2 2 ( ) 28 4 6, 216 4 (http://nalab.mind.meiji.ac.jp/~mk/lecture/tahensuu2/)

1 3 1.1............................................. 3 1.1.1.................................... 3 1.1.2.................................... 4 1.1.3........................... 4 1.2 : R 3..................................... 5 1.3.................................... 1 1.3.1........................................ 1 1.3.2 (gradient)..................................... 1 1.3.3 (divergence).................................... 11 1.3.4 (rotation)..................................... 11 1.3.5 (Laplacian)............................... 13 1.3.6.................................... 13 1.3.7 :................................... 14 2 16 2.1 :....................................... 16 2.2...................................... 18 2.3 ( ).............................. 22 2.3.1............................................ 22 2.3.2........................................... 24 2.3.3...................................... 26 2.4..................................... 3 2.4.1.................................. 3 2.4.2 (1)........................... 31 2.4.3 (2)............. 32 2.5 Green.......................................... 39 3 46 3.1........................................... 46 3.1.1 3.................................... 46 3.1.2................................. 48 3.2................. 52 3.2.1........................................... 52 3.2.2................. 56 3.3.................................... 6 3.4 Gauss........................................ 65 3.5 tokes.......................................... 69 1

A 74 A.1 k......................................... 74 A.2.............................................. 75 A.3............................................. 76 A.4 Jordan....................................... 77 A.5..................................... 77 A.6.......................................... 78 B 8 B.1 1:......................... 8 B.1.1 1:........ 8 B.1.2 2:.................. 8 B.2 2:........................... 81 B.3 3:..................... 82 Gauss, Green, tokes 85.1 Green.......................... 85.2 Gauss..................... 87.3 Green.......................................... 89.4 Gauss............................... 89.5 tokes...................................... 92 D misc 94 D.1 P....................... 94 E 95 E.1...................................... 95 E.2 Gauss........................................ 96 E.3 tokes.......................................... 96 F 97 (PDF) http://nalab.mind.meiji.ac.jp/~mk/lecture/tahensuu2/ 27 TEX (216/4) 2

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

1.1.2 a a 2 T (transpose ) a = (a 1, a 2, a 3 ) T x r r x, y, z ( x 1, x 2, x 3 ) 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 1.1: F 1 F : I R (I R ) 4

F : 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 a 3, b = b 1 b 2 b 3 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 a 2 b 3 b 2 a 3 1 = a a 2 b 2 3 b 1 b 3 a 1. a 1 b 2 b 1 a 2 a 1 b 1 e 1 a b = det a 2 b 2 e 2, a 3 b 3 e 3 e 1 = 1, e 2 = 1, e 3 = 1. 3 a 2 b 2 a 3 b 3 e 1 a 1 b 1 a 3 b 3 e 2 + a 1 b 1 a 2 b 2 e 3 = a 2 b 3 b 2 a 3 (a 1 b 3 b 1 a 3 ) a 1 b 2 b 1 a 2 = a 2 b 3 b 2 a 3 a 3 b 1 b 3 a 1 a 1 b 2 b 1 a 2. 3 (outer product) Grassmann algebra (exterior product) 5

i a i, b i 1 2 3 1 a 1 a 2 a 3 a 1 3 1 2 b 1 b 2 b 3 b 1 2 ( ) a 2 a T 3 a b = b 2 b 3,, = a 3 a 1 b 3 b 1 a 1 a 2 b 1 b 2 a 2 b 3 a 3 b 2 a 3 b 1 a 1 b 3 a 1 b 2 a 2 b 1 1.2.1 ( ) a = (1, 2, 3) T, b = (4, 5, 6) T a b (1) ( ) 1 4 e 1 2 5 a b = 2 5 e 2 = 3 6 e 1 4 1 3 6 e 1 4 2 + 2 5 e 3 = 3 6 e 3. 3 6 3. (2) a b = ( 2 3 5 6, 3 1 6 4, 1 2 4 5 1 2 3 1 3 1 2 4 5 6 4 ) T = 3 6 3. 1. e 1 e 2, e 2 e 3, e 3 e 1 ( a b ) 2. (a b) c = a (b c) a, b, c 1.2.2 ( ) 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]} (a b) a, (a b) b, det(a b a b), a b =. 6

a b b a 1.2: a b a b a, b, a b (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) (5) (3) a b 1 x s.t. a, b, x 1 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. = a b a, b 1 a, b 1 3 a, b, a b 3 ( ) V a b V = a b. V = det(a b a b). 7

a b = det(a b a b) = a b 2 a b (> ) = a b. (5) a b θ cos θ = (a, b) a b = a b sin θ = a b 1 cos 2 θ = a 2 b 2 (a, b) 2 = (a 2 1 + a2 2 + a2 3 )(b2 1 + b2 2 + b2 3 ) (a 1b 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. 3 ( ) a b a b 3. 3 a, b, c 1 (a b + b c + c a) abc 2 4. 3 4 ( ) 4 ( p. 97) 5. ( ) a (b c) = b (c a) = c (a b) = det (a b c) 6. (, Lagrange ) a (b c) = (c a) b (a b) c 7. a (b c) + b (c a) + c (a b) = (Jacobi ). ( Jacobi Lie ) 8. ( ) (a b, c d) = (a, c)(b, d) (a, d)(b, c) 1.2.1 (R n ) R n (n 4) R n n 1 a 1, a 2,, a n 1 R n x det(a 1, a 2,, a n 1, x) R 4 8

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 ) 1.2.2 ( ) 3 A(1, 2, 3), B(2, 1, ), (, 2, 1) AB 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 = AB = 1 AB A = 1 4 2 2 2 + 7 2 + ( 1) 2 = 66 2. A(1, 2, 3) (4, 7, 1) T 4x + 7y z = 15. 4(x 1) + 7(y 2) + ( 1)(z 3) = 1.2.3 ( ) 1 f(t), g(t) ( ) (f(t) g(t)) = f (t) g(t) + f(t) g (t) t m, r(t), f (Newton ) f mr (t) = f f = f(r)r f (central force) 3 a a a = ( ) d 1 dt 2 r(t) r (t) = 1 ( r (t) r (t) + r(t) r (t) ) = 1 2 2 r(t) f m = 1 2 r(t) f m r(t) = 9 4 7 1.

1 2 r(t) r (t). 5 ( Kepler ) 1.3 grad, rot, div ( ) grad ( 1 ) div, rot 1.3.1 R n ( ) := (nabla) 6 Hamilton x 1 x 2. x n 1.3.2 (gradient) R n Ω 1 F : Ω R F = x 1 x 2. x n F := F : Ω R n F Ω n F f (gradient) F grad F F x 1 F x 2. F x n 5 r(t) (mr (t)) 6 ( ) (Nebel) ( [22]) 1

: F a Ω F (a) F ( ) {x Ω; F (x) = h}, h := F (a) a F 7 3 F (x, y, z) = (a, b, c) F x (a, b, c)(x a) + F y (a, b, c)(y b) + F z (a, b, c)(z c) =. 1.3.3 (divergence) R n Ω 1 f : Ω R n x 1 f 1 f = x 2 f 2. := f 1 + f 2 + + f n x 1 x 2 x n. x n f n f : Ω R f Ω ( ) f f (divergence) f div f div f = (solenoidal) div f Gauss 1.3.1 f 2 ( ) div f = div f <, div f > 1.3.4 (rotation) n = 3 R 3 a, b a b R 3 Ω 1 f : Ω R 3 f := ( x 2 f 2 x 3 f 3, x 3 f 3 x 1 f 1, x 1 f 1 x 2 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 f : Ω R 3 f Ω 3 f f (rotation) f rot f curl f f = det x 1 f 1 e 1 x 2 f 2 e 2 x 3 f 3 e 3 7 chwarz F h F h h F 1 11

f rot f (vorticity) rot f = (irrotational) (lamellar) rot f tokes 1.3.1 ( ) rot f 1 x 2 f 2 x 3 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 1.3.1 (2 ) 2 f = (f 1, f 2 ) T f = rot f := f 2 x 1 f 1 x 2 3 f 2 f = det x 1 f 1 e 1 x 2 f 2 e 2 x 3 e 3 ψ f(r) = (f 1 (x 1, x 2 ), f 2 (x 1, x 2 ), ) T = det rot ψ := ( x 1 f 1 x 2 f 2 ) ( ψ, ψ ) T x 2 x 1 e 3 = (,, f 2 f ) T 1 x 1 x 2 ( ψ rot ψ (stream function) ) 3 rot 1.3.2 2 ( ) ( ) ( f = (f 1, f 2 ) T ) f 2 f 1 x 1 x 2 (3 x 1, x 2, x 3 ) 12

1.3.5 (Laplacian) R n Ω 2 F : Ω R F := n i=1 2 F x 2 i F F (Laplacian) = n j=1 2 (, Laplace operator) x 2 j F F (harmonic function) R n Ω 2 f : Ω R n f f := ( f 1,, f n ) T ( ) 1.3.2 ( ) F = div(grad F ) = ( F ) 2 ( ) 1.3.6 1.3.2 ( ) f R 3 2 F R 3 2 (1)-(4) (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) (1) div f (2) rot f div (grad F ) = rot f = div f = f = grad F = n j=1 x j f j = n j=1 f j x j ( F,..., F ) T x 1 x n n j=1 F = x j x j n j=1 2 F x 2 j ( f3 f 2, f 1 f 3, f 2 f ) T 1 x 2 x 3 x 3 x 1 x 1 x 2 = F. 13

f = grad F = rot (grad F ) = ( F, F, F ) T x 1 x 2 x 3 F F x 2 x 3 x 3 x 2 F F x 3 x 1 x 1 x 3 F F x 1 x 2 x 2 x 1 F 2 F 2 rot (grad F ) = =. (3), (4) ( ) (4) 1.3.3 9. 1.3.2 1. 1 grad, div, rot 2 11. 1 u, v 12. F (x, y, z) = div (u v) = (rot u, v) (rot v, u) 1 grad F, F ( ) x 2 + y 2 + z2 13. 3 f rot f = (f 1 ( ) x F f(r) = F (r)r, f(x, y, z) = F x 2 + y 2 + z 2 y z ). 1.3.7 : ( ) 8 1.3.3 (Maxwell (1873), ) E, B, ρ, j (1.1) E = ρ, E = B ε t, B =, c2 B = j + E ε t 8 14

