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............................... 5 1.3.............................. 1 1.3.1.................................. 1 1.3.2..................................... 1 1.3.3..................................... 11 1.3.4..................................... 11 1.3.5................................ 12 1.3.6.............................. 13 1.3.7...................................... 13 2 16 2.1................................ 16 2.2 (........................ 17 2.2.1...................................... 17 2.2.2..................................... 19 2.2.3................................ 2 2.3............................... 21 2.3.1............................ 21 2.3.2....................... 23 2.3.3.......................... 24 2.4 Green.................................... 27 3 32 3.1..................................... 32 3.1.1 3.............................. 32 3.1.2........................... 35 3.1.3........................... 38 3.2........... 39 3.3.............................. 42 3.4 Gauss.................................. 45 1

3.5 Stokes.................................... 47 A 5 A.1 Green........... 5 A.2 Gauss.................... 52 B 58 B.1 k -................................... 58 B.2.................................... 59 B.3........................................ 6 B.4....................................... 62 B.5 Jordan................................. 62 64.1 1:................... 64.1.1 1:... 64.1.2 2:............ 65.2 2:..................... 65.3 3:.............. 67 D 69 D.1 Green.................................... 69 D.2 Helmholtz.................................. 69 D.3........................................ 7 D.4.............................. 7 D.5......................... 7 D.6.................................. 71 D.7 Schwarz............................... 71 D.8.............................. 71 D.9............................................ 71 E 73 E.1 :................. 73 E.2.............................. 73 E.3............................... 74 E.4 de Rham...................... 74 E.5.................................... 76 E.6 de Rham........................ 76 E.7............................. 77 F 78 F.1.................................... 78 F.2 [1].................... 78 2

1 1.1 1.1.1 ( II ( 1. 2. ( (a f dr (b f n ds 3. S d dx x a f(t dt = f(x, b a F (x dx = F (b F (a div f dx = f n ds Ω Ω (Gauss S rot f ds = S f dr (Stokes 1 ( 1 (Michael Faraday, 1791 1867, (James lerk Maxwell, 1831 1879, Edinburgh ambridge (Oliver Heaviside, 185 1925, 3

2 2 O.Heaviside [2] (1893 Josiah Willard Gibbs (1839 193, (1881, 191 James lerk Maxwell (1831 1879 ([23], 1873 J.W.Gibbs [21] http://www-groups.dcs.st-and.ac.uk/~history/ Mathematicians/Gibbs.html Gibbs work on vector analysis was also of major importance in pure mathematics. He first produced printed notes for the use of his own students in 1881 and 1884 and it was not until 191 that a properly published version appeared prepared for publication by one of his students. Using ideas of Grassmann, Gibbs produced a system much more easily applied to physics than that of Hamilton. ( Hamilton a 4 (quaternion a Sir William Rowan Hamilton (185 1865, Ireland Dublin Dublin 4 (1843 http: //www.maths.tcd.ie/pub/histmath/people/hamilton/papers.html 4 ( [14] [6] 1.1.2 a a II T (transpose a = (a 1, a 2, a 3 T x r r x, y, z 2 ( 4

1.1.3 1 f : I R (I R n m f : Ω R m (Ω R n m = n n (vector field Ω r ( f(r Ω f ( f 1 f : I R f : I R (I R 1 n f : Ω R (Ω R n n 2 f n n (n f : Ω R n (n ( Ω f : Ω R Ω (scalar field 1.2 : R 3 R 3 a = a 1 a 2, b = b 1 b 2 a 3 b 3 5

a b 3 R 3 ( a 2 b 2 a b := a 3 b 3, a 3 b 3 a 1 b 1, a 1 b T 1 a 2 b 2 = = a 2 b 2 a 3 b 3 e 1 a 1 b 1 e 1 a 2 b 2 e 2 a 3 b 3 e 3 a 1 b 1 a 3 b 3 e 2 + a 1 b 1 a 2 b 2 e 3 (. e 1 = 1, e 2 = 1, e 3 = 1. i a i, b i 1 2 3 1 e 1 e 2 e 3 a b = a 1 a 2 a 3 b 1 b 2 b 3 a = (1, 2, 3 T, b = (3, 2, 1 T a b (1 ( e 1 e 2 e 3 2 3 a b = 1 2 3 = 2 1 e 1 3 1 3 1 e 1 2 4 2 + 3 2 e 3 = 8. 3 2 1 4 (2 a b = ( 2 3 2 1, 3 1 1 3, 1 2 3 2 1 2 3 1 3 1 2 3 2 1 3 T = 4 8 4. 3 6

1.2.1 ( a, b, a 1, a 2 R 3 λ R (1 a b = (b a. a a =. (2 (a 1 + a 2 b = (a 1 b + (a 2 b, (λa b = λ(a b. (3 x R 3 det (a b x = (a b, x. (4 a b 1 a b. ( (5 a, b {ta + sb; t [, 1], s [, 1]} S (a b a, (a b b, det(a b a b, a b = S. (1, (2 (3 x = (x 1, x 2, x 3 T a 2 b 2 (a b, x = a 3 b 3 x a 1 b 1 1 a 3 b 3 x a 1 b a 1 b 1 x 1 1 2 + a 2 b 2 x 3 = a 2 b 2 x 2 = det(a b x. a 3 b 3 x 3 (a b e 1, e 2, e 3 x 1, x 2, x 3 a b x (4 (3 a b 1 x s.t. a, b, x 1 (5 (3 x s.t. det(a b x x s.t. (a b, x a b. (a b, a = det(a b a =, (a b, b = det(a b b =, det(a b a b = (a b, a b = a b 2. S = a b a, b 1 a, b 1 3 a, b, a b 3 V a b V = S a b. V = det(a b a b. 7

S a b = det(a b a b = a b 2 a b (> S = a b. (a, b (5 a b θ cos θ = a b S = a b sin θ = a b 1 cos 2 θ = a 2 b 2 (a, b 2 = (a 2 1 + a 2 2 + a 2 3(b 2 1 + b 2 2 + b 2 3 (a 1 b 1 + a 2 b 2 + a 3 b 3 2 = (a 2 b 3 a 3 b 2 2 + (a 3 b 1 a 1 b 3 2 + (a 1 b 2 a 2 b 1 2 = a b. (a b c = a (b c a, b, c (a b c + (b c a + (c a b = (Jacobi. 1.2.1 (R n R n R n n 1 a 1, a 2,, a n 1 R n x det(a 1, a 2,, a n 1, x R c R n s.t. x R n det(a 1, a 2,, a n 1, x = (c, x. c a 1, a 2,, a n 1 a 1 a 2 a n 1 n n 1 2 3 a 1 a 2 a n 1 x = e i : a 1 a 2 a n 1 i = det(a 1, a 2,, a n 1, e i. ( 3 1.2.1 ( 3 A(1, 2, 3, B(2, 1,, (, 2, 1 AB 8

