436 8., {(x, y) R 2 : y = x, < x 1} (, 1] φ(t) = (t, t), (, 2] ψ(t) = (t/2, t/2), [1, ) σ(t) = (1/t, 1/t).. ψ φ, σ φ. (φ, I) φ(i) φ : I φ(i). 8.2 I =

8. 8.1 ( ).,,,.. 8.1 C I R φ : I R m φ (φ I ) φ(i) = {x R m : x = φ(t), t I} C, t, I. C C = (φ, I). x R m C C. 1 x, a R m. φ(t) := ta + x R ( 2). x a. R m. 2 φ(t) = (cos t, sin t) [, 2π].. 435

436 8., {(x, y) R 2 : y = x, < x 1} (, 1] φ(t) = (t, t), (, 2] ψ(t) = (t/2, t/2), [1, ) σ(t) = (1/t, 1/t).. ψ φ, σ φ. (φ, I) φ(i) φ : I φ(i). 8.2 I = [a, b] C = (φ, I) φ(a), φ(b) C. φ(a) = φ(b) (φ, [a, b]). φ I 1-1, (φ, I), φ [a, b) 1-1 φ(a) = φ(b), (φ, [a, b]). R 2 (φ, I) R 2, E Ω (, E = Ω = φ(i)). R m φ : R R m. [, 1] [, 1] [, 1]. 1 (pace-filling) 2.,,... m, n, p N E R n. f : E R m E C p V E V j p g : V R x E f(x) = g(x). f g. x E k = 1, 2,, n, j = 1, 2,, m f j / x k (x) = g j / x k (x). p N E f C p f : E R m E C. 1). 2).. R 2 I X.[?]

8.1 437 p. 8.3 I ( ). R m C p C = (φ, I) φ : I R m I C p. (φ, I) C p ψ(j) = φ(i) C p (ψ, J). x j = ψ j (t), t J, j = 1,, m (ψ, J) (φ, I). C p C p. 3 I f : I R C p. y = f(x), x I 1-1 C p φ : I R 2. φ(t) = (t, f(t)) φ I C p 1-1. φ(i) I y = f(x). 4 x 2 + y 2 = a 2 R 2 C. x 2 + y 2 = a 2 (x, y) R 2 φ(t) = (a cos t, b sin t) I = [, 2π] C.( 8.1). y a a x 8.1

438 8 8.4 C p C = (φ, I) k L(C) := sup φ(t j ) φ(t j 1 ) : {t, t 1,, t k } I j=1. L(C) C. C p p 1. φ 4. 8.1 C = (φ, I) C 1 C L(C) = φ (t) dt. I ɛ >. φ = (φ 1, φ 2, φ m ) I m := I I (x 1, x 2,, x m ) ( m F (x 1, x 2,, x m ) = φ l(x l ) 2) 1/2. F I m, I m. F I m. δ > l=1 x, y I m, x y < δ = F (x) F (y) < ɛ 2 I. P = {u,, u N } I. 3.5 P P = {t,, t k } P < δ/ m I φ (t) dt ɛ 2 < k φ (t j ) (t j t j 1 ) < φ (t) dt + ɛ 2. j=1 l {1, 2,, m} j {1, 2,, k}. 2.2( ) c j (l) [t j 1, t j ] φ l (t j ) φ l (t j 1 ) = φ l(c j (l))(t j t j 1 ). P < δ/ m F (t 1,, t j ) F (c j (1),, c j (m)) < ɛ/(2 I ). φ (t) = (φ 1(t),, φ m(t)) F (t 1,, t j ) = φ (t j ) F (c j (1),, c j (m))(t j t j 1 ) = I ( m φ l(c j (l)) 2) 1/2 (tj t j 1 ) l=1 = φ(t j ) φ(t j 1 ).

