2009 7 9 1 2 2 2 3 6 4 9 5 14 6 18 7 23 8 25 9 26 10 29 11 32 12 35 A 37 1
B 38 1 (x, y), (x, y, z) (x 1, x 2 ) (x 1, x 2, x 3 ) 2 : x 2 + y 2 = 1. (parameter) x = cos t, y = sin t. y = f(x) r(t) = (x(t), y(t), z(t)), a t b. x(t), y(t), z(t) t r(t) t (path) 2
(i) (curve) r(t) (ii) (iii) (iv) (v) r r 2.1 ( ). r(t) = (x(t), y(t), z(t)) (velocity) (speed) dr dt (t) = r (t) = (x (t), y (t), z (t)) r (t) = (x (t)) 2 + (y (t)) 2 + (z (t)) 2 2.2 (Cycloid). 1 x t = 0 P 1 t (t, 1). P θ θ = t π/2. r(t) = (x(t), y(t)) = (t, 1) + (cos θ, sin θ) = (t sin t, 1 cos t). P x dr dt = (1 cos t, sin t) = (0, 0) t = 0, ±2π, ±4π,.... 0 (singular point) regular point 2.3 ( ). P 0 = (a 0, b 0, c 0 ), P 1 = (a 1, b 1, c 1 ) P 0 P 1 (line segment) P (t) = tp 1 + (1 t)p 0 = ((1 t)a 0 + ta 1, (1 t)b 0 + tb 1, (1 t)c 0 + tc 1 ) 0 t 1. P 0, P 1 3
2.4. r(t) = (x(t), y(t), z(t) r(a) r(t) t = a (x, y, z) = φ(a) + tφ (a) = (x(a) + tx (a), y(a) + ty (a), z(a) + tz (a)) 2.5. r(t) = (cos t, sin t, t) (helix) r (t) = ( sin t, cos t, 1), r (t) = 2. r(a) (x, y, z) = (cos a t sin a, sin a + t cos a, a + t) 1. x 2 + y 2 = 1 (1, 0) (distance) L = 2.6. (i) (ii) (iii) (iv) b a r (t) dt. r = r(θ) (α θ β) θ (x, y) = r(θ)(cos θ, sin θ) L = β α r(θ)2 + r (θ) 2 dθ 4
2. r = sin θ (0 θ 2π) (change of variables) t = h(s) h 0 h > 0 h < 0 r(t) h r(h(s)) 9.3) ( dr(h(s)) dx = ds dt (h(s))dh ds, dy dt (h(s))dh ds, dz ) dt (h(s))dh ds = h (s) dr dt (h(s)). 2.7 (Theorem 9B). 3. 4. 2.8. r(t) (a t b) r(t 0 ), r(t 1 ),... r(t n ) (a = t 0 < t 1 < < t n = b) n r(t j ) r(t j 1 ) j=1 C 2.9. C r(t) C = b a r (t) dt. 5
2.10. r n ( 2πr 1 n π sin π ) π3 r n 3n 2. O(1/n 2 ) n 2 3 f(x, y, z) f(a, b, c) f f f (a, b, c) x + (a, b, c) y + (a, b, c) z, x y z f(a, b, c) f(a + x, b + y, c + z) f(a, b, c) df = f f f dx + dy + x y z dz (x, y, z) (x(t), y(t), z(t)) (chain rule) df dt = f dx x dt + f dy y dt + f dz z dt d f(x(t), y(t), z(t)) = dt f f f (x(t), y(t), z(t))dx(t) + (x(t), y(t), z(t))dy (t) + (x(t), y(t), z(t))dz x dt y dt z dt (t).. f(x, y, z) x = x(t), y = y(t), z = z(t) t f x ((x(t), y(t), z(t)) f x (x, y, z) ( (x, y, z) ) x = x(t), y = y(t), z = z(t) 6
3.1. 2 1 V = xyz V = yz x + xz y + xy z V V = x x + y y + z z x/x = 0.02, y/y = 0.02, z/z = 0.01 V/V = 0.03 5. 2.9996 ( ) f f f (a, b, c), (a, b, c), (a, b, c) = f f f (a, b, c)i + (a, b, c)j + (a, b, c)k x y z x y z f(a, b, c) gradf(a, b, c) f (a, b, c) (gradient) i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1). gradient grade (nabla) J.W. Gibbs (del) nabla f(x) x f f of x f(a + x, b + y, c + z) (a, b, c) + f(a, b, c) ( x, y, z) 7