AB = 2 1 1 2 3 = 1 1 3, A = 2 1 1 2 3 = 1 4 ( AB A = 1 3 4, 3 1 4 1, 1 1 1 T = 4 7 1. ( AB = 1 AB A = 1 86 42 + 7 2 2 2 + ( 1 2 = 2. A(1, 2, 3 (4, 7, 1 T 4(x 1 + 7(y 2 + ( 1(z 3 = 4x + 7y z = 15. 1.2.2 ( 1 f(t, g(t (f(t g(t = f (t g(t + f(t g (t ( t m, r(t, f mr (t = f (Newton f f = f(rr f 3 a a a = ( d 1 dt 2 r(t r (t = 1 2 (r (t r (t + r(t r (t = 1 2 r(t f m = 1 2 r(t f m r(t = 1 2 r(t r (t. 4 ( Kepler 4 r(t (mr (t 9

1.3 grad, rot, div ( grad ( I div, rot 1.3.1 R n ( := x 1 x 2. x n nabla 5 Hamilton 1.3.2 R n Ω 1 - f : Ω R f = x 1 x 2. x n f := f x 1 f x 2. f x n : Ω R n f f (gradient f grad f : f a Ω f(a f ( {x Ω; f(x = c}, c := f(a a f 5 ( (Nebel ( [17] 1

1.3.3 R n Ω 1 - f : Ω R n f := x 1 x 2. x n f 1 f 2. f n = n i=1 f i x i : Ω R f f (divergence f div f div f = (solenoidal div f Gauss 1.3.1 f f 2 div f = div f <, div f > 1.3.4 n = 3 R 3 a, b a b R 3 Ω 1 - f : Ω R 3 f := ( x 2 x 3 f 2 f 3, x 3 x 1 f 3 f 1, x 1 x 2 f 1 f 2 T = f 3 x 2 f 2 x 3 f 1 x 3 f 3 x 1 f 2 x 1 f 1 x 2 : Ω R 3 f f (rotation f rot f curl f e 1 e 2 e 3 f = x 1 x 2 x 3 f 1 f 2 f 3 f rot f (vorticity rot f = Ω (r cos Ωt, r sin Ωt (r = x 2 + y 2, t v ω := rot v rot f Stokes 11

f 1 x 2 x 3 f 2 f 3 = f 3 f 2 x 2 x 3 1 2, 2 3, 3 1 f 1 x 3 f 3 x 1, f 2 x 1 f 1 x 2 : 2 2 f = (f 1, f 2 T f = rot f := f 2 x 1 f 1 x 2 3 f f(r = (f 1 (x 1, x 2, f 2 (x 1, x 2, T 2 f x 1 f 1 e 1 ( ( x = det x 2 f 2 e 2 = det 1 f 1 e 3 =,, f 2 f T 1 x e 2 f 2 x 1 x 2 3 ψ rot ψ := ( ψ, ψ T x 2 x 1 ( ψ rot ψ (stream function 3 rot 1.3.5 R n Ω 2 - f : Ω R f := n i=1 2 f x 2 i 12

f f (Laplacian f f (harmonic function R n Ω 2 - f : Ω R n f Laplacian f := ( f 1,, f n T 1.3.1 ( f = div(grad f = ( f 2 ( 1.3.6? 1.3.1 ( f R 3 2 f R 3 2 - (1 div(grad f = f. ( f = f. (2 rot(grad f =. ( f =. (3 div(rot f =. ( f =. (4 rot(rot f = grad(div f f. ( f = ( f f. ( (1, (2, (3 ( (2, (3 (1 (4 1.3.2 1.3.1 1 - u, v div (u v = (rot u, v (rot v, u 1.3.7 ( 6 ( 6 13

II http://www.math.meiji.ac.jp/~mk/lecture/ouyoukaiseki2/pde24-long.pdf ( 1.3.2 (Maxwell (1873, E, B, ρ, j Maxwell E = ρ ε, E = B t, B =, c2 B = j + E ε t 7 (c, ε 8 (ρ, j E =, E = B t, B =, c2 B = E t 1.3.1 (4 1 2 B c 2 t 2 1 2 E c 2 t 2 = B ( B = = ( E t t = E ( E = E = E, = 1 c 2 t ( E = 1 c E 2 t = 1 c ( c 2 B = B 2 = B ( B = B = B. E, B c Maxwell (1831 1879 (1864 1887 Hertz (Heinrich Rudolph Hertz, 1857 1894 ( 1.3.3 (Fourier, Fourier (Jean-Baptiste-Joseph Fourier, 1768 183, ( u u = grad u ( ( u = div(k grad u t ( k Fourier k div(grad = u t = κ u, κ := k ( 7 Maxwell Heaviside 8 MKS ε = 17 4πc 2 8.854 1 12 F/m. 14

1.3.4 ( ( v v = ( 1.3.5 ( Gauss ( [15] 15

2 2.1 1 - : r = ϕ(t (t [a, b] ( L (2.1 L = b a ( n 1/2 ϕ (t dt, ϕ (t = ϕ i(t 2 1 t = a t = t s = σ(t (t [a, b] s = σ(t = t a i=1 ϕ (r dr ds dt = dσ dt = ϕ (t. σ(t t 1 - t [a, b] ϕ (t dσ dt > (t [a, b]. t = τ(s 2 1 - ψ := ϕ τ, ψ(s := ϕ(τ(s (s [, L] r = ϕ(t r = ψ(s s {r; r = ψ(s, s [, L]} f L (2.2 f ds := f(ψ(s ds 1 [a, b] = {t j } N j= L := N j=1 ϕ(t j ϕ(t j 1 L L (rectifiable L ϕ 1 - (2.1 II (2.1 2 σ, τ s, t 16

b (2.3 f ds = f(ϕ(t ϕ (t dt a (2.3 ds := ϕ (t dt = ϕ 1(t 2 + + ϕ n(t 2 dt ( dγ 2.2 ( f ( f dr 1 ( ω = f 1 dx 1 + f 2 dx 2 + + f n dx n ω = f 1 dx 1 + f 2 dx 2 + + f n dx n 3 dr = (dx 1,, dx n T f dr = f 1. f n dx 1. dx n (? = f 1 dx 1 + + f n dx n 2.2.1 ( ( (... 3 d ( 17

( f r W W = fr f r W = f r. f W N W f(r j r j, r j := r j r j 1. j=1 r j [a, b] = {t j } N j= r j = ϕ(t j (j =, 1,..., N W = f dr := lim N f(r i r i 1 - b a i=1 f(ϕ(t ϕ (t dt Ω Ω f : Ω f(z dz = lim N f(z i (z i z i 1 i=1 ( [a, b] = {t j } N j=1 z j = ϕ(t j f(z dz = (u + iv(dx + i dy = u dx v dy + i v dx + u dy 18

