Φ dl C 1 +C 2 C 1 C 2 = Φ dl + Φ dl + Φ dl Φ dl C 1 C C 2 C = Φ dl + Φ dl C 1 +C C 2 C 1 2 rot Φ d = Φ dl, rot Φ d = Φ dl 1 C 1 +C 2 C 2 C 2 rot Φd = Φ dl C 1 +C 2 I xyz x ξ(s, t) y = T (s, t) = η(s, t), s 1, t 1 z ζ(s, t) C 4 C 1 = {T (s, ) s 1}, C 2 = {T (1,t) t 1}, C 3 = {T (s, 1) s 1}, C 4 = {T (,t) t 1}

7 12 7 C = C 1 + C 2 C 3 C 4 C 3 C 3 Φ F (x, y, z) = F (x, y, z) G(x, y, z) H(x, y, z) I st [, 1] [, 1] rot Φ d = ( F)(T (s, t)) (T s (s, t) T t (s, t) ) dsdt I = I = I H y (T (s, t)) G z (T (s, t)) F z (T (s, t)) H x (T (s, t)) G x (T (s, t)) F y (T (s, t)) η s (s, t)ζ t (s, t) ζ s (s, t)η t (s, t) ζ s (s, t)ξ t (s, t) ξ s (s, t)ζ t (s, t) ξ s (s, t)η t (s, t) η s (s, t)ξ t (s, t) { H y (T (s, t))η s (s, t)ζ t (s, t) H y (T (s, t))ζ s (s, t)η t (s, t) dsdt G z (T (s, t))η s (s, t)ζ t (s, t)+g z (T (s, t))ζ s (s, t)η t (s, t) + F z (T (s, t))ζ s (s, t)ξ t (s, t) F z (T (s, t))ξ s (s, t)ζ t (s, t) H x (T (s, t))ζ s (s, t)ξ t (s, t)+h x (T (s, t))ξ s (s, t)ζ t (s, t) + G x (T (s, t))ξ s (s, t)η t (s, t) G x (T (s, t))η s (s, t)ξ t (s, t) } F y (T (s, t))ξ s (s, t)η t (s, t)+f y (T (s, t))η s (s, t)ξ t (s, t) dsdt F x (T (s, t))ξ s (s, t)ξ t (s, t) F x (T (s, t))ξ t (s, t)ξ s (s, t)

8 12 8 { Fx (T (s, t))ξ s + F y (T (s, t))η s + F z (T (s, t))ζ s } ξt + { G x (T (s, t))ξ s + G y (T (s, t))η s + G z (T (s, t))ζ s } ηt + { H x (T (s, t))ξ s + H y (T (s, t))η s + H z (T (s, t))ζ s } ζt { F x (T (s, t))ξ t + F y (T (s, t))η t + F z (T (s, t))ζ t } ξs (2) { G x (T (s, t))ξ t + G y (T (s, t))η t + G z (T (s, t))ζ t } ηs { H x (T (s, t))ξ t + H y (T (s, t))η t + H z (T (s, t))ζ t } ζs F x (T (s, t))ξ s + F y (T (s, t))η s + F z (T (s, t))ζ s =(F T ) s (2) (F T ) s ξ t +(G T ) s η t +(H T ) s ζ t (F T ) t ξ s (G T ) t η s (H T ) t ζ s (F T ) s ξ t (F T ) t ξ s = (G T ) s η t (G T ) t η s (3) (H T ) s ζ t (H T ) t ζ s s, t 2 f(s, t) g(s, t) F (T (s, t)) ξ s (s, t) f(s, t) = G(T (s, t)) η s (s, t) H(T (s, t)) ζ s (s, t) g(s, t) = F (T (s, t)) G(T (s, t)) H(T (s, t)) ξ t (s, t) η t (s, t) ζ t (s, t), ξ st = ξ ts (F T ) t ξ s F T f t g s = (G T ) t η s + G T (H T ) t ζ s H T (F T ) s ξ s F T (G T ) s η s + G T (H T ) s ζ s H T (F T ) t ξ s (F T ) s = (G T ) t η s (G T ) s (H T ) t ζ s (H T ) s ξ st η st ζ st ξ s η s ζ s ξ ts η ts ζ ts