( x, y, z) n f (a, b, c) h f(x, y, z) = h (a, b, c) h f(x, y, z) = h + h f(a, b, c) f = h (a, b, c) f(a, b, c) f = h + h (a, b, c) ( x, y, z) ( x, y, z) = λ f(a, b, c) h = f(a, b, c) ( x, y, z) = λ f(a, b, c) 2 f(a, b, c) = h ( x)2 + ( y) 2 + ( z) 2 f(a, b, c) 0 (a, b, c) f (regular point) f (singular point). f(x) {x; f(x) = c} level set x R (level surface) x R 2 (level curve or contour line) 8
3.2. f(x, y, z) = h (a, b, c) f x (a, b, c)(x a) + f y (a, b, c)(y b) + f z (a, b, c)(z c) = 0, x y = a + tf x(a, b, c) b + tf y (a, b, c), t R. z c + tf z (a, b, c)) (vector field) 3.3. r = x 2 + y 2 + z 2 ( x r, y r, z r ) = r r. 6. r n 3.4. x 2 + y 2 + z 2 = 1 (a, b, c) ax + by + cz = 1. 7. f(x, y, z) = x cos(xy + z) f(x, y, z) = 0 (1, π/3, π/6) 4 9
4.1. (vector field) (a + x, b + y, c + z) (a, b, c) F (a, b, c) = lim t 0 t 4.2. (i) (ii) F : R 2 R 2 F ( ) x = y ( ) F1 (x, y) F 2 (x, y) F 1, F 2 10
. F (x, y) = (F 1 (x, y), F 2 (x, y)) 4.3. F (x, y) F 2 x = F 1 y Proof. F 1 = ϕ x, F 2 = ϕ y F 2 x = ( ) ϕ = ( ) ϕ = F 1 y x x y y.. 4.4. F (x, y) = (αx + βy, γx + δy) β = γ ϕ x = αx + βy ϕ(x, y) = 1 2 αx2 + βxy + f(y). ϕ y = βx + δy f (y) = δy f(y) = δy 2 /2 + c F ϕ(x, y) = αx 2 /2 + βxy + δy 2 /2 + c ( α β F (x, y) = γ δ ) ( ) x = A y ( ) x y A 11
8. F (x 1, x 2, x 3 ) 3 L F (x 1, x 2, x 3 ) = L F L 9. F (x, y) = (x y 2, 2xy + 2y 3 ) F = f f x 1 x 2 x 3 4.5. (i) (constant vector field) F (x, y) = (u, v). (ii) F (x, y) = (x, y). (iii) F (x, y) = ( y, x). 10. (i) F (x, y) = (x, 0), (ii) F (x, y) = (x, y). (x(t), y(t)) F (x, y) (flow line, streamline) d dt ( ) x(t) = F y(t) ( ) x(t) = y(t) ( ) F1 (x(t), y(t)) F 2 (x(t), y(t)) (integral curve) 4.6. f F (a, b) 0 (a, b) F (x, y) F (a, b) 12
dx dt = F 1(a, b), dy dt = F 2(a, b) x(t) = F 1 (a, b)t + x(0), y(t) = F 2 (a, b)t + y(0) (a, b) (a, b) F 1 (x, y) F 11 (x a) + F 12 (y b), F 2 (x, y) F 21 (x a) + F 22 (y b), F 11 = F 1 x (a, b), F 12 = F 1 y (a, b), F 21 = F 2 x (a, b), F 21 = F 2 y (a, b), ( ) ( ) F11 F F (x, y) = 12 x a F 21 F 22 y b linearization, 11. F (x, y) = (x 2 + y, x + y + 2) 12. F (x, y) = (sin(x + y), sin(x y)) (π/2, π/2) 13. 4.7. A, B T B = T AT 1 13
4.8. ( ) a cos θ a sin θ A =, a > 0. a sin θ a cos θ 14. F (x, y) = (sin(x y), x + y) 15. F (x, y) = (x 1, y 1) (i) (ii) (iii). 5 line integral F C C F (r) dr = lim 0 j=1 n F (r j ) (r j r j 1 ) r 0,..., r n C = max{ r j r j 1 ; 1 j n} 5.1. C r(t) (a t b) b a F (r(t)) dr (t) dt dt 14