2.2.2 R n Ω n f : Ω R n f = (f 1,, f n T Ω 1 - r = ϕ(t = (ϕ 1 (t,, ϕ n (t T (t [a, b] f dr := b a f dr dt dt = b a f ϕ (t dt = b a [f 1 (ϕ(tϕ 1(t + + f n (ϕ(tϕ n(t] dt f ( (25 2.2.1 f(x, y, z = (y + z, z + x, x + y T j f dr j (1 1 : r = (t, t 2, t 3 T (t [, 1] (2 2 : (,,, (1,,, (1, 1,, (1, 1, 1 (1 ϕ(t = (t, t 2, t 3 f(ϕ(t = (t 2 + t 3, t 3 + t, t + t 2 T, ϕ (t = (1, 2t, 3t 2 T f(ϕ(t ϕ (t = (t 2 + t 3 1 + (t 3 + t 2t + (t + t 2 3t 2 = t 2 + t 3 + 2t 4 + 2t 2 + 3t 3 + 3t 4 = 3t 2 + 4t 3 + 5t 4. 1 f dr = 1 (3t 2 + 4t 3 + 5t 4 dt = 3 1 3 + 4 1 4 + 5 1 5 = 3. (2 (,, (1,, γ 1, (1,, (1, 1, γ 2, (1, 1, (1, 1, 1 γ 3 2 = γ 1 + γ 2 + γ 3 f dr = 2 f dr + γ 1 f dr + γ 2 f dr. γ 3 γ 1 ϕ(t = (t,, T (t [, 1] f(ϕ(t = (, t, t T, ϕ (t = (1,, T, f(ϕ(t ϕ (t = f dr =. γ 1 γ 2 ϕ(t = (1, t, T (t [, 1] f(ϕ(t = (t, 1, 1 + t T, ϕ (t = (, 1, T, f(ϕ(t ϕ (t = 1 f dr = 1. γ 2 γ 3 ϕ(t = (1, 1, t T (t [, 1] f(ϕ(t = (1 + t, t + 1, 2 T, ϕ (t = (,, 1 T, f(ϕ(t ϕ (t = 2 f dr = 2. γ 3 f dr = + 1 + 2 = 3. 2 19

2.2.1 ( 4 dr ds dx f ds, f dx t = dr ds ds f t ds (f, dr, (f dr ( 2.2.2 ( 3 f ds = f dr = b a b f(ϕ(t ds dt dt, a P dx + Q dy = f(ϕ(t dr dt dt, b a ds dt = ϕ (t, dr dt = ϕ (t, [ P (x(t, y(t dx + Q(x(t, y(tdy dt dt ] dt (n = 2 1 2 2 3 2.2.3 ( 4 ( 2

2.2.1 ( ( 1 -, (1 ( (2 (f + g dr = f dr + g dr. (3 (λf dr = λ f dr. (4 f dr = f dr + f dr. 1 + 2 1 2 (5 f dr = f dr. (6 f dr f ds. ((1 2.2.3 f ds (5 f ds = f ds 2.3 (1 R 5 2.3.1 f : Ω R n F (x = f(x (x Ω F F f (potential 5 Lagrange (1773 Green (1828 21

1 F = f F f Ω R n F : Ω R 1 - F ( F F : Ω R n 1 2.3.1 ( f(x 1, x 2, x 3 := (g g F (x 1, x 2, x 3 := gx 3 F = f F f 2.3.2 ( f(r := GM r 3 r (M, G, r R3 \ {} F (r := GM r f ( 2.3.1 ( f grad V = f V f V f 2.3.2 ( 1 ω = f 1 dx 1 + + f n dx n ω = df := F dx 1 + + F dx n F ω x 1 x n ω (exact 22

2.3.2 (1 (1 1 - f = (f 1,, f n T f i x j = f j x i (i, j = 1, 2,, n 3 f = (rot f = (2 f F, a, b f dr = F dr = F (b F (a. 1 b a F (x dx = F (b F (a (3 Ω a x a x Ω 1 - x F (x := f dr x F f ( F = f 1 d dx x a f(t dt = f(x 2.3.3 ( ( x 2 f(x 1, x 2 = x 1 F F x 1 = f 1 (x 1, x 2 = x 2, 2 F = 1, x 2 x 1 2 F x 2 x 1 23 F x 2 = f 2 (x 1, x 2 = x 1 2 F x 1 x 2 = 1 2 F x 1 x 2

(1 f F f 1 - F 2 - f k = F x k (k = 1, 2,, n f i x j = F = x j x i F = f j x i x j x i (i, j = 1, 2,, n. (2 r = ϕ(t (t [α, β] β n β n f dr = f i (ϕ(tϕ F i(t dt = (ϕ(tϕ x i(t dt i α i=1 β d = α dt = F (b F (a. α i=1 F (ϕ(t dt = [F (ϕ(t]t=β t=α = F (ϕ(β F (ϕ(α (3 2.3.3 2.3.1 R n ( Ω Ω f : Ω R n f Ω 1 - f dr = ( f F r = ϕ(t (t [α, β] ϕ(α = ϕ(β f dr = F (ϕ(β F (ϕ(α =. ( Ω 1 - f dr = Ω a Ω x a, x Ω 1 - x 6 F (x := f dr x 6 B.3.1 24

F (x x ( well-defined F i {1, 2,, n} (x = f i (x 7 γ h ϕ(t := x + the i x i ( t 1 x x + he i x+he i x + γ h ϕ (t = he i, f e i = f i F (x + he i F (x = f dr f dr = f dr x +γ h x γ h = h = 1 1 f(x 1,, x i 1, x i + th, x i+1,, x n he i dt f i (x 1,, x i 1, x i + th, x i+1,, x n dt F (x + he i F (x h = 1 1 f i (x 1,, x i 1, x i + th, x i+1,, x n dt f i (x dt = f i (x (h. F (x = f i (x F = f F f x i f 8 R n f f i x j = f j x i (i, j = 1, 2,..., n 2.3.2 ( Ω R n f : Ω R n 1 - (2.4 f i x j = f j x i (i, j = 1, 2,..., n f R 3 1 - f rot f = f 7 n = 2 F/ x = f 1 8 f dr 25

Ω f i / x j = f j / x i f ( 1 f 2.3.1 ( Ω R n ( Ω (simply connected Ω Ω ( 2.3.4 ( R n, B(a; R, 3 1 R 3 \ {a}, R 2 \ {(x, ; x } 2 1 R 2 \ {a}, R 3 \ l (l 2.3.1 (Poincaré ( (2.4 1 ω := f 1 dx 1 + + f n dx n 1 Poincaré 9 (1 ( 1 2.3.5 ( R 3 f(r = (y + z, z + x, x + y T f R 3 e 1 e 2 e 3 ( f3 f = det x y z = y f 2 z, f 1 z f 3 x, f 2 x f T 1 y f 1 f 2 f 3 y (x + y (z + x z 1 1 = (y + z (x + y = 1 1 = z x x (z + x 1 1 (y + z y 9 Jules Henri Poincaré (1854 1912, Nancy Paris 19 2 (D.Hilbert 1 2.3.2 Poincaré 26

f x = (x, y, z T x ϕ(t = (tx, ty, tz T (t [, 1] 1 1 ty + tz x F (x := f dr = f(r(t ϕ (t dt = tz + tx y dt x tx + ty z = 1 t [x(y + z + y(z + x + z(x + y] dt = xy + yz + zx F = f ( 2.3.6 ( Ω = R 2 \ {} f(x, y = ( T y x 2 + y, x 2 x 2 + y 2 f = ( f dr = 2π ( 2.3.1 dz z = 2πi (i z 1/z \ {x; x } (log z R 2 \ {(x, ; x } f R 2 \ {(x, ; x } f 2.4 Green 27