8.1 439 k φ (t j ) (t j t j 1 ) ɛ k k 2 < φ(t j ) φ(t j 1 ) < φ (t j ) (t j t j 1 ) + ɛ 2. j=1 j=1 j=1 k φ (t) dt ɛ < φ(y j ) φ(t j 1 ) < φ (t) dt + ɛ I j=1 I 8.4 k L(C) φ(t j ) φ(t j 1 ) > φ (t) dt ɛ, j=1 I, L(C) I φ (t) dt., P = {t, t 1,, t k } P N k φ(u i ) φ(u i 1 ) φ(t j ) φ(t j 1 ) < φ (t) dt + ɛ. i=1 j=1 I I {u, u 1,, u N } L(C) φ (t) dt + ɛ, I, L(C) I φ (t) dt. p φ (φ, I). 1 t I o x = φ(t ) φ (t ) x p φ (t ) x C. t I. h > φ(t + h) φ(t ) h C x C ( 8.2). p C s. s := l(t) := t a φ (u) du, t [a, b]. (8.1)

44 8 T (t h) (t ) h (t h) (t ) (I ) 8.2 ds/dt = l (t) = φ (t). s t φ (t). x p φ (t )..,. C p. 5 φ(t) = (cos 3 t, sin 3 t) I = [, 2π]. (φ, I) R 2 C. ( (astroid).) φ I C [, 2π) 1-1. x = cos 3 t, y = sin 3 t x 2 + y 2 = 3 4 cos3 (2t) + 1 4. x 2 + y 2 (t =, π/2, π, 3π/2, 2π ) 1 (t = π/4, 3π/4, 5π/4, 7π/4 ) 1/2. I, φ, φ(i). φ(i) (, 1) ( 8.3). φ(i) (x, y ) R 2 (φ, I) (x, y ) = φ(t ). t I o I, J f : J R t I I x J J y = f(x) φ(i ), y J J x = f(y) φ(i ),.? (φ, I) ( 5). t =, π/2, π, 3π/2, 2π φ (t) = (, )., (φ, I). (φ, I) φ (t ) (x, y ) = φ(t )

8.1 441 y 1 2 1 x 8.3.. 2 (φ, I) t I o (x, y ) = φ(t ) R 2 C 1, φ (t ) =. (φ 1, φ 2 ) φ. φ (t ). φ 1(t ). F (x, t) = φ 1 (t) x, x J g : J I x J φ 1 (g(x)) = x g(x ) = t. f = φ 2 g, y = f(x), x J (x, y ) g(j) φ. f x y = f(x) x... 8.5 (φ, I) C 1. (i) φ (t ) (φ, I) t I. (ii) I (φ, I), (φ, [a, b)) φ (a) = φ (b), (φ, [a, b]). (iii) C = (φ, I) x T (x ) := φ (t )/ φ (t ). = φ(t ) C

442 8 2.. 3. ψ(t) = (t 3, 3t 3 ). ψ(t) y = 3x. ψ () =.(, φ(t) = (t, 3t) φ ().)., 8.1?. 8.1 I, J φ : I R m 1-1. ψ : J R m φ(i) = ψ(j) J I τ ψ = φ τ. I φ 1-1 I φ 1 φ(i) I (5.5 9). ψ(j) = φ(i) τ = φ 1 ψ J I. τ J I ψ = φ τ J φ(i). ψ(j) = φ(i). (ψ, J), (φ, I) C 1 τ = φ 1 ψ ψ (u) = φ (τ(u))τ (u), u J. (8.2), (φ, I) (ψ, J) u J τ (u).. 8.6 p 1. C p (φ, I), (ψ, J) φ(i) = ψ(j) J I C p τ : J R ψ = φ τ u J τ (u) (smoothly equivalent). τ ψ(j) φ(i) (transition).

8.1 443 τ τ J τ J. 2.22 τ 1-1. (8.4 5). 8.2 (φ, I) (ψ, J) φ (t) dt = ψ (u) du. I J τ ψ(j) φ(i) τ(j) = I. (8.2) ( 7.16) φ (t) dt = φ (t) dt = I τ(j) J φ (τ(u)) τ (u) du = J ψ (u) du. u J τ (u) 8.2 ( 8). g. 8.7 C = (φ, I) R m g : φ(i) R. C g g ds := C φ(i) g ds := g(φ(t)) φ (t) dt. (8.3) y = f(x), x [a, b] C C g ds = b a I g(x, f(x)) 1 + f (x) 2 dx. 8.1 (8.3) g = 1 C. s ( (8.1)), ds/dt = φ (t). s ds = φ (t) dt. g. 8.5 3. 6 g(x, y) = 2xy, φ(t) = (cos t, sin t) I = [, π/2] g ds φ(i).