9 (c, ε 1 ) (ρ, j ) E =, E = B t, B =, c2 B = E t 1.3.2 (4) 1 2 E c 2 t 2 = B ( B) = = ( E) t t = E ( E) = E = E, 1 2 B c 2 t 2 = 1 c 2 t ( E) = 1 c 2 E t = 1 c 2 ( c 2 B ) = ( B) = B ( B) = B = B. E, B c (wave equation) Maxwell (1831 1879) (1864 ) 1887 Hertz (Heinrich Rudolph Hertz, 1857 1894) ( ) 14. u = u(x, y, z, t) ρ 2 u = µ u + (λ + µ) grad (div u) t2 ( ρ, µ, λ ) p := div u, s := rot u ρ 2 p = (λ + 2µ) p t2 (P ), ρ 2 s t 2 = µ s ( ) P (primary wave) (secondary wave) ( p. 94) Gauss.4 9 Maxwell Heaviside 1 MK c = 299792458 m/s ( ), ε = 17 4πc 2 8.854 1 12 F/m. 15

2 2.1 : 1 Ω R n R Ω Ω (curve) ( φ: [α, β] Ω ) 1 γ, Γ φ φ φ([α, β]) (image) (spur) φ(α) φ(β) (φ(α) = φ(β)) (closed curve) ( ) Jordan (Jordan curve) φ = φ 1. φ n m φ j m (m N {, }) m φ (t) = φ (j) (t) = φ 1 (t). φ n(t), φ (t) = φ (j) 1 (t). φ (j) n (t) φ 1 (t). φ n(t), (j =, 1,..., m) φ φ (t) φ (t) φ(t) φ (t) 1 ( ) 2.1.1 a, b φ(t) = (a cos t, b sin t) T (t [, 2π]) x 2 a 2 + y2 b 2 = 1 1 ( ) ( ) ( ) (1, ) T 1 16

(a cos t, b sin t) ( ) φ a sin t (t) = b cos t cos, sin φ. m (m 1) m m α = t < t 1 < < t l = β {t j } l j= [t j 1, t j ] φ φ [tj 1,t j ] m t j φ (k) (t j ), φ (k) (t j + ) φ (k) (t j ) [t j 1, t j ] 1 ( ) 1 ( ) 2.1.2 R 2 4 (, ), (1, ), (1, 1), (, 1) φ(t) := (t, ) T (t [, 1]) (1, t 1) T (t [1, 2]) (3 t, 1) T (t [2, 3]) (, 4 t) T (t [3, 4]) y (1,1) O 1 x 2.1: [ β, α] t φ( t) R n 1 : r = φ 1 (t) (t [α 1, β 1 ]) 2 : r = φ 2 (t) (t [α 2, β 2 ]) { φ φ(t) := 1 (t) (t [α 1, β 1 ]) φ 2 (α 2 + (t β 1 )) (t [β 1, β 1 + β 2 α 2 ]) R n 1 2 1 + 2 1 2 1 + 2 17

2 1 2.2 2.2: 1 = 2 1 + 2 ( ) 2.2.1 ( ) : r = φ(t) (t [α, β]) R n 1 f f ds := β α f(φ(t)) φ (t) dt f (line integral, path integral) a f 1 (2.1) ds = β α φ (t) dt ( ) ds := φ (t) dt (, line element) a contour integral f ds f ( ) φ(t) t φ (t) φ (t) (2.1) (2.1) 18

(1) x = f(t), y = g(t) (α t β) β (dx ) 2 ( ) dy 2 β + dt = f α dt dt (t) 2 + g (t) 2 dt. α (2) y = f(x) (a x b) b ( ) dy 2 b 1 + dx = 1 + f a dx (x) 2 dx. a (2) (1) (2.1) [α, β] = {t j } N j= L := N φ(t j ) φ(t j 1 ), j=1 L := sup{l ; [α, β] } L < (rectifiable) L φ 1 (2.1) φ(t j ) φ(t j 1 ) = φ (t j 1 )(t j t j 1 ) + o( t j t j 1 ), φ(t j ) φ(t j 1 ) = φ (t j 1 ) (t j t j 1 ) + o( t j t j 1 ) 2 (2.1) 2.2.2 ( (cycloid) ) a φ(t) = (a(t sin t), a(1 cos t)) T (t [, 2π]) ( ) φ (t) = (a(1 cos t), a sin t) φ (t) = a 2 (1 cos t) 2 + a 2 sin 2 t = a 2(1 cos t) = 2a sin 2 t 2 = 2a sin t 2. 2π φ (t) 2π dt = 2a sin t dt = 8a. 2 19

ϕ(t 2 ) ϕ(t 1 ) ϕ(t ) ϕ(t N ) ϕ(t j 1 ) ϕ(t j ) 2.3: L y (a(θ sinθ),a(1 cosθ)) (aθ, a) O x 2.4: (x, y) = (a(θ sin θ), a(1 cos θ)) (θ [, 2π]) 1.8 2 1.6 t-sin(t), 1-cos(t) 1.4 1.2.8 1.6.4.2-8 -6-4 -2 2 4 6 8 2.5: (x, y) = (t sin t, 1 cos t) (gnuplot ) set parametric;set size ratio -1;plot [-8:8] t-sin(t),1-cos(t) 2

2.2.3 ( ) r = f(θ) (θ [a, b]) ( ) ( ) ( ) x r cos θ f(θ) cos θ = = y r sin θ f(θ) sin θ b (dx ) 2 ( ) dy 2 b + dθ = f dθ dθ (θ) 2 + f(θ) 2 dθ. r = a(1 + cos θ) (θ [, 2π]) 2π a π ( a sin θ) 2 + (a(1 + cos θ)) 2 dθ =... = 2 2a cos θ dθ = 8a. 2 2.2.1 a ( ) 3 4 (elliptic integral) (elliptic function) 19 15. y = x 2 (x [, 1]) 2 ( ( 2 5 + sinh 1 2 ) /4 = [ 2 5 + log ( 2 + 5 )] /4 = 1.47894285754 ) 16. a, b x2 a 2 + y2 b 2 = 1 ( ) 2.2.2 ( f ds ) : r = φ(t) (t [α, β]) 1 ( 1 ) t [α, β] α t σ(t) := s ( ) : t [α, β] s = σ(t) = t α φ (τ) dτ. t α φ (τ) dτ. ds dt = σ (t) = φ (t). σ : [α, β] R 1 t [α, β] φ (t) σ (t) > (t [α, β]). 2 2 1 21

σ : [α, β] [, L] t = σ 1 (s) 1 ψ := φ σ 1, ψ(s) := φ(σ 1 (s)) (s [, L]) 1 ( ) ( ) r = ψ(s) (s [, L]) s ψ (2.2) ψ (s) = 1 (s [, L]) 1 = (2.2) f ds = L f(ψ(s)) ds ( ds ds ) 2.3 ( ) 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 dr = (dx 1,, dx n ) T f dr = (?) f 1. f n dx 1. dx n = f 1 dx 1 + + f n dx n 2.3.1 ( ) 22

( ) ( f ) r W W = fr f r W = f r. f W W N f(r j 1 ) r j, r j := r j r j 1. j=1 r j [α, β] = {t j } N j= r j = φ(t j ) (j =, 1,..., N) W = f dr := lim N f(r i ) r i 1 β α i=1 f(φ(t)) φ (t) dt f f r θ r 2.6: W = fr 2.7: W = f cos θ r = f r f(r j 1 ) r N r j r j r j 1 r 1 r 2.8: f r 23

Ω Ω f : Ω ( ) f(z) dz = lim f(z) dz = N f(z i )(z i z i 1 ) i=1 ( [α, β] = {t j } N j=1 z j = φ(t j ) ) u(x, y) := Re f(z), v(x, y) := Im f(z) (x = Re z, y = Im z) (u + iv)(dx + i dy) = u dx v dy + i v dx + u dy 2.3.2 2.3.1 ( ) R n 1 : r = φ(t) (t [α, β]) f : R n f dr := β α f(φ(t)) φ (t) dt f ( ) f = (f 1,..., f n ) T f 1 dx 1 + f 2 dx 2 + + f n dx n 2.3.2 f(x, y, z) = y + z z + x x + y ( ) φ(t) := (t, t 2, t 3 ) T, : r = t t 2 t 3 (t [, 1]) f dr f(φ(t)) = f(t, t 2, t 3 ) = (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. f dr = 1 (3t 2 + 4t 3 + 5t 4 ) dt = 3 1 3 + 4 1 4 + 5 1 5 = 3. 24

2.3.3 P (x, y) = P dx + Q dy ( ) y x x 2, Q(x, y) = + y2 x 2, : r = φ(t) = + y2 ( cos t sin t ) (t [, π]) sin t P (φ(t)) = P (cos t, sin t) = cos 2 t + sin 2 = sin t, t cos t Q(φ(t)) = Q(cos t, sin t) = cos 2 t + sin 2 = cos t, t dx dt = sin t, dy dt = cos t P dx dt + Qdy dt = sin t ( sin t) + cos t cos t = 1. P dx + Q dy = π 1 dt = π. 2.3.1 ( ) 3 dr ds dx t := f ds, f dx 1 φ (t) φ (t) ds f t ds 4 (f, dr), (f dr), f dr ( ) 2.3.4 ( ) 3 f ds = f dr = β α β α P dx + Q dy = f(φ(t)) ds dt dt, f(φ(t)) dr dt dt, β α ds dt = φ (t), dr dt = φ (t), [ P (x(t), y(t)) dx + Q(x(t), y(t))dy dt dt ] dt (n = 2 ) 3 ( ) β 4 φ (t) f t ds = f(φ(t)) φ (t) dt = α φ (t) β α f(φ(t)) φ (t) dt = f dr. f n dσ 25