Green (2.6 11 Green 2.4.1 (Green auchy-green R 2 1 - Jordan D D D Ω 1 - f = (P, Q T ( P x (2.5 f dr = rot f dx dy, rot f := det ( f = det = Q Q x P y. D ( Q (2.6 P dx + Q dy = D x P dx dy. y 2.4.1 ( Green Gauss-Green Green-Stokes ( Gauss 2 f n ds = div f dx dy D f = (Q, P T n ds = (dy, dx T (2.6 (2.5 Stokes 2 2 Gauss Stokes Green Green (Gauss, Stokes 1. Jordan ( 2. 12 ( 99% ( y 11 1 12 ( x y 28

2.4.2 (x y Green R 2 D D = {(x, y; x (a, b, ϕ 1 (x < y < ϕ 2 (x} = {(x, y; y (c, d, ψ 1 (y < x < ψ 2 (y} ϕ j [a, b] 1 - ψ j [c, d] 1 - x (a, b ϕ 1 (x < ϕ 2 (x, y (c, d ψ 1 (y < ψ 2 (y D 1 - P, Q ( Q P (x, ydx + Q(x, ydy = x P dx dy y D 1, 2, 3, 4 1 : r = (t, ϕ 1 (t T (t [a, b], 2 : r = (b, t T (t [ϕ 1 (b, ϕ 2 (b], 3 : r = (t, ϕ 2 (t T (t [a, b], 4 : r = (a, t T (t [ϕ 1 (a, ϕ 2 (a]. D y ( P ( b ϕ2 (x dx dy = P D y a ϕ 1 (x y dy dx b b = P (x, ϕ 1 (xdx P (x, ϕ 2 (xdx a a = P (x, y dx P (x, y dx 1 3 = P (x, y dx = P (x, y dx. 1 + 2 + 3 + 4 2, 4 dx/dt = P dx = P dx = 2 4 D x Q dx dy = Q(x, y dy D x ( A.1.1 ( ( 29

2.4.1 ( D {(x, y; 1 < x 2 + y 2 < 2} D 1 := {(x, y D; x >, y > }, D 2 := {(x, y D; x <, y > }, D 3 := {(x, y D; x <, y < }, D 4 := {(x, y D; x >, y > } D j (j = 1, 2, 3, 4 2.4.2 13 D = D 1 D 2 D 3 D 4. ( = D 1 + D 2 + D 3 + D 4, D j D j D j ( Q x P dx dy = y D = 4 j=1 D j ( Q x P 4 dx dy = P dx + Qdy y j=1 D j P dx + Qdy. 1 + 2 + 3 + 4 P dx + Qdy = 1 Jordan D 2.4.2 ( 2.4.2 2.4.1 ( ( [9] ( Green 3 2.4.2, A.1.1 2.4.2 ( 1 - D x dy = y dx = 1 x dy y dx = µ(d 2 (µ(d D 13 A.1.1 3

Q P (x, y =, Q(x, y = x x P y = 1, Q P (x, y = y, Q(x, y = x P y = 1, P (x, y = 1 2 y, Q(x, y = 1 2 x Q x P y = 1 1 Green x dy, y dx, x dy y dx 2 1 dxdy = µ(d D 2.4.3 (auchy 1 - Jordan D D 1 - f f(z = u(x, y+iv(x, y, z = x + iy (x, y R, u(x, y R, v(x, y R f(z dz = (u(x, y + iv(x, y(dx + i dy = u(x, ydx v(x, ydy + i v(x, ydx + u(x, ydy ( = v x u ( u dx dy + i y x + v dx dy. y D f auchy-riemann D u x = v y, u u = v x f(z dz = D dx dy + i dx dy =. D 2.4.4 (2 2 Ω f rot f = f 2 x f 1 y = Ω Ω D D Ω Green ( f2 f dr = x f 1 dx dy = dx dy = y D 2.3.1 f 2.3.2 2 ( 14 D 14 Jordan 31

3 3 1. 2. 3. Gauss ( Stokes ( 3.1 3.1.1 3 R (> ( (a z = f(x, y x = g(y, z y = h(z, x. z = R 2 x 2 y 2 ((x, y Ω := {(x, y; x 2 + y 2 < R 2 } R (b ( F (x, y, z = c. c x 2 + y 2 + z 2 = R 2 R ( 32

(c ϕ = (ϕ 1, ϕ 2, ϕ 3 T : D R 3 x = ϕ 1 (u, v, r = ϕ(u, v i.e. y = ϕ 2 (u, v, z = ϕ 3 (u, v x = R sin θ cos φ (3.1 y = R sin θ sin φ ((θ, φ (, π (, 2π z = R cos θ R ( φ = ( 3 3 (a (b z = f(x, y F (x, y, z := f(x, y z, c := F (x, y, z = c (a (c z = f(x, y ϕ 1 (u, v := u, ϕ 2 (u, v := v, ϕ 3 (u, v := f(u, v x = ϕ 1 (u, v, y = ϕ 2 (u, v, z = ϕ 3 (u, v (b F (a = c, F (a a (a F x (a F y (a F z (a 1 x = ϕ(y, z x 33

(c ϕ u (u, v ϕ v (u, v (u, v (a y u y v z u z v x u z u z v x u x v y u x v y v ( ( ( ( u v = ξ(y, z, u v = η(z, x, u v = ζ(x, y u, v (u, v T = ξ(y, z x = ϕ 1 (u, v = ϕ 1 (ξ(y, z (b F F ϕ (c r = ϕ(u, v u ϕ v 3.1.1 ( F F = c I F (x, y, z = c (x, y, z F (x, y, z F (x, y, z := ax + by + cz = d (a, b, c, d (a, b, c (,, F = (a, b, c T (a, b, c T R F (x, y, z := (x a 2 + (y b 2 + (z c 2 = R 2 (x, y, z T F (x, y, z = 2(x a, y b, z c T 2 2 34

3.1.2 3.1.1 ( D R 2 Jordan U D U R 2 ϕ: U R 3 r - ( 1 r S := ϕ(d = {ϕ(u, v; (u, v D} 2 r - S := ϕ( D = {ϕ(u, v; (u, v D} ( D := D \ D S S (i, (ii (i ϕ D p, q D, p q = ϕ(p ϕ(q. (ii p D (3.2 ϕ u ϕ (p (p. v 3.1.1 ( D U R 2 D, U ( ( I u v (u, v D {ϕ(u, v ; (u, v D} ϕ(u, v u {ϕ(u, v; (u, v D} ϕ(u, v v ϕ(u, v u v ϕ 1 ϕ 1 u v ϕ u = ϕ 2 ϕ u, v = ϕ 2 v ϕ 3 ϕ 3 u v 35

u v ϕ u ϕ v = ϕ 1 u ϕ 2 u ϕ 3 u ϕ 1 v ϕ 2 v ϕ 3 v = ϕ 2 u ϕ 3 u ϕ 1 u ϕ 3 v ϕ 3 ϕ 2 u v ϕ 1 v ϕ 1 ϕ 3 u v ϕ 2 v ϕ 2 ϕ 1 u v = (x 2, x 3 (u, v (x 3, x 1 (u, v (x 1, x 2 (u, v ( ϕ(u, v = (x 1, x 2, x 3 T S S ϕ(u, v (3.3 ϕ u ϕ 2 v = 1 i<j 3 ( 2 (xi, x j (u, v 3.1.2 ( (3.2 (3.2 (3.4 rank ϕ (p = 2. ϕ (p ϕ ϕ 1 ϕ 1 u v ϕ = ϕ 2 ϕ 2 u v ϕ 3 ϕ 3 u v 2 2 1 2 ( 2 (xi, x j (3.5 > (u, v 1 i<j 3 ( (3.3 (3.2 (3.2, (3.4, (3.5 36