444 8 φ (t) = ( sin t, cos t) = 1 π/2 g ds = (2 cos t sin t) dt = π/2 φ(i) sin 2t dt = 1.. C j C = N j=1 C j. C I. j k C j C k. ( < a 2 < x 2 + y 2 < b 2.) II. C j. C j = (φ j, [a, b]) j = 2,, N φ j 1 (b j 1 ) = φ j (a j ) ( 8.4) (,,.) (piecewise smooth curve). C II φ(a 1 ) φ(b N ) C. 2(b 2 ) 3(a 3 ) 1(a 1 ) 1(b 1 ) 2(a 2 ) 3(b 3 ) 4(a 4 ) 4(b 4 ) 8.4 C = N j=1 C j. C C j. C C j (φ j, I j ). N j=1 I j (8.2 4). C N j=1 (φ j, I j ) N j=1 (ψ j, I j ) j {1, 2,, N} (φ j, I j ) (ψ j, I j ). C j C, N L(C) = L(C j ) j=1

8.1 445. g C C g N g ds = g ds C j C j=1. C C,. 7 [, 1] [, 1] C g(x, y) = x 3 + y 2 g ds. C C t [, 1] φ 1 (t) = (t, ), φ 2 (t) = (1, t), φ 3 (t) = (1 t, 1), φ 4 (t) = (, 1 t). φ j (t) = 1 C g ds = 1 t 3 dt + 1 (1 + t 2 ) dt + 1 ((1 t) 2 + 1) dt + 1 (1 t) 2 dt = 13 4. 8 ( 8.5) C g(x, y) = x + xy 2 C g ds. C t [, 1] C 1 : φ 1 (t) = (2t, ), C 2 : φ 2 (t) = (2 t, 3(2 t), C 3 : φ 3 (t) = (t, 3t). C = C 1 + C 2 + C 3 g ds = C g ds + C 1 g ds + C 2 g ds C 3. φ i (t) = 2, i = 1, 2, 3 C 1 g ds = 2 1 (2t) dt = 2,

446 8 1 g ds = 2 (2 t) + 3(2 t) 3 dt = 51 C 2 2, 1 g ds = 2 (t + 3t 3 ) dt = 5 C 3 2 g dt = 3. C y (1, 3 ) 3(t ) 2(t ) O 1(t ) 2 x 8.5 (8.1) 1. ψ(t) = (a sin t, a cos t), σ(t) = (a cos 2t, a sin 2t), I = [, 2π), J = [, π). (ψ(t), I) (σ(t), J). 2. x, a R m, a φ(t) = ta + x. C = (φ, R) x x + a. t 1, t 2 φ(t 1 ) φ() φ(t 2 ) φ() π. 3. I f : I R θ I f(θ) 2 + f (θ) 2. r = f(θ) R 2.

8.1 447 4. y = sin(1/x), < x 1. 8.1 C. 5.,. (a) φ(t) = (sin t, cos t, e t ), t [, 2π] (b) y 3 = x 2, ( 1, 1) (1, 1) (c) φ(t) = (t 3, t 2, t), t [, 2] (d) 5 6. C ( ) C g ds. (a) C y = 9 x 2, x g(x, y) = xy (b) C x 2 /a 2 + y 2 /b 2 = 1, a, b > g(x, y) = xy. (c) C x 2 + z 2 = 4 y = x 2 g(x, y, z) = 1 + yz 2. (d) C (,, ), (1,, ), (, 2, ) g(x, y, z) = x + y + z 3. 7. (φ, I) g k : φ(i) R k N. (a) φ(i) g k g k C g k ds g ds. C (b) {g k } k φ(i) g k g. g φ(i) k C g k ds g ds C. 8. (φ, I) R m τ : J R J I 1-1, C 1. u J τ (u) ψ = φ τ g : φ(i) R. g(φ(t)) φ (t) dt = g(ψ(u)) ψ (u) du. I J

448 8 9. C I 1 = (, 1), I 2 = ( 1, ) (φ, I 1 ) (φ, I 2 ), φ(t) = ( 3t 1 + t 3, 3t 2 ) 1 + t 3. (x, y) = φ(t) x 3 + y 3 = 3xy. C. 1. x = ψ(t ) (ψ, I) θ(t) κ(x ) = lim t t l(t). θ(t) ψ (t) ψ (t ), l(t) ψ(t) ψ(t ) (ψ, I). ( κ θ(t).) (a) a, b R n, ψ(t) := ta + b I := (, ) Λ = (ψ, I) Λ x. (b) (a, b) R 2 ψ(t) = (r cos t, r sin t) C C x 1/r. 11. C = (φ, [a, b]) R m (1) s = l(t). C (natural parametrization) (ν, [, L]). ν(s) = (φ l 1 )(s), L = L(C) (a) s [, L] ν (s) = 1 C (ν, [c, d]) d c. (b) s [, L] ν (s) ν (s). (c) x = ν(s ) (ν, [, L]) ( 1) κ(x ) = ν (s ). (d) x = φ(t ) = ν(s ). κ(x ) = ν (s ) ν (s ) = φ (t ) φ (t ) φ (t ) 3