β α d d dt dt 1 2 5 2 3 2.3.3 ( ) 2.3.5 ( ) ( 1, ) (1) (f + g) dr = f dr + g dr. (2) (λf) dr = λ f dr. (3) f dr = 1 + 2 f dr + 1 f dr. 2 (4) f dr = f dr. (5) (3) f dr f ds. (1), (2) c a = b a c + b (4) r = φ(t) (t [α, β]) r = φ( t) (t [ β, α]) f dr = α β f(φ( t)) d (φ( t)) dt = dt α β f(φ( t)) φ ( t) dt. t = u (u [α, β]) dt = du, t = β u = β, t = α u = α α α f dr = f(φ(u) φ (u) ( 1)du = f(φ(u) φ (u) du = β β α f(φ(u) φ (u) du = f dr. β 5 t 1 2 26

(5) chwarz f(φ(t)) φ (t) f(φ(t)) φ (t) 2.3.6 f ds (5) f ds = (a) ( ) f ds f f dr f f ds ( ) ( 1 ) (b) t := 1 φ (t) φ (t) f dr = f t ds f t t t 1 ( y 2.3.7 f(x, y) = x 2 ), 1 : r = φ(t) = ( f dr, f dr 1 2 ( ) ( ) f(φ(t)) = f(t, t 3/2 ) = 1 f dr = 1 f(ψ(u)) = f(u 2, u 3 ) = 2 f dr = 1 t 3/2 t 2, φ (t) = t t 3/2 ) 1 3 2 t1/2 (t [, 1]), 2 : r = ψ(u) = f(φ(t)) φ (t) = t 3/2 1 + t 2 3 2 t1/2 = t 3/2 + 3 2 t5/2. (t 3/2 + 32 ) [ 2 t5/2 dt = 5 t5/2 + 2 7 3 ( u 3 (u 2 ) 2 ) = ( u 3 u 4 ] 1 ( u 2 u 3 2 t7/2 = 2 5 + 3 7 = 2 7 + 3 5 35 ) ( ), ψ 2u (u) = 3u 2 f(ψ(u)) ψ (u) = u 3 2u + u 4 3u 2 = 2u 4 + 3u 6. ( 2u 4 + 3u 6) du = ) = 29 35. [ 2 5 u5 + 3 ] 1 7 u7 = 2 5 + 3 7 = 2 7 + 3 5 = 29 35 35. ( ) 27 (u [, 1])

y 5 4 1 O 1 x 2.9: 5 = 4 ( ) 2.3.8 5 ( ) cos t 1 : r = (t [, π]) sin t ( ) cos(πt) 2 : r = (t [, 1]) sin(πt) ( ) cos(πt 2 ) 3 : r = sin(πt 2 (t [, 1]) ) ( ) t 4 : r = (t [ 1, 1]) 1 t 2 ( ) t 5 : r = (t [ 1, 1]) 1 t 2 x 2 + y 2 = 1 (y ) 1, 2, 3, 4 (1, ) ( 1, ) 5 ( 1, ) (1, ) 5 = 4 5 4 f = (f 1, f 2 ) T f dr = j f dr 1 (j = 1, 2, 3, 4) ( ) 5 = 4 f dr = 5 f dr = 4 f dr. 1 28

2.3.9 ( ( ) ) f R n Ω : r = φ(t) (t [α, β]) Ω 1 η : [a, b] [α, β] 1 ψ(τ) := φ(η(τ)) (τ [a, b]) f dr = f dr. ψ (τ) = φ (η(τ))η (τ) 6 f dr = = b a b a f(ψ(τ)) ψ (τ) dτ f(φ(η(τ))) φ (η(τ))η (τ) dτ. t = η(τ) (α = η(a), β = η(b) ) f dr = β α f(φ(t)) φ (t) dt = f dr. ( 1 Jordan ψ φ ψ = φ η 1 η ) ( ) x 2 + y 2 = 1 (y ) y + z 2.3.1 f(x, y, z) = z + x, : (,, ), (1,, ), (1, 1, ), (1, 1, 1) x + y f dr ( ) (,, ) (1,, ) γ 1, (1,, ) (1, 1, ) γ 2, (1, 1, ) (1, 1, 1) γ 3 = γ 1 + γ 2 + γ 3 f dr = 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. 6 dr dτ = dt dτ dr dt 29

2.4 (1 R ) 7 2.4.1 Ω R n F : Ω R 1 F ( ) F F : Ω R n 1 ( ) 2.4.1 R n Ω f : Ω R n F (x) = f(x) (x Ω) 1 F F f (potential) 2.4.1 ( ) 1 ω = f 1 dx 1 + + f n dx n ω = df := F dx 1 + + F dx n F ω ω x 1 x n (exact) 2.4.2 ( ) f(x 1, x 2, x 3 ) := g (g ) F (x 1, x 2, x 3 ) := gx 3 F = f F f 2.4.3 ( ) f(r) := GM r 3 r (M, G, r R3 \ {}) f F (r) := GM r 7 Lagrange (1773) Green (1828) 3

17. 2.4.3 2.4.1 ( ) f V = f V f (conservative force) V f ( ) 1 R I f : I R F = f ( 1 ) F f F f F (x) := F (x) + F F f f : I R (1), (2) ( ) (1) F f a, b I b a f(x) dx = [F (x)] b a = F (b) F (a). (2) a I F (x) := x a f(t) dt (x I) F F f ( 1 ) F f F (x) := F (x) + (x Ω) F f 2.4.2 (1) (1), (2) (1) ( ) 2.4.4 ( ) Ω R n f : Ω R n F f Ω 1 f dr = F (b) F (a) (a, b ) 31

r = φ(t) (t [α, β]) f dr = = β α i=1 β α n f i (φ(t))φ i(t) dt = β α n i=1 F x i (φ(t))φ i(t) dt d F (φ(t)) dt = [F (φ(t))]t=β t=α = F (φ(β)) F (φ(α)) dt = F (b) F (a). 2.4.5 ( ) Ω R n f : Ω R n 1 (2.3) f i x j = f j x i (i, j = 1, 2,, n) n = 3 rot f = n = 2 rot f = f 2 f 1 = x 1 x 2 f F f i = F x i (i = 1, 2,, n) f 1 F 2 2 2 f i x j = F = x j x i F = f j x i x j x i (i, j = 1, 2,, n). n = 3 2 F rot (grad F ) = 1.3.2 ( ) 2.4.6 n 2 (2.3) f(x 1, x 2 ) := ( x 2 x 1 ) (2.3) f 1 x 2 = 1, f 2 x 1 = 1 2.4.3 (2) (2) 32

2.4.7 R n ( ) Ω Ω f : Ω R n f Ω 1 f dr = a Ω F (x) := f dr (x Ω), x (x Ω 1 a, x ) F f ( ) 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 8 F (x) := f dr x F (x) x ( well-defined) x x a x Ω 1 x + ( x) Ω x +( x f dr =. f dr = f dr ) x x i {1, 2,, n} F (x) = f i (x) 9 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 8 A.2.1(p.75) 9 n = 2 F (p, q) = f1(p, q) ε > s.t. B((p, q); ε) Ω. x x (p ε, p + ε) x = (x, q) x a (p, q) x (x, q) F (x, q) = F (p, q) + x p f 1(t, q) dt F (x, q) = f1(x, q) ( 1 (2) ) x 33

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 i (x) = f i (x) F = f F f f 1 R n f f i x j = f j x i (i, j = 1, 2,..., n) 2.4.8 ( ) Ω R n f : Ω R n 1 (2.4) f i x j = f j x i (i, j = 1, 2,..., n) f n = 3 (2.4) rot f = n = 2 ( ) rot f = f 2 f 1 = x 1 x 2 B Ω f i / x j = f j / x i f ( 1 ) f 2.4.9 ( ( )) Ω R n ( ) Ω (simply connected) Ω Ω ( A.3.1 (p.76) ) 2.4.1 ( ) R n, B(a; R), 11 12 3 1 R 3 \ {a}, R 2 \ 1 f dr ( 2.4.13 ) F (x) ( ) F = f ( ) 11 Ω Ω 2 Ω ( a Ω, b Ω, t [, 1] (1 t)a + tb Ω) 12 Ω (star-like, star-shaped) p Ω, x Ω, t [, 1] (1 t)p + tx Ω 34

{(x, ); x } R 2 Jordan ( Jordan ) 2 1 R 2 \ {a}, 2 { (x, y) R 2 r1 2 < (x a)2 + (y b) 2 < r2 2 }, R 3 \ l (l ) ( ) Ω Ω 2.1: : 1 2.4.2 (Poincaré ) ( ) (2.4) 1 ω := f 1 dx 1 + + f n dx n (closed form, dω = ) 1 Poincaré 13 14 (1 ) ( ) 15 Poincaré = 2.4.11 ( ) R 3 f(r) = (y + z, z + x, x + y) T 13 Poincaré 14 Jules Henri Poincaré (1854 1912, Nancy Paris ) 19 2 (D. Hilbert ) 15 2.4.8 Poincaré 35

f R 3 x f 1 e 1 f = det y f 2 e 2 = z f 3 e 3 = y (x + y) (z + x) z (y + z) (x + y) z x x (z + x) (y + z) y ( f3 y f 2 z, f 1 z f 3 x, f 2 x f ) T 1 y = 1 1 1 1 1 1 = f x = (x, y, z) T x f(φ(t)) = f(tx, ty, tz) = F (x) := f dr = x = 1 φ(t) = (tx, ty, tz) T (t [, 1]) 1 t(y + z) t(z + x) t(x + y) f(φ(t)) φ (t) dt = t [x(y + z) + y(z + x) + z(x + y)] dt = xy + yz + zx, φ (t) = 1 ty + tz tz + tx tx + ty x y z x y z dt F = f ( ) 2.4.2 ( ) E B Maxwell (1.1) (2.5) div E = ρ ε, rot E =, div B =, c 2 rot B = j ε E B rot E = E 1 ϕ E = grad ( ϕ). Maxwell (2.5) 1 div (grad( ϕ)) = ρ ε, (2.6) ϕ = ρ ε. ϕ (2.6) Poisson (Poisson equation) Poisson ( ) 36