3.1.2 (2 D Jordan U D U R 2 f : U R r - (1 r f D grad f D = {(x, y, f(x, y; (x, y D} x r - ϕ(x, y def. = y f(x, y 3 ϕ x = ϕ x ϕ y = 1 f x 1 f x, ϕ y = 1 f y = 1 f y f x f y 1 graph f D 3.1.3 ( R (> S := {(x, y, z; x 2 + y 2 + z 2 = R 2 } M := {(x, y, z S; x, y = } S \ M R sin θ cos φ U = R 2, D = (, π (, 2π, ϕ: U (θ, φ R sin θ sin φ R 3 R cos θ S \ M = ϕ(d - ϕ - ϕ D 1 1 ( (3.6 ϕ θ = ϕ θ ϕ φ = R2 R cos θ cos φ R cos θ sin φ R sin θ, sin 2 θ cos φ sin 2 θ sin φ sin θ cos θ ϕ φ = = R 2 sin θ < θ < π ϕ θ ϕ φ = R2 sin θ >. R sin θ sin φ R sin θ cos φ sin θ cos φ sin θ sin φ cos θ S \ M - 3 (x 1, y 1 (x 2, y 2 (x 1, y 1, f(x 1, y 1 (x 2, y 2, f(x 2, y 2. 37.

( S ϕ ϕ S ϕ ϕ θ =, π D θ =, π D θ φ ϕ(θ, φ = (,, ±R T ϕ ϕ θ ϕ φ = R2 sin θ θ =, π 3.1.3 ((3.6 (3.6 sin θ cos φ n := sin θ sin φ cos θ n = ϕ(θ, φ ϕ(θ, φ = ϕ(θ, φ R n ϕ(θ, ψ R 2 sin θ ϕ θ ϕ φ = ( ( 3.1.3 ( S = {(x, y, z T ; x 2 + y 2 + z 2 = R 2 } (3.1 (θ, φ (x, y, z T 1 1 (θ = φ (x, y, z T = (,, R T (a, (b, (c 38

R 3 S k - x S R 3 U 2 (i x U (ii U S k - (manifold S R 3 2 k - S = {(x, y, z T ; x 2 +y 2 +z 2 = R 2 } z > z = R 2 x 2 y 2 z < z = R 2 x 2 y 2 x > x = R 2 y 2 z 2 x < x = R 2 y 2 z 2 y > y = R 2 z 2 x 2 y < y = R 2 z 2 x 2 ( S - R 3 ( II 4 3.2 S = ϕ(d R 3 2 r - ϕ ϕ(u + t, v + s = ϕ(u, v + t ϕ u + s ϕ v + O(t2 + s 2 ((t, s (, D ( t, s A = {(u + t, v + s; (t, s [, t] [, s]} { x = ϕ(u, v + t ϕ ϕ(a = {ϕ(u + t, v + s; (t, s [, t] [, s]} u + s ϕ v } ; (t, s [, t] [, s] 4 39

ϕ u ϕ v t s ϕ(a ϕ u ϕ v t s. 3.2.1 (, U, D, ϕ, S r - S f : S R S f ds := f(ϕ(u, v ϕ u ϕ v du dv S µ c (S µ c (S := 1 ds = S S D D ϕ u ϕ v dudv ( ds := ϕ u ϕ v dudv 3.2.1 ( ds dγ dσ 3.2.2 ( ϕ xy 2 Jordan ( ϕ 3 =, ϕ = (ϕ 1, ϕ 2, T ϕ u ϕ v = ϕ 1 u ϕ 2 v ϕ 1 v ϕ 2 u R 3 S µ c (S = ϕ 1 ϕ 2 D u v ϕ 1 ϕ 2 v u dudv. ( ( R 2 x S µ(s = Φ(u, v def. = y µ(s = dx dy = det Φ (u, v du dv = S D 4 D ϕ 1 u ϕ 2 v ϕ 1 v ϕ 1 (u, v ϕ 2 (u, v ϕ 2 u du dv.

3.2.1 ( x 2 + y 2 + z 2 = R 2 3.1.3 ds = R 2 sin θ dθ dφ π µ c (S = R 2 sin θ dθ dφ = R 2 θ [,π],φ [,2π] 2π sin θ dθ dφ = 4πR 2. 3.2.2 (2 R 2 1 - f : U R x ϕ(x, y := y f(x, y ϕ x ϕ y = f x f y 1 D U D graph f D ds = ϕ x ϕ y 1 dx dy = + (f x 2 + (f y 2 dx dy µ c (graph f D = D. 1 + (f x 2 + (f y 2 dx dy. 3.2.3 ( xy 1 f : [a, b] R x S f(x > (x [a, b] ( x x ϕ: [a, b] R f(x cos θ R 3, θ f(x sin θ D := (a, b (, 2π S = ϕ(d 1 ϕ x = f ϕ (x cos θ, θ = f (x sin θ f(x sin θ f(x cos θ, ϕ x ϕ θ = f(xf (x f(x cos θ f(x sin θ ϕ x ϕ θ = f(x 1 + f (x 2 >. 41

S µ c (S = f(x b 1 + f (x 2 dx dθ = 2π f(x 1 + f (x 2 dx. x [a,b],θ [,2π] ( : a S = ϕ(d ( ϕ E := u, ϕ ( ϕ, F := u u, ϕ v, G := ϕ u ϕ v = EG F 2 ( ϕ v, ϕ v 3.3 ϕ: U R 3 n = ϕ u ϕ v ϕ u ϕ v n ds = ϕ u ϕ du dv v ds (3.7 ds := n ds = ϕ u ϕ v dudv. S f f S S f ds = S f n ds := D f(ϕ(u, v ( ϕ u ϕ dudv v ( 3.3.1 ( f n f 5 f f n ds ds 5 n e a a e a e ( R 3 i = 1, j = 1, k = a j = a 2, a k = a 3 42 1 a i = a 1,

( f n ds S (n S 6 S S Gauss 3.3.2 ( f = (f 1, f 2, f 3 T ( f ds = f 1 (ϕ(u, v (x 2, x 3 (u, v + f 2(ϕ(u, v (x 3, x 1 (u, v + f 3(ϕ(u, v (x 1, x 2 dudv. (u, v S D ( f 1 dx 2 dx 3 + f 2 dx 3 dx 1 + f 3 dx 1 dx 2. D 3.3.3 ( (x 2, x 3 n ds = ds = ϕ u ϕ (u, v dx du dv = (x 3, x 1 2 dx 3 v (u, v du dv = dx 3 dx 1. (x 1, x 2 dx 1 dx 2 (u, v ( 2 3.3.1 ( S S x 2 + y 2 + z 2 = R 2 f S (1 f(x, y, z := (α, β, γ T (. (2 f(x, y, z := (y, z, x T. (3 f(x, y, z := 1 (x 2 + y 2 + z 2 3/2 (x, y, zt. S (3.8 ds = R 2 sin θ dθ dφ. ( 3.1.3 ϕ θ ϕ φ = R2 sin θ sin θ cos φ sin θ sin φ cos θ 6 II 43