8.2 449 (e) p 1. (x, y ) C p y = f(x). κ = y (x ) (1 + (y (x )) 2 ) 3/2 8.2 C (φ, I) C. φ(t) t I φ (t). (φ, I) C (orientation) ( 8.1, 8.3, 8.4, 8.5 ). (φ, I) (ψ, J) τ. τ u J τ (u) > τ (u) <,., φ (τ(u)) ψ (u) (8.1 (8.2))., φ (τ(u)) ψ (u).. 8.8 (φ, I) (ψ, J) (orientation equivalent) ψ(j) φ(i) τ u J τ (u) >. 1 z-, R 3 x 2 + 5y 2 = 5 z = x 2 C. x 2 + 5y 2 = 5 z = x 2 ( 8.6 x 2 + 5y 2 = 5 z = x 2 ). x = 5 sin t, y = cos t x 2 + 5y 2 = 5 z = x 2 = 5 sin 2 t. C I = [, 2π] φ(t) = ( 5 sin t, cos t, 5 sin 2 t).

45 8 z x 5 1 y 8.6 2 xy- y = x R 3 z = x 2 y 2 x + y = 1 C. z = x 2 y 2 x + y = 1. x = t y = 1 t z = t 2 (1 t) 2 = 2t 1. C I = R φ(t) = (t, 1 t, 2t 1)., C (, 1, 1) (1, 1, 2).,. 8.9 C = (φ, I) R m F : φ(i) R m. C F (orientated line integral) F T ds := F T ds := F dφ := F (φ(t)) φ (t) dt. (8.4) C φ(i) φ(i) I T = φ (t)/ φ (t) C ds = φ (t) dt C. F T ds = F φ (t) dt. F F T F., T., C F (x, y) = ( y, x). (x, y) C ( y, x) F T ( 8.7). F T = 1.,

8.2 451 G(x, y) = (y, x) H(x, y) = (x, y) G T = 1 T H T = (, )( 8.7). C F T ds C F. T T. y H G T F (x, y) H (x, y) F T C (x, y) G (x, y) G T F x H F T G H 8.7 8.9. 3. 3 φ(t) = (cos t, sin t, t), t I = [, 4π] F (x, y, z) = (sin z, cos z, xy) C = (φ, I). F T ds C (x, y, z) = φ(t). x 2 + y 2 = 1 C x 2 + y 2 = 1, z 4π. t (x, y) x 2 + y 2 = 1. φ x 2 + y 2 = 1 ( 8.8). t 4π z

452 8 4π. φ (t) = ( sin t, cos t, 1) F T ds = C = = 4π 4π 4π F (φ(t)) φ (t) dt (sin t, cos t, cos t sin t) ( sin t, cos t, 1) dt ( sin 2 t + cos 2 t + sin t cos t) dt =. z 4 2 x 1 1 y 8.8 C g ds F T ds C. 1 (φ, I) (ψ, J) I F (φ(t)) φ (t) dt = F (ψ(u)) ψ (u) du. J τ ψ(j) φ(i). τ τ J. φ(i) ψ(j) τ J., u J τ (u) = τ (u). I F (φ(t)) φ (t) dt = F (φ(τ(u)) φ (τ(u)) τ (u) du J = F (ψ(u)) ψ (u) du. J

8.2 453 C C C 1 F T ds = F T ds C C. (8.4) ( 5). 4 F (x, y) = (xy, x) C x 2 + y 2 = 1. C F T ds C φ(t) = (cos t, sin t), t [, 2π] (8.1 2). 1 2π F T ds = (sin t cos t, cos t) ( sin t, cos t) dt C = 2π (sin 2 t cos t cos 2 t) dt = π. (8.4). x j = φ j (t) dx j = φ j (t) dt. F (φ(t)) φ (t) dt (F 1 (φ 1 (t))φ 1(t) + + F m (φ(t))φ m(t)) dt = F 1 dx 1 + + F m dx m. R m 1 (1-form) F j. 1 E 1 E. R m C = (φ, I) 1 F = (F 1,, F m ) F 1 dx 1 + + F m dx m := F T ds C φ(i). 5 C (, ) (π, π 2 ) y = x 2 + sin x y dx + cos x dy. C