Proof. r j r j 1 = r(t j ) r(t j 1 ) dr dt (t j)(t j t j 1 ) lim F (φ(tj )) dr 0 dt (t j)(t j t j 1 ) = b a F (φ(t)) dr (t) dt dt. r = (x, y, z) dr = (dx, dy, dz) F 1 (x, y, z)dx + F 2 (x, y, z)dy + F 3 (x, y, z)dz C b ( F 1 (x(t), y(t), z(t)) dx dt + F 2(x(t), y(t), z(t)) dy dt + F 3(x(t), y(t), z(t)) dz ) dt dt a Leibniz 5.2. ( t, t + 1) (0 t 1) C 1 (x y)dx + xydy = ( t (t + 1))( 1)dt + C 0 15 1 0 ( t)(t + 1)dt =
(0, 0) (1, 1) 0 16. 5.3 ( ). (af (r) + bg(r)) dr = a F (r) dr + b G(r) dr C C C F (r) dr = F (r) dr + C 1 +C 2 C 1 F (r) dr, C 2 F (r) dr = F (r) dr. C F C F, C af + bg, C = a F, C + b G, C, F, C 1 + C 2 = F, C 1 + F, C 2 C (piece-wise smooth) 16
17. F (x, y, z) = (y, z, x) r C r,φ : (x, y, z) = (r sin t cos φ, r sin t sin φ, r cos t), 0 t π 5.4. C F (r) dr C F C. C C F C = max{ F (r) ; r C} 5.5. f dr = f(r f ) f(r i ) C F F (potential energy) 5.6. 1/r ( ) 1 = r r r 3 17
5.7. F (x, y, z) = (f(x), g(y), h(z)) f(x) dx + g(y) dy + h(z) dz 6 C {r(t)} (a t b) r(a) = r(b) (closed curve) (circulation) F (r) dr C (contour integral) 0 6.1. f(x)dx + g(y)dy + h(z)dz = 0. C 6.2. ( ) ( ) cos θ sin θ x F (x, y) = sin θ cos θ y C : (x, y) = (cos t, sin t) (0 t 2π) 2π 0 ( cos(t + θ) sin t + sin(t + θ) cos t) dt = 2π 0 sin(t + θ t) dt = 2π sin θ. 18. 18
19. F (x, y) = r n ( y, x) C : (a cos t, a sin t) (0 t 2π) a > 0 n D D 6.3. D xdy = ydx = D. D Proof. 6.4 ( ). lim D p 1 F (r) dr = F 2 D D x (p) F 1 y (p). Proof. p = (a, b) (x, y) (a, b) F (x, y) F (a, b) + F (a, b) ( ) ( ( x a = F (a, b) F a (a, b) + F y b b) x (a, b) y) F (a, b) = ( F1 x (a, b) ) F 1 ( ) y (a, b) α β x (a, b) F 2 = + y (a, b) β α F 2 ( ) 0 ω ω 0 ( ( ( ) G(x, y) = F (a, b) F a α β x (a, b) + b) β α) y 0 F (r) dr ω ( ydx + xdy) = 2ω D D D 2ω = (β + ω) (β ω) = F 2 x (a, b) F 1 (a, b) y 20. n A 19
y+ y f y (x, y) dy = f(x, y +) f(x, y ). D = {(x, y); a x b, ϕ (x) y ϕ + (x)} ( b ) ϕ+ (x) f(x, y) dxdy = f(x, y) dy dx 6.5 (George Green, 1828). F (r) dr = D D Proof. a D ϕ (x) ( F2 x F ) 1 dxdy. y (i) (ii) D y = ϕ ± (x) (a x b) D = {(x, y); ϕ (x) y ϕ + (x)} D ϕ ± D F 1 (x, y) dx = b = a b (F 1 (x, ϕ (x)) F 1 (x, ϕ + (x))) dx a = D ϕ+ (x) dx ϕ (x) F 1 (x, y)dy y F 1 (x, y) dxdy. x 6.6. F F F 2 x = F 1 y 6.7. F (x, y) = (xy, x) D = {(x, y); 0 x, y 1} D xydx + xdy = (xydx + xdy) = 0 + C 1 +C 2 +C 3 +C 4 D 1 ( x x (xy) ) 1 1 dxdy = dx dy(1 x) = y 0 0 20 0 dy 1 0 1 0 xdx + 0 = 1 2. (1 x) dx = 1 2.