2.4.12 ( ) R 3 Ω f ( ) T x f(x, y, z) = (x 2 + y 2 + z 2 ) 3/2, y (x 2 + y 2 + z 2 ) 3/2, z (x 2 + y 2 + z 2 ) 3/2 rot f = Ω f 16 F (x, y, z) := 1/ x 2 + y 2 + z 2 17 p = (p,, ) T q = (q,, ) T ( pq > ) f dr = 1 p 1 q q x ( dx ) p x3 1 γ f dr = γ γ φ(t) (t I) φ(t) 2 = r 2 (r ) φ(t) φ (t) = f(φ(t)) φ (t) = (t I) f dr = f(φ(t)) φ (t) dt =. γ I e := (1,, ) T x = (x, y, z) T R 3 x e (q,, ) T ( q := x 2 + y 2 + z 2 ) Γ q (q,, ) x q γ q,x Γ q + γ q,x f n dσ = f n dσ + f n dσ = x Γ q γ q,x f F (x, y, z) := 1 ( 1 1 1 ) + = 1 1 q q = 1 1 x 2 + y 2 + z. 2 1 (1 ) x 2 + y 2 + z2 18. ( f f i = f j n 3 R n \ {} x j x i n = 2 R n \ {} ( ) ( ) ) 2.4.13 ( ) Ω = R 2 \ {} f(x, y) = ( ) y x 2 + y 2, x T x 2 + y 2 f = ( ) f dr = 2π 16 17 F = f 37

( ) 2.4.7 dz z = 2πi (i ) z 1/z \ {x; x } (log z ) R 2 \ {(x, ); x } f 19. R 2 \ {(x, ); x } f ( : tan 1 y x ) 2.4.3 1 n f (1) ( ) f i x j = f j x i (i, j = 1, 2,..., n) ( ( ) n = 3 rot f = n = 2 rot f = f 2 f 1 = ) x 1 x 2 ( ) (2) (2) f Ω 18 (3) ( ) F (x) := f dr (x Ω) x x Ω a x Ω 1 19 F = f (3) (( ) Ω ) f dr Ω Ω ( ) ( ) 18 R n ( ) 2 Ω = R 2 \ {a} 19 Ω = R n a =, a x φ(t) = tx (t [, 1]) 38

2. Ω := R 2 \ {} f : Ω R 2 R 2 ( : ) 21. f R 3 rot f = ( ) F (x) F = f ( 1 ) 2.5 Green 1 Green D 2.11: D D 2.5.1 (Green auchy-green ) R 2 1 Jordan D D D Ω 1 f = (P, Q) T (2.7) f dr = D rot f dx dy, rot f := det ( f) = det ( x y ) P Q = Q x P y. ( Q (2.8) P dx + Q dy = x P ) dx dy. y D ( ) 39

2.5.1 1. Jordan (A.4 ) ( ) D R 2 D 1 Jordan D D 2. 2 ( 99% ) ( ) 2.5.2 (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, y)dx + Q(x, y)dy = x P ) dx dy y 1, 2, 3, 4 D 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 = y D a b φ 1 (x) ) P y dy dx = P (x, φ 1 (x))dx P (x, φ 2 (x))dx 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 2 ( ) x y b 4

y 3 y = ϕ 2 (x) 4 D 2 1 y = ϕ 1 (x) O a b x 2.12: 1 + 2 + 3 + 4 D D x Q dx dy = Q(x, y) dy x D (.1.1(p.85)) 2.13 ( ) 3 2 D 1 2.13: 3 f dr = j j=1 D rot f dx dy 2.5.2 41

( ) 2.5.3 ( ) 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 > }, 1 := x 2 + y 2 = 2, 2 := x 2 + y 2 = 1 D j (j = 1, 2, 3, 4) 2.5.2 21 y 1 D 2 D 1 2 O x D 3 D 4 2.14: + = + + + 1 2 D 1 D 2 D 3 D 4 D = D 1 D 2 D 3 D 4. ( ) D 1 + 2 = D 1 + D 2 + D 3 + D 4, D j D j D j ( Q x P ) dx dy = y 4 j=1 D j ( Q x P ) dx dy = y = P dx + Q dy. 1 + 2 21.1.1 4 j=1 D j P dx + Q dy 42

1 Jordan D x y 22. 2.5.2 ( ) 2.5.1 2.5.3 ( ) ( ) [14] ( ) Green 3 2.5.2,.1.1 2.5.4 ( ) 1 D x dy = y dx = 1 x dy y dx = µ 2 (D) 2 (µ 2 (D) D ) Green P (x, y) =, Q(x, y) = x Q x P y = 1, P (x, y) = y, Q(x, y) = Q x P y = 1, P (x, y) = 1 2 y, Q(x, y) = 1 2 x dy, y dx, D x Q x P y = 1 1 2 1 dx dy = µ 2 (D) x dy y dx 23. Γ : r = (a(t sin t), a(1 cos t)) T (t [, 2π]) y 22 ( : y dx = 3πa 2 ) Γ 2.5.5 ( ) n Ω P j (x j, y j ) (j = 1,..., n) Ω µ 2 (Ω) (2.9) µ 2 (Ω) = 1 2 n (x j y j+1 x j+1 y j ) = 1 2 j=1 P n+1 = P 1, P = P n n (x j+1 x j 1 ) y j. j=1 22 ( ) 3πa 2 ( [26] ) 43

( ) Green µ 2 (Ω) = Ω x dy = n j=1 x dy. P j P j+1 P j P j+1 P j P j+1 ( ) φ(t) := (1 t)p j + tp j+1 (t [, 1]) φ (t) = P j + P j+1 = P j P j+1, x = (1 t)x j + tx j+1, x dy = P j P j+1 1 dy dt = y j+1 y j. ((1 t)x j + tx j+1 ) (y j+1 y j ) dt = 1 2 (x j + x j+1 ) (y j+1 y j ). µ 2 (Ω) = 1 n (x j + x j+1 ) (y j+1 y j ) 2 j=1 = 1 n n n x j y j+1 x j y j + x j+1 y j+1 2 j=1 j=1 j=1 n x j+1 y j. j=1 x n+1 y n+1 = x 1 y 1 2 3 µ 2 (Ω) = 1 2 n (x j y j+1 x j+1 y j ). j=1 (x n y n+1 = x y 1 ) µ 2 (Ω) = 1 n n x j y j+1 x j+1 y j = 1 n x j 1 y j 2 2 j=1 j=1 j=1 n x j+1 y j = 1 2 j=1 n (x j 1 x j+1 ) y j. ( : (2.9) Green n n ( ) 3 = 1 2 det x 2 x 1 x 3 x 1 = 1 y 2 y 1 y 3 y 1 2 [(x 2y 3 x 3 y 2 ) + (x 3 y 1 x 1 y 3 ) + (x 1 y 2 x 2 y 1 )]. (2.9) n = 3 (2.9) n n + P 1 P n P n+1 = 1 n 1 (x j y j+1 x j+1 y j ) + (x n y 1 x 1 y n ) 2 j=1 + 1 2 [(x ny n+1 x n+1 y n ) + (x n+1 y 1 x 1 y n+1 ) + (x 1 y n x n y 1 )] = 1 n (x j y j+1 x j+1 y j ) + (x n+1 y 1 x 1 y n+1 ). 2 j=1 o (2.9) n + 1 Green j=1 44

2.5.6 (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, y)dx v(x, y)dy + i v(x, y)dx + u(x, y)dy = D ( v x u ) dx dy + i y f auchy-riemann D ( u x + v ) dx dy. y u x = v y, u u = v x f(z) dz = D dx dy + i dx dy =. D 2.5.7 (2 ) 2 Ω f rot f = f 2 x f 1 y = Ω Ω D D Ω Green ( f2 f dr = D x f ) 1 dx dy = dx dy = y D 2.4.7 f 2.4.8 2 ( 23 ) 2.5.2 ( ) Green Gauss-Green Green-tokes ( ) Gauss 2 f n ds = D div f dx dy f = (Q, P ) T n ds = (dy, dx) T (2.8) 24 (2.7) tokes 2 2 Gauss tokes Green Green (Gauss, tokes ) 23 Jordan 8 ( ) ( ) ( ) ( ) ( ) cos( π/2) sin( π/2) φ 24 n φ (t) π/2 1 (t) 1 φ sin( π/2) cos( π/2) φ = 1 (t) φ 2(t) 1 φ = 2 (t) 2(t) φ ( ) 1(t) 1 φ n = 2 (t) (φ 1 (t)) 2 + (φ 2 (t))2 φ. ds = ( ) φ (φ 1(t) 1 (t))2 + (φ 2 (t))2 dt n ds = 2 (t) φ dt. 1(t) f n ds = Q dy ( P )dx = P dx + Q dy. 45

3 ( ) 3 1. 2. 3. Gauss ( ) tokes ( ) 3.1 3.1.1 ( ) ( ) 2 2 ( 3 ) ( ) 3.1.1 3 3 R (> ) ( ) (a) 2 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) 3 ( ) F (x, y, z) = h. h x 2 + y 2 + z 2 = R 2 R ( ) 46

(c) φ = (φ 1, φ 2, φ 3 ) T : D R 3 x = φ 1 (u, v), y = φ 2 (u, v), i.e. r = φ(u, v) z = φ 3 (u, v) 1 (3.1) x = R sin θ cos ϕ y = R sin θ sin ϕ z = R cos θ ((θ, ϕ) [, π] [, 2π]) R 3 3 (a) (b) F (x, y, z) := f(x, y) z, h := z = f(x, y) F (x, y, z) = h (a) (c) φ 1 (u, v) := u, φ 2 (u, v) := v, φ 3 (u, v) := f(u, v) z = f(x, y) x = φ 1 (u, v), y = φ 2 (u, v), z = φ 3 (u, v) (b) F (a) = h, F (a) a (a) F x (a) F y (a) F z (a) F (x, y, z) = h x = φ(y, z) x (c) φ u (u, v ) φ v (u, v ) (u, v ) (a) y u y v z u z v z u z v x u x v x u y u x v y v 47