454 8 y = x 2 + sin x dy = (2x + cos x) dx π y dx + cos x dy = (x 2 + sin x) dx + C = π3 3 + π 2 2. π cos x(2x + cos x) dx ( ) ( 7)... ( 8).. 3. C = N j=1 C j R m E (8.1 5). F : E R m C F N F T ds = F T ds C j C j=1. ω E 1, C ω N ω = C j=1 C j ω. 6 C Q = [, 1] [, 1] (x 2 + y 2 ) dx + xy dy. C C = Q ( 8.9). C 1 ( x = ), C 2 ( y = ), C 3 ( x = 1), C 4 ( y = 1). C 1 x = y 1 3) (simply connected domain).[?]

8.2 455 y 1 C 4 C 1 C 3 O C 2 1 x 8.9 (C ). (x 2 + y 2 ) dx + xy dy = y 2 dy = 1 C 1 1 3., C 2, C 3, C 4, 1/2, 2/3. F T ds = 1 3 + + 1 2 2 3 = 1 2. C (8.2) 1. (φ, R),,. (a) φ(t) = (3t, 2 sin t, cos t), = {(x, y, z) : y 2 + 4z 2 = 4}. (b) φ(t) = (t 3, t 2, t 3 ), = {(x, y, z) : z = x}. (c) φ(t) = (t, t 3, sin t), = {(x, y, z) : y = x 3 }. (d) φ(t) = (sin t, sin t, cos t), = {(x, y, z) : y 2 + z 2 = 1}. (e) φ(t) = (cos t, cos t, t), = {(x, y, z) : y = x}.

456 8 2. C ( ) F T ds C. (a) C (1, 1) (2, 8) y = x 3 F (x, y) = (xy, y x). (b) C y 2 + 2z 2 = 1 x = 1, x-, F (x, y, z) = ( x 3 + y 3 + 5, z, x 2 ). (c) C y = x x 2 + 3z 2 = 1, y- F (x, y, z) = (z, z, x + y). 3. ω. C (a) C (1, 1) (2, 1) (2, 1) (2, 3) ω = y dx + x dy. (b) C z = x 2 + y 2 x 2 + y 2 + z 2 = 1 z- ω = dx + (x + y) dy + (x 2 + xy + y 2 ) dz. (c) C R = [a, b] [c, d] ω = xy dx + (x + y) dy. (d) C y = x y = z 2, z 1 y- ω = x dx + cos y dy dz. (e) C y = 2x 3 (, ) (1, 2) ω = x 3 y 2 dx 2x 4 y dy. (f) C (, ) (a, b) ω = (y 2 y) dx + x dy. 4. (a) c R, δ > u R τ(u) = δu + c. (φ, I), J = τ 1 (I) τ(u) = δu + c ψ = φ τ (ψ, J) (φ, I). (b) (φ, I) (ψ, [, 1]). (c) b). (d) C [, 1] [, 1]. [, 1] C, I j = [(j 1)/4, j/4] (φ 1, I 1 ), (φ 2, I 2 ), (φ 3, I 3 ), (φ 4, I 4 ).

8.2 457 5. (φ, I), τ u J τ (u) > J I 1-1 C 1. ψ = φ τ F : φ(i) R m F (φ(t)) φ (t) dt = F (ψ(u)) ψ (u) du. I J 6. (8.5.) f : [a, b] R [a, b] C 1 t [a, b] f (t). y f(a) f(b) x = f 1 (y) x a b y = f(x). 7. 5 C y = x 2 (, ) (π, π 2 ). C y dx + cos x dy 8. V R 2. F : V R 2 V (conservative) V F = f f : V R. (x, y) V F = (P, Q) : V R 2 V. (a) C(x) V (x, y), L((x 1, y); (x, y)) (x 1, y) (x, y) V. F T ds = P (x, y) x C(x). V (x, y) / y. (b) (x, y ) V. C V ( ) F T ds = (x, y) V f(x, y) := F T ds C C(x,y)

458 8 C(x, y). C(x, y) (x, y ) (x, y), V. (c) F V V C (*). (d) V C F (*) P y = Q x. : V. (8.6 7) (e) R 2 \{(, )} 1 ω =. y x 2 + y 2 dx + x x 2 + y 2 dy 9. f : [, 1] R [, 1]. T (, f()), (1, f()), (1, f(1)). c T, a b T, L y = f(x), x [, 1], c L a + b. 8.3 8.1..? R 2. 2.. 8.1 m E R m V E = V E R m.