(3.9 ds = n ds = ϕ θ ϕ sin θ cos φ dθ dφ = sin θ sin φ R 2 sin θ dθ dφ φ cos θ ds = nds S r = (x, y, z T n = r r = 1 x sin θ cos φ y = sin θ sin φ R z cos θ ( α (1 f β ( γ 2π ( (2 f = S (3 f = S cos φ dφ = y z x f n ds = f n ds = 2π θ [,π] φ [,2π] sin φ dφ = π = θ [,π] φ [,2π] θ [,π] φ [,2π] = =. 1 (x 2 + y 2 + z 2 3/2 x y z α β γ R sin θ sin φ R cos θ R sin θ cos φ sin θ cos φ sin θ sin φ cos θ R 2 sin θ dθdφ = =. cos θ sin θ dθ = sin θ cos φ sin θ sin φ cos θ R 2 sin θ dθdφ R 3 (sin 2 θ cos φ sin φ + cos θ sin θ sin φ + cos θ sin θ cos φ sin θ dθdφ S f = 1 R n 2 S f n ds = 1 R 2 S ds = 4π. Gauss Gauss Gauss 44

3.4 Gauss 7 (Green 3.4.1 (Gauss Ω R 3 S = Ω 1 - Ω 1 - f div f dx 1 dx 2 dx 3 = f ds. Ω Ω S (flux 3.4.1.F.Gauss 8 (1839 G.Green 9 (1828, M.V.Ostrogradskii 1 (1831 ( 3.4.1 ( 3 1 f = F, Ω = (a, b b div f dx = F (x dx = F (b F (a Ω a 1 f n ds Ω = {a, b} S n = { 1 (x = b 1 (x = a Gauss 3.4.2 (Gauss u dx = un i ds (1 i 3 x i Ω S (n i n i 11 7 8 Johann arl Friedrich Gauss (1777 1855. 17 ( 2 1839 1813 9 George Green (1793 1841, Sneinton Sneinton. 1 Mikhail Vasilevich Ostrogradski (181 1862, Ukraine Poltava Poltava. 11 S 45

3.4.1 ( 3.3.1 3.3.1 (1, (2 Ω = B(; R Gauss (1 f div f =. f n ds = div f dx dy dz =. (2 f = (3 f = y z x S div f =. 1 (x 2 + y 2 + z 2 3/2 S x y z f n ds = Ω Ω div f dx dy dz =. div f = f S ( Gauss S f n ds S ( 4π 3.4.2 ( Gauss E Gauss E ρ div E = ρ/ε. S S Q/ε Q S Gauss 12 E ( E ds E ds Gauss S = E ds = div E dx dy dz = S Ω Ω ρ ε dx dy dz = Q ε 12 Gauss Gauss 46

3.4.3 ( Ω x S = Ω ds p(x n ds. p(x p(x = ρgx 3 + ρ g Ω S P := p n ds (3.1 P = S ρµ(ωg 13 ( (3.1 P =: (P 1, P 2, P 3 T P j = p n j ds (j = 1, 2, 3 P 3 f := (,, p T P 3 = f n ds = div f dx dy dz = ρg dx dy dz = ρg S = ρgµ(ω. Ω S Ω Ω dx dy dz P 1 = P 2 = 14 = Ω 3.5 Stokes ( 13 µ(ω ρ g ( 14 47

3.5.1 (Stokes, rot f ds = f dr. S S 3.5.1 George Gabiriel Stokes (1819 193, Skreen ambdridge William Thomson (1824 197, Lord Kelvin, Belfast Netherhall (185 7 2 Stokes ( [1] Green Gauss Stokes ω = dω M M Stokes R 2 1 - γ: (u, v T = ψ(t (t I = [a, b] γ ψ(i R 2 U 1 - ϕ: U R 3 ϕ ψ : I R 3 R 3 1 - Γ ϕ V f : U R 3 f dr f dr = Γ b a f(ϕ(ψ(t (ϕ (ψ(tψ (t dt x R 2, A R 2 3, u R 3 x (Au = (Au T x = (u T A T x = u ( T A T x = (A T x u (3.11 Γ f dr = b ϕ 1 (V f a [ ϕ (ψ(t T f(ϕ(ψ(t ] ψ (tdt. f(u := ϕ (u T f(ϕ(u (u ϕ 1 (V ϕ f f (3.11 f dr = f dr. ( ( (3.12 rot f ϕ = (rot f ϕ u ϕ v Γ γ Γ 48

( ϕ rot f n ds = rot f(ϕ(u, v S D u ϕ du dv = rot v f(u, v du dv D = f dr = f dr. D S (3.12 15 rot f = f 2 u f 1 ( v 3 ( 3 = f i (ϕ(u, v ϕ i f i (ϕ(u, v ϕ i u v v u i=1 i=1 3 = f i (ϕ(u, v 2 ϕ i u v + f i ϕ j ϕ i 3 x i=1 i,j j u v i=1 = ( f i ϕj ϕ i (ϕ(u, v x j u v ϕ j ϕ i v u i j ( fj = (ϕ(u, v f ( i ϕi (ϕ(u, v x i x j u (i,j=(1,2,(2,3,(3,1 ( ϕ = (rot f ϕ u ϕ. v f i (ϕ(u, v 2 ϕ i v u i,j ϕ j v ϕ j u f i ϕ j ϕ i x j v u ϕ i v 3.5.1 (Faraday ( Maxwell rot E = B t Faraday B n ds = E dr t S S Stokes 3.5.2 ( 15 49

A A.1 Green A.1.1 (y Green R 2 D D = {(x, y; x (a, b, ϕ 1 (x < y < ϕ 2 (x} ϕ j [a, b] 1 - x (a, b ϕ 1 (x < ϕ 2 (x D 1 - P, Q ( Q P (x, ydx + Q(x, ydy = x P dx dy y D 1, 2, 3, 4 1 : r = (t, ϕ 1 (t T (t [a, b], 2 : r = (b, t T (t [ϕ 1 (b, ϕ 2 (b], 3 : r = (t, ϕ 2 (t T (t [a, b], 4 : r = (a, t T (t [ϕ 1 (a, ϕ 2 (a]. [9] 2.4.2 ( P dx dy = P (x, y dx D y F (x, u, v := v u Q(x, y dy F x (x, u, v = v u Q x (x, y dy, F u(x, u, v = Q(x, u, F v (x, u, v = Q(x, v 5