( ) u = ξ(y, z) v u, v x = φ 1 (u, v) = φ 1 (ξ(y, z)) (b) F F (c) r = φ(u, v) φ u φ v 3.1.1 ( F F = h ) 1 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 1 (x a)(x a) + (y b)(y b) + (z c)(z c) = 24. 1 f z = f(x, y) (x, y, z ) (z = f(x, y )) ( ) 3.1.2 ( ) (c), 1 48

3.1.2 ( ) D R 2 1 r, φ: D R 3 r φ r : r = φ(u, v) ((u, v) D) φ ( ) φ(d) = {φ(u, v); (u, v) D} : := φ(d). (i), (ii) (i) φ p, q D, p q = φ(p) φ(q). (ii) p D (3.2) φ u φ (p) (p). v R n R n D R n R n U (U = D) U ( U ) R n Ω Ω f : Ω R m k Ω U U k f : U R m f Ω f ( f Ω = f) 3.1.2 ( ) ( ) ( ) (i) (ii) φ u φ v φ (i), (ii) φ φ ( ) u v (u, v ) D u φ(u, v ) : r = φ(u, v) ((u, v) D) 1 ( {u; (u, v ) D} u ) φ(u, v ) u v φ(u, v) ( {v; (u, v) D} v ) 49

φ(u, v ) v 25. φ(θ, ϕ) = (R sin θ cos ϕ, R sin θ sin ϕ, R cos θ) T ((θ, ϕ) [, π] [, 2π]) θ ϕ φ(u, v ) u v φ 1 u (u, v ) φ u (u, v ) = φ u (u, v ) = φ 2 u (u, v ) φ, v (u, v ) = φ v (u, v ) = φ 3 u (u, v ) φ 1 v (u, v ) φ 2 v (u, v ) φ 3 v (u, v ) 2 φ 2 u (u, v ) φ 3 v (u, v ) φ 3 u (u, v ) φ 2 v (u, v ) φ φ u (u, v ) φ v (u, v ) = 3 u (u, v ) φ 1 v (u, v ) φ 1 u (u, v ) φ 3 v (u, v ) = φ 1 u (u, v ) φ 2 v (u, v ) φ 2 u (u, v ) φ 1 v (u, v ) (φ 2, φ 3 ) (u, v) (φ 3, φ 1 ) (u, v) (φ 1, φ 2 ) (u, v) ( (u, v ) ) v z y D (u+ u,v + v) ϕ(u+ u,v + v) ϕ v (u,v) ϕ u ϕ(u, v) u x 3.1: φ u φ v φ(u, v ) ( F = h F ) : r = φ(u, v) ((u, v) D) φ u φ v 3.1.1 ( (3.2) ) (3.2) (3.3) rank φ (p) = 2. 5

φ (p) φ φ 1 u (p) φ 1 v (p) φ (p) = φ 2 u (p) φ 2 v (p) φ 3 u (p) φ 3 v (p) 2 2 1 2 ( ) (φi, φ j ) 2 (3.4) > (u, v) 1 i<j 3 ( ) (3.5) φ u φ 2 v = 1 i<j 3 ( ) (φi, φ j ) 2 (u, v) (3.4) (3.2) 3 (3.2), (3.3), (3.4) 3.1.3 (2 ) D R 2 ( ) f : D R r (1 r ) f grad f = {(x, y, f(x, y)); (x, y) D} r φ(x, y) := 2 φ x = 1 f x, φ y = 1 f y x y f(x, y) φ x φ y = 1 f x 1 f y = f x f y 1 φ r graph f 3.1.4 ( ) R (> ) R := {(x, y, z); x 2 + y 2 + z 2 = R 2 } D := [, π] [, 2π], φ: D (θ, ϕ) R R sin θ cos ϕ R sin θ sin ϕ R cos θ R 3 φ φ φ D (1 1) φ (ϕ φ(, ϕ) = (,, R), φ(π, ϕ) = (,, R). θ φ(θ, ) = φ(θ, 2π)) 2 (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)). 51

φ θ = (3.6) φ θ φ ϕ = R 2 ( θ π ) (3.7) 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 φ θ φ ϕ = R 2 sin θ. R sin θ sin ϕ R sin θ cos ϕ sin θ cos ϕ sin θ sin ϕ cos θ < θ < π φ θ φ ϕ θ =, π φ θ φ ϕ =. ( ) 3.1.5 ((3.6) ) n = n := sin θ cos ϕ sin θ sin ϕ cos θ φ(θ, ϕ) φ(θ, ϕ) = φ(θ, ϕ) R. n φ(θ, ψ) (3.7) D Λ = [θ, θ + θ] [ϕ, ϕ + ϕ] φ φ(λ) = R sin θ ϕ, = R θ 3 R 2 sin θ θ ϕ R 2 sin θ 4. φ θ φ ϕ = ( ) ( ) 3.2 3.2.1 : r = φ(u, v) ((u, v) D) R 3 r φ ( ) φ(u + h, v + k) = φ(u, v) + hφ u (u, v) + kφ v (u, v) + o h 2 + k 2 ((h, k) (, )) 3 1 2 1 2 θ R sin θ 4 3 r θ ϕ r 2 sin θ r sin θ cos ϕ r sin θ sin ϕ r cos θ 52

D ( u, v) A = [u, u + u] [v, v + v] φ(a) = {φ(u + h, v + k); (h, k) [, u] [, v]} {r = φ(u, v) + hφ u (u, v) + kφ v (u, v); (h, k) [, u] [, v]} φ u (u, v) φ v (u, v) u v φ(a) φ u (u, v) φ v (u, v) u v. φ u (u, v) φ v (u, v). 3.2.1 (, ) D R 2 Jordan : r = φ(u, v) ((u, v) D) 1 f f : R f ( ( ) ) (surface integral) f dσ := f(φ(u, v)) φ u (u, v) φ v (u, v) du dv D f 1 dσ = D φ u (u, v) φ v (u, v) du dv (surface area) µ c () ( ) dσ := φ u (u, v) φ v (u, v) du dv ( ) (surface element) 3.2.2 ( ) dσ d dγ, dγ, da ( 25 d ) dσ 3.2.3 ( ) 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 53

R 3 µ c () = φ 1 φ 2 D u v φ 1 φ 2 v u du dv. ( ) ( R 2 µ() x y = Φ(u, v) := µ() = dx dy = det Φ (u, v) du dv = φ 1 u D D φ 1 (u, v) φ 2 (u, v) φ 2 v φ 1 v ) φ 2 u du dv. 3.2.4 ( ) x 2 + y 2 + z 2 = R 2 3.1.4 µ c () = θ [,π] ϕ [,2π] dσ = R 2 sin θ dθ dϕ R 2 sin θ dθ dϕ = R 2 π 2π sin θ dθ dϕ = 4πR 2. 3.2.5 (2 ) R 2 Jordan D 1 f : D R φ(x, y) := x y f(x, y) φ x φ y = f x f y 1. graph f : r = φ(x, y) ((x, y) D) dσ = φ x φ y 1 dx dy = + (f x ) 2 + (f y ) 2 dx dy µ c () = D 1 + (f x ) 2 + (f y ) 2 dx dy. 3.2.6 ( ) xy 1 f : [a, b] R x f(x) > (x [a, b]) D := [a, b] [, 2π], φ: D ( x θ ) x f(x) cos θ f(x) sin θ R 3 54

: r = φ(x, θ) ((x, θ) D) 1 φ x = f φ (x) cos θ, f θ = f(x) sin θ, (x) sin θ f(x) cos θ φ x φ θ = φ x φ θ = f(x) 1 + f (x) 2 >. µ c () = f(x) 1 + f (x) 2 dx dθ = 2π x [a,b],θ [,2π] b a f(x)f (x) f(x) cos θ f(x) sin θ f(x) 1 + f (x) 2 dx. 26. ( ) = φ(d) ( φ E := u, φ ) ( φ, F := u u, φ ) ( φ, G := v v, φ ) v φ u φ v = EG F 2 3.2.7 ( ) xz (c,, ), r ( c > r > ) z (, ) ( ) xy { x = c + r cos θ (θ [, 2π]) z = r sin θ z φ(θ, ϕ) = (c + r cos θ) cos ϕ (c + r cos θ) sin ϕ r sin θ r sin θ cos ϕ φ θ = r sin θ sin ϕ r cos θ φ θ φ ϕ =, φ ϕ = ((θ, ϕ) [, 2π] [, 2π]) (c + r cos θ) sin ϕ (c + r cos θ) cos ϕ (c + r cos θ)r cos ϕ cos θ (c + r cos θ)r sin ϕ cos θ (c + r cos θ)r sin θ φ θ φ ϕ 2 = (c + r cos θ) 2 r 2 cos 2 θ(cos 2 ϕ + sin 2 ϕ) + (c + r cos θ) 2 r 2 sin 2 θ = (c + r cos θ) 2 r 2 φ θ φ ϕ = (c + r cos θ)r. µ c () = (c + r cos θ)r dθ dϕ = 2πr θ [,2π] ϕ [,2π] 2π. (c + r cos θ)dθ = (2π) 2 rc = 4π 2 rc. 55

1.5 -.5-1 -2 2 2-2 3.2: (Mathematica ) c=2; r=1; ParametricPlot3D[{(c+r os[t])os[p],(c+r os[t])in[p], r in[t]},{t,,2pi},{p,,2pi}] 3.2.8 5 6 ( 2π ) ( ) ( [8] ) 2πc 2πr = 4π 2 rc (paraboloid of revolution) z = x 2 +y 2 (x 2 +y 2 1) ( : π ( 5 5 1 ) /6) 3.2.2 5 (Pappus of Alexandria, 32 ) ynagoge of Mathematical collection (, 8, 325 ) 6 Paul Guldin (1577 1643). 164 Guldin (Pappus ) 56