8.3 459 1. 2 2 2. 8.11 (i) R 3 C p E 2 φ : E R 3 E C p = (φ, E). φ(e). (x, y, z) R 3. (ii) (φ, E) φ E 1-1. iii) (ψ, B) (φ, E) C p (ψ, B) C p ψ(b) = φ(e). x = ψ 1 (u, v), y = ψ 2 (u, v), z = ψ 3 (u, v), (u, v) B (φ, E) (ψ, B). z = f(x, y). f C p 8.11. 1 E 2 f : E R C p. z = f(x, y) C p. φ(u, v) = (u, v, f(u, v)) φ C p E 1-1. φ z = f(x, y). ( z = f(x, y).) x = f(y, z) y = f(x, z). x = f(y, z), (y, z) E φ(u, v) = (f(u, v), u, v) (φ, E).,,, C. 2 x 2 + y 2 = 1, z 2 C. φ(u, v) = (cos u, sin u, v) E = [, 2π] [, 2] φ E C. x = cos u, y = sin u, z = v. x 2 + y 2 = 1. φ(e) x 2 + y 2 = 1, z 2. E φ(e). φ(e) E

46 8 z (, 2) (2, 2) v v (, ) (2, ) x (1,, ) (, 1, ) y 8.1. v = v (,, v ) 1 ( 8.1). v 2 v = v x 2 + y 2 = 1, z 2. 3 x 2 + y 2 + z 2 = a 2 C. φ(u, v) = (a cos u cos v, a sin u cos v, a sin v) E = [, 2π] [ π/2, π/2]. φ E C. x = a cos u cos v, y = a sin u cos v, z = a sin v. x 2 + y 2 = a 2 cos 2 v x 2 + y 2 + z 2 = a 2. φ(e) a. v = v z = a sin v (,, a sin v ) a cos v ( 8.11). z (, 2) (2, 2) v v (,, a) v (, 2) (2, 2) x (a,, ) (, a, ) y 8.11

8.3 461 E, v = π/2(, v = π/2), (, ) (,, a)( (,, a)). v π/2 π/2 v = v x 2 + y 2 + z 2 = a 2. C xz- (a,, ) b. a > b. a > b (torus) z- C ( 8.12). (, ) (, ) z v v u (, ) (, ) x (a b,, ) y (, a b, ) 8.12 4 a > b C. φ(u, v) = ((a + b cos v) cos u, (a + b cos v) sin u, b sin v) E = [ π, π] [ π, π] φ E C. u = xz- (a,, ) b. v = v xy- (,, b sin v ) a + b cos v. v = ±π xy- (,, ) a b. φ(e). 5 z = x 2 + y 2, z b C. (x, y, z) = φ(u, v) = (v cos u, v sin u, v) E = [, 2π] [, b]. φ E C. x 2 + y 2 = z 2 z b. φ(e). v b v = v z = v (,, v ) v ( 8.13). φ(e) z = x 2 + y 2 = z 2, z b. v = (,, ). (x, y, z ) = (φ, E) (x, y, z ) = φ(u, v ).

462 8 (, b) (2, b) z v v b (, ) (2, ) x b y 8.13 (u, v ) E o (u, v ) E E E V f : V R : I. (x, y ) V, z = f(x, y) φ(e ) ; II. (x, z ) V y = f(x, z) φ(e ) ; III. (y, z ) V x = f(y, z) φ(e ) ;., f (x, y, z ). 6.1 1 6.25. 6 x = f(y, z) (1, f y (y, z ), f z (y, z ))..,,,. x, y R 3 x y x y (5.1 7).. 1 = (φ, E) p 1 C p φ = (φ 1, φ 2, φ 3 ). (u, v ) E o (x, y, z ) = φ(u, v ) (x, y, z ) N φ := φ u (u, v ) φ v (u, v ).