( d ϕ2 (x Q(x, y dy dx ϕ 1 (x = d dx (F (x, ϕ 1(x, ϕ 2 (x = F x (x, ϕ 1 (x, ϕ 2 (x + F u (x, ϕ 1 (x, ϕ 2 (xϕ 1(x + F v (x, ϕ 1 (x, ϕ 2 (xϕ 2(x = ϕ2 (x ϕ 1 (x Q x (x, y dy Q(x, ϕ 1(xϕ 1(x + Q(x, ϕ 2 (xϕ 2(x. x [a, b] ( b ϕ2 (x Q (x, y dy dx a ϕ 1 (x x [ ] x=b ϕ2 (x b = Q(x, y dy + Q(x, ϕ 1 (xϕ 1(x dx = ϕ 1 (x ϕ2 (b ϕ 1 (b b a Q(b, y dy x=a ϕ2 (a ϕ 1 (a Q(x, ϕ 2 (xϕ 2(x dx. a Q(a, y dy + b a b Q(x, ϕ 1 (xϕ 1(x dx a Q(x, ϕ 2 (xϕ 2(x dx Q(x, y dy + Q(x, y dy + Q(x, y dy + Q(x, y dy = Q(x, y dy 2 4 1 3 2 r = (b, t T, t [ϕ 1 (b, ϕ 2 (b] dx/dt =, dy/dt = 1 ϕ2 (b ϕ2 (b Q(x, y dy = Q(b, t 1 dt = Q(b, y dy. 2 ϕ 1 (b ϕ 1 (b 4 Q(x, y dy = ϕ2 (a ϕ 1 (a Q(a, y dy. 1 r = (t, ϕ 1 (t T, t [a, b] dx/dt = 1, dy/dt = ϕ (t b b Q(x, y dy = Q(t, ϕ 1 (t ϕ 1(t dt = Q(x, ϕ 1 (xϕ 1(x dx. 1 a a b Q(x, y dy = Q(x, ϕ 2 (xϕ 2(x dx 2 a 4 A.1.1 (Green P, Q 1 Q x, P y Q x P y (Goursat-Bochner 51

A.2 Gauss ( Gauss [9] Green (x, y, z Gauss A.2.1 D R 2 1 - Jordan D ϕ 1, ϕ 2 D 1 - ϕ 1 < ϕ 2 (on D Ω := {(x, y, z; (x, y D, ϕ 1 (x, y < z < ϕ 2 (x, y} Ω 1 - f f := (,, f T ( f (A.1 dx dy dz = f n ds = f dx dy. Ω z S S f n ds 3 S T, S B, S S S (i S T ϕ(u, v := (u, v, ϕ 2 (u, v T ((u, v D (ii S B S B ϕ(u, v := (u, v, ϕ 1 (u, v T ((u, v D (iii S S r = (ξ(t, η(t T (t [a, b] ϕ(t, s := (ξ(t, η(t, s ((t, s {(t, s; t [a, b], ϕ 1 (ξ(t, η(t s ϕ 2 (ξ(t, η(t} D ( f ϕ2 (x,y f (x, y, z dx dy dz = (x, y, z dz dx dy Ω z D ϕ 1 (x,y z = f(x, y, ϕ 2 (x, y dx dy f(x, y, ϕ 1 (x, y dx dy. S T ϕ u ϕ ( v = ϕ T 2 u, ϕ 2 v, 1 ( ϕ f u ϕ = f(u, v, ϕ 2 (u, v. v f n ds = f(u, v, ϕ 2 (u, v du dv. S T D D D 52

( ϕ S B f u ϕ = f(u, v, ϕ 1 (u, v v f n ds = f(u, v, ϕ 1 (u, v du dv. S B D S S ϕ t ϕ s = ξ (t η (t ( ϕ f t ϕ =. s 1 S S f n ds =. = η (t ξ (t f (x, y, z dx dy dz = z Ω = = f(u, v, ϕ 2 (u, v du dv f(u, v, ϕ 1 (u, v du dv + D D f n ds f n ds + f n ds S T S B S S f n ds + f n ds + f n ds. S T S B S S Ω z x, y x y z Ω ( f x + g y + h dx dy dz = f dy dz + g dz dx + h dx dy. z Ω f := (f, g, h T div f dx dy dz = Ω S S f n ds A.2.1 ( Bourbaki Stokes Gauss 53

1 ( ϕ 1, ϕ 2 ϕ f u ϕ f(u, v, ϕ j (u, v ϕ j v ϕ j 1 54

2 [4] [15] [7] ( 3 [6] [1] (dx dy [dx, dy] artan 55

[1],, (196. [2],,, (25 [3],,, 23 7, 45 49. [4], 2, (22, [5],, (23. [6], I, II,, (1994, 1995. [7], 3, (199. [8], I, (198. [9], II, (1985. [1],, (1972. [11],, (1987. [12],, (. [13],, (1989, 199, 199, 1991. 2 3 [14],, (1997. [15],,,, III, (1986. [16],, (24. [17], Nebel (,, No.57 (22. [18], (,, (1976. [19],, (25. 56

[2] Oliver Heaviside, Electromagnetic Theory, Vol.I, The Electrician, London, 1893. [21] J.W.Gibbs and E.B.Wilson, Vector Analysis, Yale Univ.Press (191. [22] G.Green, An essay on the application of mathematical analysis to the theories of electricity and magnetism, 1828. W.Thomson 184 [23] J..Maxwell, A Treatise on Electricity and Magnetism, larendon Press (1873. [24] Georges de Rham, Variétés Defférentiables, Hermann, Paris (1955, 196. G.,, :, (1974. 57

B B.1 k - k - II ϕ: [a, b] R n k - l N, {t j } l j= s.t. j {1, 2,..., l} ϕ [tj 1,t j ] k ([t j 1, t j ] B.1.1 k I = [a, b], ϕ: I R n ϕ I k - l N, {t j } l j=1 s.t. j = 1, 2,..., l ϕ j a = t < t 1 < < t l = b ϕ [tj 1,t j ] k i =, 1,..., k lim ϕ(t (t (t j 1, t j ϕ j (t := ϕ(t j 1 + (t = t j 1 ϕ(t j (t = t j 1 t t j 1 + ϕ(i (t lim t t j 1 + ϕ(i (t ϕ j k ([t j 1, t j ] B.1.1 ϕ I ϕ j := ϕ [tj 1,t j ] ϕ(a +, ϕ(b lim x a+ ϕ(x, lim ϕ(x x b ϕ: [a, b] R n ϕ (a, ϕ (b ϕ (a = lim h + ϕ(a + h ϕ(a, ϕ ϕ(b + h ϕ(b (b = lim h h h 58

B.1.1 ϕ: [a, b] R n c [a, b] lim ϕ (t = A t c ϕ c ϕ (c = A. n = 1 c + h [a, b] h c c + h ξ h ϕ(c + h ϕ(c h = ϕ (ξ h. h ξ h c A ϕ c ϕ (c = A. B.1.1 ϕ: [a, b] R n [a, b] k 1 c [a, b] k lim ϕ (k (t = A t c ϕ (k 1 c ϕ (k (c = A. B.2 R I = [a, b] R n ϕ: I R n R n ( γ 1 ϕ: [a, b] R n : r = ϕ(t (t [a, b] ϕ k - k - ( k (R n R n ϕ(i = {ϕ(t; t I} ϕ(a ϕ(b (ϕ(a = ϕ(b (closed curve [ b, a] t ϕ( t R n 1 : r = ϕ 1 (t (t [a 1, b 1 ] 2 : r = ϕ 2 (t (t [a 2, b 2 ] 59