1.75.5 1.25-1 -.5 -.5.5.5 1-1 3.3: z = x 2 + y 2 (x 2 + y 2 1) (Mathematica ) ParametricPlot3D[{r os[t],r in[t],r*r},{r,,1},{t,,2pi}] 3.2.9 : r = φ(u, v) ((u, v) D) : r = φ(ũ, ṽ) ((ũ, ṽ) D) 1 ( = ) (1), (2) (1) Φ: D (ũ, ṽ) (u, v) = φ 1 ( φ(ũ, ṽ)) D Φ 1 1 φ (3.8) ũ φ ( (u, v) φ = ṽ (ũ, ṽ) u φ ). v (u, v) (ũ, ṽ) (= det Φ ) D ( det Φ > det Φ < ) (2) D D Jordan f : R f dσ = f dσ. (1) φ: D, φ: D Φ := φ 1 φ: D D Φ 1 = φ 1 φ. 1 7 7 (y, z) (z, x) (x, y) : φ u φ v,, (u, v) (u, v) (u, v) (x, y) (u, v) (u, v) (x, y) 1 : ψ s.t. (u, v) = ψ(x, y). Φ(ũ, ṽ) = ψ( φ 1(ũ, ṽ), φ 2(ũ, ṽ)) Φ 1 57

rũ = u ũ r u + v ũ r v, ( u rũ rṽ = ũ r u + v ) ( u ũ r v ṽ r u + v ṽ r v = u u ũ ṽ r u r u + u v ( ũ u v = ũ ṽ u ) v r u r v ṽ ũ (u, v) = (ũ, ṽ) r u r v. rṽ = u ṽ r u + v ṽ r v ) ṽ r u r v + v ũ u ṽ r v r u + v ũ v ṽ r v r v Φ Φ 1 1 Φ ( Φ 1) = I (I ). det Φ = D 8 (2) f dσ = D f(φ(u, v)) φ u φ v du dv. (u, v) = Φ(ũ, ṽ) = φ 1 φ(ũ, ṽ) (ũ, ṽ) D du dv = (u, v) (ũ, ṽ) dũ dṽ, φ(ũ, ṽ) = φ(u, v) (u, v) (ũ, ṽ) (1) f dσ = D f( φ(ũ, ṽ)) φ u φ v (u, v) (ũ, ṽ) dũ dṽ. φ u φ v (u, v) (ũ, ṽ) = φ ũ φṽ f dσ = f( φ(ũ, ṽ)) φũ φṽ dũ dṽ = D f dσ. 3.2.1 (2) ( ) φũ φṽ dũ dṽ = (u, v) (ũ, ṽ) φ u φ v (ũ, ṽ) (u, v) du dv = φ u φ v du dv 1 dσ = φ u φ v du dv 3.2.11 (3.8) φ u φ v φũ φṽ φũ φṽ = uv R 3, φ u φ v = ũṽ R 3 8 (det Φ )( D) R 58

φũ φṽ = (ũ, ṽ) (u, v) = φ u φ v (u, v) (ũ, ṽ) (u, v) φũ φṽ = ± (ũ, ṽ) (φ u φ v ) ± + OK 3.2.12 ( ) : r = φ(u, v) ((u, v) D) 1 n := φ u(u, v) φ v (u, v) φ u (u, v) φ v (u, v) ((u, v) D) ( ) φ(u, v) 1 2 : r = φ(u, v) ((u, v) D), : r = φ(ũ, ṽ) ((ũ, ṽ) D) (3.8) n = φ u φ v φ u φ v, ñ = φ ũ φṽ φ u φ v, ñ = ( sign ) (u, v) n (ũ, ṽ) sign : 1 (x > ) sign x := (x = ) 1 (x < ). Φ := φ 1 φ (a) det Φ > on D (b) det Φ < on D (a) n = ñ, (b) n = ñ (a) 2 (b) ( ) 2 59

3.3 : r = φ(u, v) ((u, v) D) n = φ u(u, v) φ v (u, v) φ u (u, v) φ v (u, v). n dσ = φ u(u, v) φ v (u, v) φ u (u, v) φ v (u, v) φ u(u, v) φ v (u, v) du dv = φ u (u, v) φ v (u, v) du dv. dσ : (3.9) dσ := n dσ = φ u (u, v) φ v (u, v) du dv. 3.3.1 ( ) 1 : r = φ(u, v) ((u, v) D) D Jordan f f ( ) f n dσ = f dσ := f(φ(u, v)) (φ u (u, v) φ v (u, v)) du dv f n f 9 D ( ) 3.3.2 ( ) ( ) f V (1) f f 3.4 (, f ) V = f. (2) f f (, f cos θ, θ f ) V = ( f cos θ). n V = f n (3) = l j=1 j j n j, j f f j V l f j n j j j=1 9 n e a a e a e ( ) a e j = a j 6

f f f θ n f cosθ 3.4: V = f 3.5: V = f cos θ = f n f n dσ V = f n dσ Ω Ω Ω 3.3.1 ( ) f = (f 1, f 2, f 3 ) T f n dσ = D ( f 1 (φ(u, v)) (x 2, x 3 ) + f 2 (φ(u, v)) (x 3, x 1 ) + f 3 (φ(u, v)) (x 1, x 2 ) (u, v) (u, v) (u, v) ( ) f 1 dx 2 dx 3 + f 2 dx 3 dx 1 + f 3 dx 1 dx 2. D (exterior product) dx i dx j = (x i, x j ) du dv. (u, v) ) du dv. n dσ = dσ = φ u (u, v) φ v (u, v) du dv = (x 2, x 3 ) (u, v) (x 3, x 1 ) (u, v) (x 1, x 2 ) (u, v) du dv = dx 2 dx 3 dx 3 dx 1 dx 1 dx 2. 61

3.3.3 ( ) 2 1 : r = φ(u, v) : r = φ(ũ, ṽ) ((u, v) D), ((ũ, ṽ) D), f : R 3 f n dσ = f ñ dσ n, ñ φ, φ 3.2.9 (ũ, ṽ) > ((u, v) D) (u, v) dũ dṽ = (ũ, ṽ) du dv. (u, v) f ñ dσ = f ( φ(ũ, ṽ)) ( φũ φṽ) dũ dṽ D ( ( )) = f φ φ 1 (u, v) φ(u, v) D (ũ, ṽ) (φ (ũ, ṽ) u φ v ) du dv (u, v) = f(φ(u, v)) (φ u φ v ) du dv D = f n dσ. ñ = n f n = f ñ 3.2.9 f n dσ = f ñ dσ 3.3.2 Möbius 1 Möbius (Möbius strip Möbius band) Möbius ( ) 2 3.3.4 ( ) x 2 + y 2 + z 2 = R 2 f 1 August Ferdinand Möbius (179 1868, chulpforta Leipzig ). Gauss 62

2 5-2 -5-2.5 2.5-5 3.6: Möbius ( ) a=4;b=3; ParametricPlot3D[ {(a-r in[p/2])os[p],(a-r in[p/2])in[p],os[p/2]}, {r,-b,b},{p,,2pi}] 63

(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, z)t. (3.1) dσ = R 2 sin θ dθ dϕ. ( 3.1.4) φ θ φ ϕ = R2 sin θ (3.11) n dσ = φ θ φ dθ dϕ = ϕ sin θ cos ϕ sin θ sin ϕ cos θ sin θ cos ϕ sin θ sin ϕ cos θ R 2 sin θ dθ dϕ n dσ 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 = cos ϕ dϕ = y z x f n dσ = 2π f n dσ = = sin ϕ dϕ = θ [,π] ϕ [,2π] θ [,π] ϕ [,2π] = =. θ [,π] ϕ [,2π] π α β γ sin θ cos ϕ sin θ sin ϕ cos θ cos θ sin θ dθ = ) 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 = =. 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 64

(3) f = 1 (x 2 + y 2 + z 2 ) 3/2 x y z f = 1 R 2 n f n dσ = 1 R 2 dσ = 4π. Gauss Gauss Gauss 3.4 Gauss 11 3.4.1 (Gauss ) Ω R 3 = Ω 1 Ω 1 f div f dx 1 dx 2 dx 3 = f n dσ. Ω n 3.4.1 (Gauss ) (3.12) Ω u dx = un i dσ (1 i 3) x i (n i n i n 1, n 2, n 3 12 l, m, n ) 13 (3.13) (3.12) 3.4.2 ( ) Gauss ( ) Ω Ω = {(x, y, z); (x, y) D, φ 1 (x, y) < z < φ 2 (x, y)} Fubini ( ) f φ2 (x,y) 3 f 3 dx dy dz = (x, y, z) dz dx dy Ω z D φ 1 (x,y) z = (f 3 (x, y, φ 2 (x, y)) f 3 (x, y, φ 1 (x, y))) dx dy. D 11 A.6 12 x, y, z α, β, γ l = cos α, m = cos β, n = cos γ l = n 1, m = n 2, n = n 3 13 65

1, 2 Ω : f 3 n 3 dσ = f 3 (x, y, φ 2 (x, y))dx dy, f 3 n 3 dσ = f 3 (x, y, φ 1 (x, y))dx dy. D D Ω f 3 z dx dy dz = f 3 n 3 dσ. f 1, f 2 Gauss 27. ( ) z = φ(x, y) φ x n dσ = φ y dx dy, 1 n 3 dσ = dx dy 3.4.2 ( ) 3.3.2 div f dx 1 dx 2 dx 3 = f n dσ. Ω Ω Ω (flux) 14 div f = f n dσ = 15 Ω ( ) div f = div E div E = div E = δ (δ Dirac ) ( ) 3.4.3 ( ) 3 2 Green 1 f = F, Ω = (a, b) 1 Ω div f dx = b a F (x) dx = F (b) F (a) f n dσ Ω = {a, b} n = { 1 (x = b) 1 (x = a) Gauss 1 14 (electric flux), (magnetic flux) 15 2 ( ) div f 3 F div f 66