8.3 463 8.1 1 φ u (u, v ) (x, y, z ) φ(u, v ) φ v (u, v ) (x, y, z ) φ(u, v). (x, y, z ) ( 8.14). φ u (u, v ) φ v (u, v ) (x, y, z ). N u v (E ) 8.14 z = f(x, y) φ(x, y) = (x, y, f(x, y)) N φ = ( f x, f y, 1). 6.25 6.1 1. φ N φ. 8.3. 8.12 p 1 (φ, E) C p C p. (i) N φ (u, v ) (, N φ (u, v ) > ) (φ, E) (u, v ) E. (ii) (φ, E) E (φ, E). (iii) (φ, E) E \ E E E. z = f(x, y). (u, v ) E o φ(u, v ) (φ, E) (u, v ) ( 7).

464 8., 3 φ φ u φ v = (a 2 cos u cos 2 v, a 2 sin u cos 2 v, a 2 sin v cos v) = a 2 cos v. (φ, E) v = ±π/2 ( v = ±π/2 1-1 ).,. N φ. 8.2 E, B R 2 2- (φ, E), (ψ, B) R 3 C p p 1. τ ψ = φ τ B E C p, u, v B N ψ (u, v) = τ (u, v)n φ (τ(u, v)). φ = (φ 1, φ 2, φ 3 ), ψ = (ψ 1, ψ 2, ψ 3 ). 1 N ψ = ( (ψ2,ψ 3), (ψ3,ψ 1), (ψ1,ψ 2)). ψ = φ τ, i, j = 1, 2, 3 (ψ i, ψ j ) = (φ i, φ j ) τ u, v B (ψi,ψ j)(u, v) = τ (u, v) (φi,φ j)(τ(u, v)) B N ψ = τ (N φ τ). ( 8.6 ). 8.13 p 1. C p (φ, E), (ψ, B) ψ(b) = φ(e) B E 1-1 C p τ ψ = φ τ (u, v) B τ (u, v). τ ψ(b) φ(e). (8.4 5) 8.1, 8.7.

8.3 465 8.14 = (φ, E) N φ = φ u φ v. (i). σ() := N φ (u, v) d(u, v) E (ii) g : φ(e) R g g dσ := g dσ := g(φ(u, v)) N φ (u, v) d(u, v) (8.5). φ(e) 7.9, (φ, E) 1-1 N φ (u, v) (8.5). (φ, E) Z (8.5).. z = f(x, y), (x, y) E C p (8.5). gdσ = E E g(x, y) fx(x, 2 y) + fy 2 (x, y) + 1 d(x, y). (8.6),.., [?]. τ ( 5). R 2 8.12 7.2 ( 4). 8.14. 7 z = a 2 x 2 y 2 g(x, y, z) = 3 z g dσ.

466 8 φ 3 E = [, 2π] [, π/2]. (φ, E ) N φ = a 2 cos v. v [, 2π) cos v φ(e ) [, 2π] {π/2}. g dσ = a 2 cos v 3 a sin v du dv E π/2 = 2πa 7/3 cos v 3 sin v dv = 3π 2 a7/3. g (8.5) 8.14.. 7. : z = a 2 x 2 y 2 N = ( z x, z y, 1) = (x/z, y/z, 1). ( B a (, ) B a (, ) B a (, ).) N = a/z. (8.6) g dδ = = B a(,) 2π a a 3 z z d(u, v) ( r.) r(a 2 r 2 ) 1/3 dr dθ = 3π 2 a7/3.. ( ) ( ). 2 (φ, E). (x, y, z) < z < 2 z = z = 2. ( 8.8 ). (x, y, z) φ(e o ) (x, y, z) (φ, E).., ( 1,, 1) = φ(π, 1)

8.3 467 φ(e o ). (φ, E). R 3 C p. (x, y, z). (x, y, z).,. Int() ( ) := \ Int(). ( 5.7).. E (E \ E o ) E (topological boundary). m-. =., a > x 2 + y 2 + z 2 = a 2, z = a 2 x 2 y 2 (, z = x 2 + y 2, z 1). x 2 + y 2 = a 2, z = (, x 2 + y 2 = 1, z = 1). C R 2,, C C.., ( ) R 3. R n n 1 = {x R n : x = 1} R n. j = (φ j, E j ) = N j=1 j. I. j k j k (, < a (x, y, z) b ). II. j, φ j (Ej o) φ k(ek o) = j k (, 3 ).. j.