21. D = {(x, y); 0 x 1, 0 y 1, x + y 1} 6.8. F (x, y) = ( ) y x 2 + y 2, x x 2 + y 2 x x x 2 + y 2 y y x 2 + y 2 = 0 0 x 2 + y 2 1 x 2 + y 2 = 1 2π 22. 0 ( sin t(cos t) + cos t(sin t) ) dt = 2π F (x, y) (a, b) F (a, b) F (x, y) (x, y) = (a, b) II 21
f(x, y)dxdy = f(x(u, v), y(u, v)) (x, y) (u, v) dudv, (x, y) (u, v) = x u y D E 6.9. (x, y) = (r cos θ, r sin θ) (x, y) (r, θ) = r. 6.10. a 2 x 2 + y 2 b 2 (0 < a < b) (x, y) = (r cos θ, r sin θ) (a r b, 0 θ 4π) b 4π a 0 rdθdr = 2π(b 2 a 2 ) π(b 2 a 2 ) 22 u x v y v
6.11. (x, y) = (x(u, v), y(u, v)) (u, v) = (u(s, t), v(s, t)) (s, t) (x, y) (x, y) (s, t) (x, y) (u, v) = (u, v) (s, t) (u, v) u(s, t), v(s, t) s t 7 e v e v e v v e = (e v)e, v = v (e v)e. 23. ( 1, 1, 1) C: (t, t 2, t) t = 1 1 D (area vector) D = D n D D n a, b n 23
a b a b a b (outer product) a b = 0 a b b a = a b 24. 25. a, b θ (0 θ π) a b = a b sin θ. 7.1. a, b, c (a 1, a 2, a 3 ), (b 1, b 2, b 3 ), (c 1, c 2, c 3 ) a 1 a 2 a 3 (a b) c = b 1 b 2 b 3 c 1 c 2 c 3. Proof. [a, b, c] [a, b, c] 1 7.2. a b a, b (a 2 b 3 a 3 b 2, a 3 b 1 a 1 b 3, a 1 b 2 a 2 b 1 ).. 26. 27. (a b) c = (b c) a = (c a) b. a (b c) = (a c)b (a b)c. a (b c) + b (c a) + c (a b) = 0. 7.3. O, A, B, C 1 [ OA, OB, OC] 6 24
28. O A(1, 1, 1), B( 1, 1, 2), C(3, 1, 1) 7.4 ( ). Proof. O, A, B, C a = OA, b = OB, c = OC OAB = a b, OBC = b c, OCA = c a, ABC = (b a) (c a) 7.5. D D e e D D D 29. A(a, 0, 0), B(0, b, 0), C(0, 0, c) ABC 30. O(0, 0, 0), A(1, 1, 1), B(1, 2, 1) H a b C(2, 1, 1) H 8 x, y, z f(x, y, z) = 0 8.1. (i) x 2 + y 2 + z 2 a 2 = 0 a > 0 (ii) x 2 + y 2 z 2 a 2 = 0 (hyperboloid of one sheet). (iii) x 2 y 2 + z 2 a 2 = 0 (hyperboloid of two sheets). (iv) x 2 + y 2 z = 0 (paraboloid). f(x, y, z) = h. 3 f(x, y, z) (a, b, c) 25
f(x, y, z) f(a, b, c) + f f f (a, b, c)(x a) + (a, b, c)(y b) + (a, b, c)(z c) x y z f(a, b, c) (0, 0, 0) f z 0 f(x, y, z) = f(a, b, c) z z = h(x, y) f(x, y, z) = C z = g(x, y) g g 9 (surface) (u, v) r(u, v) = (x(u, v), y(u, v), z(u, v)), (u, v) D D uv- 9.1. a = (a 1, a 2, a 3 ), b = (b 1, b 2, b 3 ) r = ua + vb, 0 u, v 1 31. OAB A(a 1, a 2, a 3 ), B(b 1, b 2, b 3 ) 32. u, v 26