= 3.4.4 ( 3.3.4 ) 3.3.4 (1), (2) Ω = B(; R) Gauss (1) f div f =. f n dσ = div f dx dy dz =. (2) f = y z x div f =. f n dσ = Ω Ω div f dx dy dz =. x 1 (3) f = (x 2 + y 2 + z 2 ) 3/2 y div f = f z ( ) Gauss f n dσ ( 4π) 3.4.1 (3) f n dσ = 4π Gauss 16 Gauss Gauss E n dσ Gauss Gauss Gauss ( ) E n dσ Gauss ( 17 ) ( ) ( Gauss = ) 16 Gauss Gauss [3] Gauss Gauss, Green, Kelvin 3 (Kelvin tokes ) 17 67

3.4.5 ( Gauss ) E Gauss E ρ div E = ρ/ε. Q/ε Q Gauss E ( ) E dσ E n dσ Gauss = E n dσ = div E dx dy dz Ω ρ = dx dy dz = ρµ 3(Ω) = Q ε ε ε 3.4.6 ( ) Ω x = (x, y, z) T = Ω dσ Ω p(x) n dσ. p(x) p(x) = ρgz + ρ g Ω P := p n dσ (3.13) P = ρµ 3 (Ω)g 18 ( ) (3.13) (P 1, P 2, P 3 ) T := P P j = p n j dσ (j = 1, 2, 3) P 3 f := (,, p) T P 3 = f n dσ = div f dx dy dz = = ρgµ 3 (Ω). Ω Ω ρg dx dy dz = ρg dx dy dz Ω 18 µ 3(Ω) ρ g ( ) 68

P 1 = P 2 = 19 = Ω ( ) 3.5 tokes tokes Green 3.5.1 (tokes, ) : r = ψ(t) (t [α, β]) R 2 1 Jordan D D : r = φ(u, v) ((u, v) D) R 3 2 = φ(d) 1 f rot f n dσ = f dr n r = φ(ψ(t))) (t [α, β]) φ ψ.5 3.5.2 : z = 1 x 2 y 2, z z n 3 > n f = (z 3 y 3, x 3 z 3, xyz) rot f n dσ ( ) xy φ(t) = (cos t, sin t, ) T (t [, 2π]) rot f n dσ = f dr. f(φ(t)) = ( sin 3 t, cos 3 t, ) T, φ (t) = ( sin t, cos t, ) T f(φ(t)) φ (t) = sin 4 t + cos 4 t = 1 4 cos 4t + 3 4 rot f n dσ = 2π f(φ(t)) φ (t) dt = 2π 3 4 = 3 2 π. 19 ( Archimedes, B 287 B 212, ( ) yracuse yracuse ) 69

3.5.3 ( ) 1 rot E = B t B n dσ = E dr t 7

2 [8] [13], [14] [1] (dx dy [dx, dy] artan ) ( ) ( ) [12] ( ) [2] 2 [21] ( ) [11] ( 3 ) [1] ( ) [24] 2 71

[1],, (196). [2],, (22). [3],,, (25) [4],,, 23 7, 45 49. [5],,, BP (27). Donal O hea, The Poincaré onjecture: In earch of the hape of the Universe, Walker Publishing ompany, Inc. (27). [6],,, (27). [7], 2 1, http://nalab.mind.meiji.ac.jp/~mk/ lecture/tahensuu2/tahensuu2-p1.pdf. [8], 2, (22). [9],, (23). [1], I, II,, (1994, 1995). [11], 3, (199). [12],, (26). [13], I, (198). [14], II, (1985). [15],,, (1972). 27 ( ) [16],, (1987). [17],,, (1962, 1963). [18],, (1989, 199, 199, 1991). 2 3 [19],, (1997). [2],,,, III, (1986). 72

[21],, (24). [22], Nebel ( ),, No. 57 (22). [23], (, ), (1976). [24],, (27). [25],, (25). [26], ( ), G5, (196). [27] Georges de Rham, Variétés Defférentiables, Hermann, Paris (1955, 196). G.,, :, (1974). ( ) [28] J. W. Gibbs and E. B. Wilson, Vector Analysis, Yale Univ. Press (191). [29] G. Green, An essay on the application of mathematical analysis to the theories of electricity and magnetism, 1828. W. Thomson 184 George Green, Mathematical Papers: Edited by N. M. Ferrers, Dover Publications (January 27, 25) [3] Oliver Heaviside, Electromagnetic Theory, Vol. I, The Electrician, London, 1893. ( ) [31] J.. Maxwell, A Treatise on Electricity and Magnetism, larendon Press (1873). ( ) 73

A A.1 k k 2 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 ]) A.1.1 k I = [a, b], φ: I R n φ I k a = t < t 1 < < t l = b l N, {t j } l j=1 s.t. φ (tj 1,t j ) k i =, 1, 2,..., k lim t t j 1 + φ(i) (t) lim t t j φ(i) (t) j = 1, 2,..., l φ j 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 ) φ j k ([t j 1, t j ]) A.1.2 φ(a + ), φ(b ) lim φ(x), lim φ(x) x a+ x b φ I φ j := φ [tj 1,t j ] 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 74

A.1.3 φ: [a, b] R n c [a, b] φ c lim t c φ (t) = A φ (c) = A. n = 1 c + h [a, b] h c c + h ξ h φ(c + h) φ(c) = φ (ξ h ). h h ξ h c A φ c φ (c) = A. A.1.4 φ: [a, b] R n [a, b] k 1 c [a, b] k φ (k 1) c lim t c φ(k) (t) = A φ (k) (c) = A. A.2 X (connected) 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 2 A.2.1 R n 1 75

( ) R n A.2.2 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) ) A.3 A.3.1 X (simply connected) φ: I = [α, β] X ( φ(α) = φ(β)) Φ: I [, 1] X (i) Φ(, ) = φ. t [α, β] Φ(t, ) = φ(t). (ii) s [, 1] Φ(, s) s [, 1] Φ(α, s) = Φ(β, s). (iii) Φ(, 1) ( 1 ) t [α, β] Φ(t, 1) = Φ(α, 1). 76

A.3.2 φ [, 1] A.3.1 ( ) : a Ω π 1 (Ω, a) Ω R 2 Ω Ω R 2 \ Ω compact ( ) R 2 ( ) A.4 Jordan Jordan (Jordan curve theorem) ( ) ( ) 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 ( ) Jordan Jordan ( ) Riemann ( Ω {z ; z < 1} ) Jordan (arathéodory ) ( ) Jordan A.5 3.1.2 77

3.1.4 = {(x, y, z) T ; x 2 + y 2 + z 2 = R 2 } 1 (3.1) (θ, ϕ) (x, y, z) T 1 1 (θ = ϕ (x, y, z) T = (,, R) T ) 3.1.1 (a), (b), (c) R 3 k x R 3 U 2 (i) x U (ii) U k (manifold) R 3 2 k = {(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 ( ) R 3 2 ( ) 1 A.6 ( ( ) ( ) ) 2 2 (closed surface) ( ) (2 ) R 4 R 3 R 3 1 78

R 3 R 3 R 3 \ 2 ( Jordan ) R 3 ( ) 1. g ( g g ) 2 2g g 2. k 2 2k k 1 R 3 g ( ) 3 3 ( ) 2 2 [5] 79

B 1 f (B.1) f i x j = f j x i in Ω (i, j = 1, 2,..., n) [1] 2 5 [14] ( ) B.1 B.1 1: (B.1) B.1.1 1: x x F (x) := f dr x x a x R n ( n [a j, b j ] ) (2 Green ) 1 grad F = f 2.4.7 (B.1) ( [1] 2 Green ) j=1 B.1.2 2: F (x) := f dr x 1 Fubini 8

x φ(t) := a + t(x a) (t [, 1]) F (x) = 1 f(a + t(x a)) (x a) dt = (B.1) F x k (x) = = N 1 δ jk f j (a + t(x a))dy + j=1 1 f k (a + t(x a))dt + N (x j a j ) j=1 N (x j a j ) j=1 N (x j a j ) j=1 (δ jk Kronecker ) 2 2 = 1 1 d dt (f k(a + t(x a))) t dt 1 1 f j (a + t(x a))dt. f j x k (a + t(x a)) t dt f k x j (a + t(x a)) t dt. 1 = [f k (a + t(x a)) t] 1 f k (a + t(x a))dt = f k (x) 1 f k (a + t(x a))dt. F x k (x) = f k (x). B.2 2: 3 1 R n Ω (B.1) f Ω : r = φ(t) (t I = [α, β]) L(f; ) 1 L(f; ) = f dr 2 Ω : r = φ(t) (t I = [α, β]) E() E() φ(i) U U 1 F grad F = f in U E() f (B.2) L(f; ) := F (φ(β)) F (φ(α)) 2 f dr f ds = 81

(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 φ(i) F (φ(β)) F (φ(α)) = F (φ(β)) F (φ(α)) (B.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 ] V j := B(φ(t j 1 ); d) φ([t j 1, t j ]) V j Ω j V j V j N 1 V j f F j = j ( ) j E( j ) (B.2) : j=1 L(f; j ) = F j (φ(t j )) F j (φ(t j 1 )) (j = 1, 2,..., N). = 1 + 2 + + N (B.3) L(f; ) := N L(f; j ) = j=1 N (F j (φ(t j )) F j (φ(t j 1 ))) ( 1 ) j=1 I (B.3) 1, 2 1 < δ, 2 < δ 1, 2 i i B.3 3: Ω f (B.1) Ω 1 f dr = 82

[, 1] (i) (iv) Φ: [, 1] [, 1] Ω (i) Φ(, ) = φ. t [, 1] Φ(t, ) = φ(t). (ii) s [, 1] Φ(, s): [, 1] t Φ(t, s) s [, 1] Φ(, s) = Φ(1, s). (iii) Φ(, 1) ( 1 ) t [, 1] Φ(t, 1) = Φ(, 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) γ, Γ, γ r = Φ(1, s) (s [, 1]), r = Φ(1 t, 1) (t [, 1]), r = Φ(, 1 s) = Φ(1, 1 s) (s [, 1]), (Γ 1 ) s y 1 Γ O 1 t O B.1: Γ f dr = L(f, ) = L(f, ) + L(f, γ) + L(f, Γ ) + L(f, γ) = L(f, ) + L(f, γ) + L(f, γ) = L(f, ) = f dr. x 83

s y 1 Γ γ γ O t 1 O x B.2: = + γ + Γ + ( γ) auchy Goursat ( ) ts 2 [, 1] [, 1] { n } n N a [, 1] [, 1] s.t. n = {a}, n=1 n N L(f, n ) ( n Φ( n ) ) s 1 O 1 t B.3: 4 a Ω n Φ( n ) F L(f, n ) = F ( n ) F ( n ) =. f dr = 84