9 12 9 (3) rot Φ d = (f t (s, t) g s (s, t))dsdt (4) I (4) f t (s, t) f(s, t) s t 1 f t (s, t)dsdt = I g t (s, t)dsdt = I ( 1 f t (s, t)dt = f(s, 1) f(s, ) ( 1 ) f t (s, t)dt ds = ) g s (s, t)ds dt = 1 (4) 1 1 rot Φ d = g(,t)dt + f(s, 1)ds 1 f(s, 1)ds 1 g(1,t)dt g(1,t)dt f(s, )ds g(,t)dt f(s, )ds f(s, t) g(s, t) 1 1 F (T (,t)) ξ t (,t) g(,t)dt = G(T (,t)) η t (,t) dt H(T (,t)) ζ t (,t) 1 = F (T (,t)) T t (,t)dt = Φ dl C f(s, 1)ds = Φ dl C 2 g(1,t)dt = Φ dl C 3 f(s, )ds = Φ dl C 4 rot Φ d = Φ dl + Φ dl Φ dl Φ dl C 1 C 2 C 3 C 4 = Φ dl = Φ dl C 1 +C 2 C 3 C 4 C

10 12 1. xyz (1). Φ (y + 1)(z 2 1) (x + 1)(z 2 +1) (x + 1)(y +1) rot Φ (2). z = {(x, y, z) x 2 + y 2 + z 2 =1,z } (1) Φ rot Φ d Φ grad ϕ = Φ ϕ rot Φ = rot(grad ϕ) = rot Φ = Φ Φ rot Φ = Φ C Φ dl = C C Φ dl = rot Φ d C

11 12 11 rot Φ = Φ dl = Φ C C C Φ Φ 1 x 2 + y 2 y x rot Φ = C xy 2π sin θ sin θ 2π Φ dl = cos θ cos θ dθ = 1dθ =2π C Φ Φ z C z Φ 1. X X C C X 3 Φ Φ rot Φ = 3 3

12 ρ E B J 4 4 xyz t 4 ε µ D D 4 D H D = ε E B = µ H E H

13 12 13 E d = 1 ρdv ε 1 µ D D D B d = B E dl = t d B dl = ( ) E ε t + J d 4 4 div E = 1 ε ρ div B = rot E = B t 1 E rot B = ε µ t + J

14 12 14 E d = 1 ρdv ε D D E E D div EdV = E d D E D div EdV = 1 ρdv ε D E D D D div E(P,t ) > 1 ε ρ(p,t ) P t E ρ 5 P D P div E(P, t ) > 1 ε ρ(p, t ) t div EdV > 1 ρdv D ε D 5 ρ

15 12 15 div E = 1 ε ρ D D B d = div B = B E dl = t d E rot E d = E dl E B rot E d = t d

16 12 16 E t B t = F rot E(P,t ) F (P,t ) P t xyz rot E = R 1 R 2 R 3, F = R 3 (P,t ) >F 3 (P,t ) rot E F 6 P D P R 3 (P, t ) >F 3 (P, t ) D P xy xy z t rot E d > F d rot E = F = B t 6 F 1 F 2 F 3

17 12 17 ( ) 1 E B dl = ε t + J d µ 1 E rot B = ε µ t + J 4.3 ρ J div E = (5) div B = (6) rot E = B t (7) E rot B = ε µ t (8) (8) t rot B t = ε 2 E µ t 2 (7) rot(rot E) =ε µ 2 E t 2 (9)

18 12 18 E rot(rot E) rot(rot E) = rot = = x E 3 E 2 y z E 1 E 3 z x E 2 E 1 x y 2 E 2 2 E 1 2 E E 3 y x y 2 z 2 z x 2 E 3 2 E 2 2 E E 1 z y z 2 x 2 x y 2 E 1 2 E 3 2 E E 2 x z x 2 y 2 y z ( ) ( E 1 y z ( E 1 + E 2 + E 3 x y z ( E 1 x + E 2 + E 2 + E 3 x y z = grad(div E) E + E 3 y z ) ( ) ( x 2 y x 2 y 2 z 2 ) E z 2 ) E 2 2 x y z 2 ) E 3 2 = 2 x y + 2 z 2 1 (5) rot(rot E) = E (9) E = ε µ 2 E t 2 (1) (7) t (8) (6) B B = ε µ 2 B t 2 (11) 2 / t 2 E B 2 f f(x, y, z, t)=ε µ (x, y, z, t) t2 f(x, y, z, t) 1/ ε µ

19 ε µ = c 3 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 k, l, e, b,v,u f g (1) (11) f (k x vt)e = ε µ v 2 f (k x vt)e g (k x ut)b = ε µ u 2 f (l x ut)b k l v = u = 1 ε µ = c 4 f (k x ct)k e = (12) g (l x ct)l b = (13) f (k x ct)k e = cg (l x ct)b (14) cg (l x ct)l b = f (k x ct)e (15)

20 12 2 f g f g (x,t) (12) (13) k e = l b = k e l b E B (14) (15) b k e e l b e b k, l e, b e b k l l = k l = k (14) l f (k x ct)(k e) l = cg (l x ct)b l (15) 1 f (k x ct)(k e) l = f (k x ct)e f a = k e (16) (k e) l = e (16) k, e, a

21 12 21 a, l, e l, e, a l k l = k E B E B

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 )