9.2. r > 0 x = r sin θ cos φ, y = r sin θ cos φ, z = r cos θ, 0 θ π, 0 φ 2π r = r(u, v) r u (u, v) = (x u (u, v), y u (u, v), z u (u, v)), r v (u, v) = (x v (u, v), y v (u, v), z v (u, v)) r u r v r(u, v), r(u + u, v), r(u, v + v), r(u + u,, v + v) S = r u r v u v S = r u r v u v (surface area) 9.3. r D r u r v dudv r = (x, y, z) = (r sin θ cos φ, r sin θ sin φ, r cos θ), 0 θ π, 0 φ 2π. r θ = (r cos θ cos φ, r cos θ sin φ, r sin θ), r φ = ( r sin θ sin φ, r sin θ cos φ, 0) r θ r φ = r 2 sin θ(sin θ cos φ, sin θ sin φ, cos θ) r 2 π 0 2π dθ dφ sin θ = 4πr 2. 0 33. r = (1 s)(0, 0, t) + s(cos t, sin t, t) (0 s 1, 0 t π) 34. r = (u, v, u 2 + v 2 ) (u 2 + v 2 1) 35. x 2 + y 2 + (z/c) 2 = r 2 5.0995 10 8 km 2 27
36. (x/a) 2 + (y/b) 2 + (z/c) 2 = 1 (a, b, c) = (r, r, r) 37. r = (u + v, u 2 v, u v 2 ) r(1, 2) 38. 39. y = f(x) (a x b) x R 2 D, D D D φ (i) φ(u, v) = (φ 1 (u, v), φ 2 (u, v)) φ 1, φ 2 (ii) φ (φ 1, φ 2 ) 0 (u, v) φ 1 : D D (φ 1, φ 2 ) > 0 (u, v) 9.4. 9.5. a 4πa 2 (x, y, z) = (r cos θ, r sin θ, a 2 r 2 ), 0 r a, 0 θ 2π z = r 2 x 2 y 2 (x 2 + y 2 r 2 ) (Schwarz lantern) 28
10 r u r v ds ds = r u r v dudv 10.1. F (x, y, z) S:(x(u, v), y(u, v), z(u, v)) (surface integral) F (r) ds = S D F (r(u, v)) (r u r v ) dudv F (flux) 10.2. F (r) = v D v D v D D 40. F (r) = c r = ua + vb (0 u, v 1) det(a, b, c) 10.3. D f(x, y) z = f(x, y) S f S f z F S f ( F 3 (u, v, f(u, v)) F 1 (u, v, f(u, v)) f u (u, v) F 2(u, v, f(u, v)) f ) (u, v) dudv v D 29
S f r(u, v) = (u, v, f(u, v)) ((u, v) D) r u = (1, 0, f u ), r v = (0, 1, f v ), r u r v = ( f u, f v, 1). 41. f(x, y) = c(1 x/a y/b) (x 0, y 0, x/a + y/b 1) a > 0, b > 0, c > 0 F (x, y, z) = (x, y, z) S f 10.4. Proof. 10.5. F S F, S af + bg, S = a F, S + b G, S, F, S 1 + S 2 = F, S 1 + F, S 2, F, S = F, S. 10.6. (a, 0, 0), (0, b, 0), (0, 0, c) F (x, y, z) = (yz, zx, xy) 10.7. S 1 r 3 r ds S (solid angle) S 1 r(u, v) ρ(u, v) = r(u, v)/r(u, v) ρ u = 1 r r u r r u r 3 r, ρ v = 1 r r v r r v r 3 r ρ u ρ v = 1 r 2 r u r v r u r r 3 30 r r v + r v r r 3 r r u
ρ u ρ v = ρ (ρ u ρ v ) = 1 r 3 r (r u r v ) 42. ABC (A = (1, 0, 0), B = (0, 1, 0), C = (0, 0, 1)) dudv (u 2 + v 2 + (1 u v) 2 ) = π 3/2 2 D D = {(u, v); u 0, v 0, u + v 1} 43. F (x, y, z) = r n (x, y, z) a > 0 a n dx dy = (x, y) (u, v) du dv r u r v dudv (dydz, dzdx, dxdy) dxdy = dydx dudv = dvdu dx dy = dy dx dx dy = (x, y) du dv (u, v) r u r v du dv = (dy dz, dz dx, dx dy) (F 1 (x, y, z)dy dz + F 2 (x, y, z)dz dx + F 3 (x, y, z)dx dy) S dx = x x du + u v dv, y y dy = du + u v dv 31
dx dy du du = dv dv = 0 ( ) x x dx dy = du + u v dv ( ) ( y y x du + u v dv y = u v y u ) x du dv v (differential form) (F 1 (x, y)dx + F 2 (x, y)dy) = D D (df 1 dx + df 2 dy) = D ( F2 x F ) 1 dx dy y r = (x, y, z) x = (x 1, x 2, x 3 ) dx = (dx 1, dx 2, dx 3 ), d 2 x = (dx 2 dx 3, dx 3 dx 1, dx 1 dx 2 ), d 3 x = dx 1 dx 2 dx 3 F (x) d 2 x = F 1 (x)dx 2 dx 3 + F 2 (x)dx 3 dx 1 + F 3 (x)dx 3 dx 1 S S 1 x y F (y) d 2y S 11 (closed surface) V 32