ϕ(t := { ϕ 1 (t (t [a 1, b 1 ] ϕ 2 (a 2 + (t b 1 (t [b 1, b 1 + b 2 a 2 ] R n 1 2 1 + 2 ϕ: [a, b] R n 1 - a = t < t 1 < < t N = b {t j } N j= ϕ [t j 1, t j ] ϕ j := ϕ [tj 1,t j ] 1 - t [t j 1, t j ] ϕ j(t 1 [t j 1, t j ] t ϕ j(t : r = ϕ(t (t [a, b] 1 - ϕ: [a, b] R n 1 - ( ϕ I D(I L( := sup ={t j } l j= D(I l ϕ(t j ϕ(t j 1 (rectifiable L( j=1 B.3 X X U 1, U 2 X = U 1 U 2 U 1 U 2 = U 1 = X, U 2 = U 1 =, U 2 = X X X X ( x, y X, ϕ: [, 1] X s.t. ϕ ϕ( = x, ϕ(1 = y R I I R n II B.3.1 R n 1 1 ϕ j (t j 1 ϕ j (t j 6

( R n B.3.2 Ω R n Ω 2 a, b Ω Ω := {x Ω; a x Ω }, Ω 1 := {x Ω; a x Ω } Ω Ω 1 = Ω, Ω Ω 1 =, a Ω. Ω x Ω x Ω Ω ε > s.t. B(x; ε Ω. B(x; ε y x a (a x x y y Ω. B(x; ε Ω Ω Ω 1 x Ω 1 x Ω Ω ε > s.t. B(x; ε Ω. B(x; ε y x a ( y a y x a x y Ω 1. B(x; ε Ω 1 Ω 1 Ω Ω = Ω Ω 1 =. a Ω Ω Ω 1 - Ω 1 - Ω 1 - Ω ( B(x; ε 61

B.4 B.4.1 X (simply connected ϕ: I = [a, b] X ( ϕ(a = ϕ(b Φ: I [, 1] X (i Φ(, = ϕ. t [a, b] Φ(t, = ϕ(t. (ii s [, 1] Φ(, s s [, 1] Φ(a, s = Φ(b, s. (iii Φ(, 1 ( 1 t [a, b] Φ(t, 1 = Φ(a, 1. B.4.1 ϕ [, 1] B.4.1 ( : a Ω π 1 (Ω, a Ω R 2 Ω Ω R 2 \ Ω compact ( R 2 ( B.5 Jordan Jordan (Jordan curve theorem ( ( 62

Jordan ( R 2 Jordan R 2 R 2 \ Ω 1, Ω 2 : R 2 \ = Ω 1 Ω 2, Ω 1 Ω 2 = Ω 1, Ω 2 Ω i p p Jordan p Ω i (i = 1, 2 (1993 ( : p.534 Jordan Ω Jordan Jordan Ω Jordan ( Jordan ( Jordan Riemann Jordan arathéodory Jordan Jordan 63

f (.1 f i x j = f j x i in Ω (i, j = 1, 2,..., n [6] 2 5 [9] (.1.1 1: (.1.1.1 1: x x F (x := f dr x n x a x R n ( [a j, b j ] (2 Green 1 grad F = f 2.3.1 (.1 ( [6] 2 Green 1 Fubini j=1 64

.1.2 2: x F (x = 1 F (x := f dr x ϕ(t := a + t(x a (t [, 1] f(a + t(x a (x a dt = N (x j a j j=1 (.1 F x k (x = = N 1 δ jk f j (a + t(x ady + j=1 1 f k (a + t(x adt + N (x j a j j=1 N (x j a j j=1 1 1 1 f j (a + t(x adt. f j x k (a + t(x a t dt f k x j (a + t(x a t dt. 2 2 = 1 d dt (f k(a + t(x a t dt 1 = [f k (a + t(x a t] 1 f k (a + t(x adt = f k (x 1 f k (a + t(x adt. F x k (x = f k (x..2 2: 3 1 - R n Ω (.1 f Ω : r = ϕ(t (t I = [a, b] L(f; 1 - L(f; = f dr 65

2 Ω : r = ϕ(t (t I = [a, b] E( E( ϕ(i U U 1 - F grad F = f in U E( f (.2 L(f; := F (ϕ(b F (ϕ(a (U, F (Ũ, F Ũ ϕ(i grad F = f in Ũ U Ũ grad(f F = grad F grad F = f f =. F F U Ũ F (ϕ(b F (ϕ(a = F (ϕ(b F (ϕ(a (.2 (U, F E( 1 - E( = 1 + 2 + + N j (j = 1, 2,..., N E( j ϕ(i ( Ω R n \ Ω ( d d > ( ϕ I δ > s.t. ( t, t I : t t δ ϕ(t ϕ(t < d. I = {t j } N j= < δ j ϕ [tj 1,t j ] t,e V j := B(ϕ(t j 1 ; d ϕ([t j 1, t j ] V j Ω j V j V j 1 V j f F j N = j ( j=1 j E( j (.2 : L(f; j = F j (ϕ(t j F j (ϕ(t j 1 f dr f ds 2 (j = 1, 2,..., N. = 66

= 1 + 2 + + N (.3 L(f; := N L(f; j = j=1 N (F j (ϕ(t j F j (ϕ(t j 1 j=1 ( 1 - I (.3 1, 2 1 < δ, 2 < δ 1, 2 i i.3 3: Ω f (.1 Ω 1 - f dr = [, 1] (i (iv Φ: [, 1] [, 1] Ω (i Φ(, = ϕ. t [, 1] Φ(t, = ϕ(t. (ii s [, 1] Φ(, s s [, 1] Φ(, s = Φ(1, s. (iii Φ(, 1 ( 1 t [, 1] Φ(t, 1 = Φ(a, 1. ψ(r := Φ(r, (r [, 1] Φ(1, r 1 (r [1, 2] Φ(3 r, 1 (r [2, 3] Φ(, 4 r (r [3, 4] = + γ + + ( γ γ r = Φ(1, t (t [, 1] 67

f dr = L(f, = L(f, + L(f, γ + L(f, Γ + L(f, γ e = L(f, + L(f, γ + L(f, γ = L(f, = f dr. auchy Goursat ( ts 2 [, 1] [, 1] {S n } n N a [, 1] [, 1] s.t. S n = {a}, n=1 n N L(f, n ( n Φ( S n a Ω n Φ(S n F L(f, n = F ( n F ( n =. f dr = 68

D ( TPO D.1 Green II ( D.1.1 (Green (f g + f g dx dy dz = Ω Ω (f g g dx dy dz = Ω Ω (f f + f 2 dx dy dz = Ω f dx dy dz = Ω f g n ds. ( f g n g f Ω Ω f n ds. n f f n ds ds. Gauss (Gauss f n = f n II http://www.math.meiji.ac.jp/~mk/lecture/ouyoukaiseki2/pde24-long.pdf D.2 Helmholtz R 3 Ω Ω - f u = f u 69