Gauss, Green, tokes.1 Green.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, y)dx + Q(x, y)dy = x P ) dx dy y 1, 2, 3, 4 D 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)]). y 3 y = ϕ 2 (x) 4 D 2 1 y = ϕ 1 (x) O a b x.1: Green D 85

[14] 2.5.2 ( P ) dx dy = P (x, y) dx y D F x (x, u, v) = v ( ) d φ2 (x) Q(x, y) dy dx φ 1 (x) u F (x, u, v) := v u Q(x, y) dy Q x (x, y) dy, F u(x, u, v) = Q(x, u), F v (x, u, v) = Q(x, v) = 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 (x))φ 1(x) + F v (x, φ 1 (x), φ 2 (x))φ 2(x) = φ2 (x) φ 1 (x) x [a, b] ( b ) φ2 (x) Q (x, y) dy dx a φ 1 (x) x [ ] x=b φ2 (x) = Q(x, y) dy + = φ 1 (x) φ2 (b) φ 1 (b) b a Q x (x, y) dy Q(x, φ 1(x))φ 1(x) + Q(x, φ 2 (x))φ 2(x). Q(b, y) dy b a x=a φ2 (a) φ 1 (a) Q(x, φ 2 (x))φ 2(x) dx. Q(x, φ 1 (x))φ 1(x) dx Q(a, y) dy + b a b Q(x, φ 1 (x))φ 1(x) dx Q(x, y) dy + 2 Q(x, y) dy + 4 Q(x, y) dy + 1 Q(x, y) dy = 3 2 r = (b, t) T, t [φ 1 (b), φ 2 (b)] dx/dt =, dy/dt = 1 2 Q(x, y) dy = φ2 (b) φ 1 (b) 4 Q(x, y) dy = Q(b, t) 1 dt = φ2 (a) φ 1 (a) φ2 (b) φ 1 (b) Q(a, y) dy. a Q(b, y) dy. Q(x, φ 2 (x))φ 2(x) dx Q(x, y) dy 1 r = (t, φ 1 (t)) T, t [a, b] dx/dt = 1, dy/dt = φ (t) 1 Q(x, y) dy = b a Q(t, φ 1 (t)) φ 1(t) dt = 86 b a Q(x, φ 1 (x))φ 1(x) dx.

2 Q(x, y) dy = b a Q(x, φ 2 (x))φ 2(x) dx 4.1.1 (Green ) P Q 1 Q x, P y Q x P y (Goursat-Bochner ).2 Gauss ( ) Gauss [14] Green ( x, y, z ) Gauss.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 (.1) Ω f z dx dy dz = f n dσ ( ) = f dx dy. f n dσ 3 T, B, (i) T φ(u, v) := (u, v, φ 2 (u, v)) T ((u, v) D) (ii) B B φ(u, v) := (u, v, φ 1 (u, v)) T ((u, v) D) (iii) 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 (x, y, z) dx dy dz = z Ω = D D ( ) φ2 (x,y) f (x, y, z) dz dx dy φ 1 (x,y) z f(x, y, φ 2 (x, y)) dx dy f(x, y, φ 1 (x, y)) dx dy. 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 dσ = f(u, v, φ 2 (u, v)) du dv. T D D 87

( φ B f u φ ) = f(u, v, φ 1 (u, v)) v f n dσ = f(u, v, φ 1 (u, v)) du dv. B D φ t φ s = ( φ f t φ ) =. s ξ (t) η (t) 1 f n dσ =. = η (t) ξ (t) Ω f (x, y, z) dx dy dz = z = = f(u, v, φ 2 (u, v)) du dv f(u, v, φ 1 (u, v)) du dv + D D f n dσ f n dσ + f n dσ T B f n dσ + f n dσ + f n dσ. T B Ω z x, y x y z Ω ( f x + g y + h ) dx dy dz = z Ω f dy dz + g dz dx + h dx dy. f := (f, g, h) T div f dx dy dz = Ω f n dσ.2.1 Gauss ( ) Bourbaki tokes Gauss 1 ( φ 1, φ 2 φ f u φ ) f(u, v, φ j (u, v)) v φ j φ j 1 88

.3 Green 2 ( ).3.1 (Green ) Ω Gauss R n Γ (1) u, v Ω 2, 1 v u dx = v u n dσ (2) u, v Ω 2 (v u u v) dx = u, v Ω Ω Γ Γ Γ Ω grad u grad v dx. ( v u n u v n ( v u n u v ) dσ =. n ) dσ. (3) u Ω 2 u dx = Ω u u dσ =. Γ n Γ u n dσ. (1) f := v grad u div f = grad v grad u + v u Gauss (2) (1) (1) u v (3) (1) v 1.4 Gauss ( ).4.1 (Fourier, ) ( ) Fourier (Jean Baptiste Joseph Fourier, 1768 183, ) u = u(x, y, z, t) grad u = (u x, u y, u z ) T V d u dx dy dz = k grad u n dσ dt V 89

2 k Fourier V n Gauss (.2) u t dx dy dz = div (k grad u) dx dy dz. V V (.3) u t = div(k grad u) 3 ( ) (heat equation) k div(grad) = u t = κ u, V κ := k ( ).4.2 ( ) ( 216/7/1 ) ρ(x, t), v(x, t) (.4) ρ + div(ρv) =. t (.5) V V d ρ dx dy dz = ρv n dσ. dt V V Gauss ρ dx dy dz = div (ρv) dx dy dz. t V V (.4) ( ) V (.6) D Dt := t + v = t + u x + v y + w z Lagrange (, material derivative) (.4) (.7) Dρ + ρ div v =. Dt Lagrange Dρ Dt Dρ Dt = (.8) div v = 2 V 3 x V x (.2) x (.3) ( 2 1 [7] B.3.1 ) 9

Lagrange t x(t) d 3 f (x(t), t) = dt j=1 f ẋ j (t) + f x j t = f t + ẋ(t) f = f t + v f = Df Dt..4.3 (, Navier-tokes ) Ω ( R 3 ) v = v(x, t) (x Ω, t R), ρ = ρ(x, t) Ω V ( V Ω ) D (.9) (ρv) dx = P n dσ Dt D Dt V V D Dt := t + v = t + (material derivative, Lagrange ) P = (p ij ) (stress tensor) n V P i p i p 11 p 12 p 13 n 1 p T 1 n P n = p 21 p 22 p 23 n 2 = p T 2 n p 31 p 32 p 33 n 3 p T 3 n Gauss p T 1 n dσ V P n dσ = p T 2 n dσ = V V p T 3 n dσ V div p T 1 div P := div p T 2 = div p T 3 (.9) D (ρv) dx = Dt V (.1) V D (ρv) = div P Dt 3 j=1 v j V V V x j div p T 1 dx div p T 2 dx. div p T 3 dx p 11 x 1 + p 12 x 2 + p 13 x 3 p 21 x 1 + p 22 x 2 + p 23 x 3 p 31 x 1 + p 32 x 2 + p 33 x 3 V P = pi + 2µE div P dx. (in Ω). p = p(x, t) I ( ) µ (, coefficient of viscosity) E = (e ij ) e ij = 1 ( vi + v ) j, 2 x j x i 91

v 1 1 ( x 1 2 E = 1 v2 + v ) 1 2 ( x 1 x 2 1 v3 + v ) 1 1 2 x 1 x 3 2 ( v1 + v 2 x 2 x 1 v 2 1 x 2 ) 2 ( v3 x 2 + v 2 x 3 ) ( 1 v1 2 + v ) 3 ( x 3 x 1 v2 + v ) 3 x 3 x 2 v 3 x 3 (, ) (.1) div P = grad p + µ ( v + grad div v) D (ρv) = grad p + µ ( v + grad div v). Dt ρ D/Dt = / t + v v t + (v ) v = 1 grad p + ν ( v + grad div v). ρ ν := µ/ρ ν ρ div v = (.11) v t + (v ) v = 1 grad p + ν v. ρ 4 Navier-tokes (Navier-tokes equation) v t + (v ) v = 1 ρ grad p Euler (Euler equation) 5 tokes (tokes equation) v t = 1 grad p + ν v ρ ( tokes tokes tokes ).5 tokes tokes Green 4. L. H. Navier (1823), George G. tokes (1845). 5 92

3.5.1 φ ψ : I R 3 R 3 1 Γ b f dr = f(φ(ψ(t))) (φ (ψ(t))ψ (t) ) dt. Γ 2 x, 2 3 A, 3 u x (Au) = (Au) T x = (u T A T )x = u T ( A T x ) = (A T x) u a 6 (.12) f Γ f dr = b a [ φ (ψ(t)) T f(φ(ψ(t))) ] ψ (t) dt. f(u) := φ (u) T f(φ(u)) ( φ f ) (.12) f dr = f dr. (.13) rot f = rot (f φ) Γ γ ( φ u φ ) v Green γ D ( φ rot f n dσ = rot f(φ(u, v)) u φ ) du dv = v D i=1 rot f n dσ = Γ f dr i=1 D rot f(u, v) du dv = (.13) rot f = f 2 u f ( 3 ) ( 3 ) 1 v = f i (φ(u, v)) φ i f i (φ(u, v)) φ i u v v u = 3 i=1 f i (φ(u, v)) 2 φ i u v + i,j f i φ j φ i x j u v 3 i=1 f i (φ(u, v)) 2 φ i v u i,j φ 2 1 3 rot f = ( 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 φ j v φ j u γ f dr. f i φ j φ i x j v u. ) φ i v 6 (x, Au) = (A T x, u) 93