V V V = {(x, y, z); x 2 + y 2 + z 2 1} V = {(x, y, z); x 2 + y 2 + z 2 = 1} 44. {(x, y, z); x a, y b, z c} 11.1. V F (x) d 2 x = V ( F1 + F 2 + F ) 3 dx 1 dx 2 dx 3. x 1 x 2 x 3 11.2. F (divergence) F div F Proof. V f(x, y) z g(x, y) ((x, y) D) S g S f F 3 e 3 ds = V = D D = (F 3 (x, y, g(x, y)) F 3 (x, y, f(x, y))) dxdy g(x,y) dxdy f(x,y) V F 3 (x, y, z) dz z F 3 (x, y, z) dxdydz. z x y. divergence F convergence 11.3. 1 F (p) = lim F (r) ds V p V V 33
45. S S ds = 0 ρ ( ρgz + p 0 a (ρgz + p 0 )ds = (ρgz + p 0 )a ds = ρga 3 dxdydz = a 3 ρg V V ρg V V V ρ(t, r) F (t, r) V ρ(t, r)f (t, r) ds. V d dt V V ρ(t, r) dxdydz = V (ρf ) dxdydz = V V ρ t dxdydz ρ t dxdydz V (ρf ) + ρ t = 0. ρ ρf (density function) ρ (current density) J J + ρ t = 0 (continuity equation) (conservation law) 34
46. V 1 r 3 r ds V x x 3 = 4πδ(x). 47. f(x, y) G f dxdy = f(0, 0) 2 R G(x, y) r = x 2 + y 2 12 0 A = 0 γ β γ 0 α β α 0 ABC (A(a, 0, 0), B(0, b, 0), C(0, 0, c)) ( ABC) ( ABC) ABC F (r) dr = αbc + βca + γab ABC = 1 (bc, ca, ab) 2 2α = F 3 y F 2 z, 2β = F 1 z F 3 x, 2γ = F 2 x F 1 y ( ABC) F (r) dr = ( F ) ABC 35
F = ( F3 y F 2 z, F 1 z F 3 x, F 2 x F ) 1 y F (rotation, curl) S (boundary) S S 12.1 (Stokes). F (r) dr = S S ( F ) ds. Proof. S F 3 (x, y, z)dz = D ( F3 y (y, z) (u, v) F 3 x ) (z, x) dudv (u, v) f(u, v) = F 3 (x(u, v), y(u, v), z(u, v)) u-v C: (u(t), v(t)) S : (x(u(t), v(t)), y(u(t), v(t)), z(u(t), v(t))) S b F 3 (x, y, z)dz = a = f(u, v) C = ( z f(u(t), v(t)) u D ( z u ( f z u v f v du dt + z v ) z du + v dv ) z dudv. u ) dv dt dt f u = F 3 x x u + F 3 y y u + F 3 z z u, f v = F 3 x x v + F 3 y y v + F 3 z z v 36
48. ( F1 F 1 (x, y, z)dx = S D z ( F2 F 2 (x, y, z)dy = x S D (z, x) (u, v) F 1 y (x, y) (u, v) F 2 z ) (x, y) dudv, (u, v) ) (y, z) dudv (u, v) 12.2. F (x, y, z) D F (r) dr = ( F ) D. D. F rotf, curlf rot curl rot rot rot Gibbs curl Maxwell Gibbs 12.3. F (i) F = G G (ii) F S (i) (ii) Stokes (ii) (i) A F (x, y) ( ) ( ) ( ) x u x + L y v y d dt ( ) x = y ( ) u + L v ( ) x y 37
L T t (x, y) = L 1 (e tl I) ( ) ( ) u + e tl x v y L L = S + A, t S = S, t A = A L 1 (e tl I) = ti + O(t 2 ), e tl = e ta e ts + O(t 2 ) T t (x, y) = t ( ) ( ) u + e ta e ts x + O(t 2 ) v y e ts S ( ) e ts x = y ( ( ) e λt 0 x 0 e µt) y x e λt y e µt e ta ( e ta cos(ωt) sin(ωt) = sin(ωt) cos(ωt) ), A = ω ( ) 0 ω ω 0 38