468 8. I, II. = N j=1 j j. j. j. 8 [, 1] [, 1] [, 1] z = 1 z = 1, x 2 + y 2 = 1, 3 z z = 1 x 2 y 2 z = 3 = N j=1 j j = (φ j, E j ) (φ j, E j ). N σ() = σ( j ), g N g dσ = g dσ j. j=1 j=1 9 x =, y =, z = x + y + z = 1 g(x, y, z) = x + y 2 + z 3 g dσ.. (u, v) E φ 1 (u, v) = (u, v, ), φ 2 (u, v) = (, u, v), φ 3 (u, v) = (u,, v), φ 4 (u, v) = (u, v, 1 u v) (, ), (1, ), (, 1). j = 1, 2, 3

8.3 469 N φj = 1 N φ4 = 3 = = = g dσ 1 1 u + 1 1 u 1 1 u 1 (u + v 2 ) dv du + (u + v 3 ) dv du + 3 1 1 u 1 1 u (u 2 + v 3 ) dv du (u + v 2 + (1 u v) 3 ) dv du ((2 + 3)u + u 2 + (1 + 3)v 2 + 2v 3 + 3(1 u v) 3 ) dv du ((2 + 3)u (1 + 3)u 2 u 3 + 1 + 3 3 = 3 1 (2 + 3). (1 u) 3 + 2 + 3 (1 u) 4 ) du 4 (8.3) 1.. (a) a z b z = x 2 + y 2 (b) 3 (c) 4 2. ( ), g dσ. (a) xy-, x = 1 x = 1 z = x 2 y 2 g(x, y, z) = 1 + 4x 2 + 4y 2 (b) y = x 3, y 8, z 4 g(x, y, z) = x 3 z (c) 2x 2 + 2y 2 = 9 z = 9 x 2 y 2 g(x, y, z) = x + y + z

47 8 3. E (φ, E). x 2 a 2 + y2 b 2 + z2 c 2 = 1 4. (a) E 2 = {(x, y, z) R 3 : (x, y) E, z = }. Area(E) = dσ g : R gdσ = g(x, y, ) d(x, y). E (b) f : [a, b] R p 1 C p C z = f(x), a x b R 2, z = f(x), a x b, c y d R 3. σ() = (d c)l(c). (c) f : [a, b] R p 1 C p y = f(x), a x b y.. σ() = 2π b a f(x) 1 + f (x) 2 dx 5. (ψ, B) (φ, E) C p Z B (ψ, B) Z τ : B R 2 B E C 1. τ 1-1 B \ Z τ ψ = φ τ, g : φ(e) R g(φ(u, v)) N φ (u, v) du dv = E. B g(ψ(s, t)) N ψ (s, t) ds dt 6. (x, y) B 3 (, ) f : B 3 (, ) R f(x, y) 1. 2z = x 2 + y 2, z 4 f(x, y) f(, ) dσ 4π.

8.4 471 7. (φ, E) C p (u, v ) E o (x, y, z ) = φ(u, v ). (φ, E) (x, y, z ) N φ (u, v ) =. 8. = (ψ, B). E = ψ u, F = ψ u ψ v, G = ψ v. E2 G 2 F 2 d(u, v). B 9. = (φ, E) (x, y, z ) = φ(u, v ) C 1. C = (ψ, I) (u, v ) E C 1. ( ψ(t ) = (u, v ) t I.) (φ ψ) (t ) (φ u φ v )(u, v ) =. 8.4 (φ, I) φ (t).., (φ, E) N φ.,.... 1 ( ) (φ, E). φ(u, v) = ((2 + v sin(u/2)) cos u, (2 + v sin(u/2) sin u, v cos(u/2)), E = [ π, π] [ 1, 1].

472 8 φ v = (2 cos u, 2 sin u, ). xy- 2. u = u R 3., u = (2,, v), 1 v 1. u = ±π (2 v,, ), 1 v 1. := {(x,, ) : 3 x 1}. (φ, E) 8.15. 8.15,. = (φ, E) (x, y, z ) n(x, y, z ) = N φ (u, v )/ N φ (u, v ). φ(u, v ) = (x, y, z ). n j =, 1 φ(u j, v j ) = (x, y, z ) (u j, v j ) E N φ (u, v ) N φ (u, v ) = N φ(u 1, v 1 ) N φ (u 1, v 1 ). φ E. φ E n φ E ( ). = (φ, E) n, φ(u, v ) = φ(u 1, v 1 ) N φ (u, v ) N φ (u 1, v 1 ), (u 2, v 2 ) (u, v ) N φ (u 2, v 2 ) N φ (u, v ) (orientable). N φ.

8.4 473, (n ) ( )., z = f(x, y).. 8.15 (φ, E) (ψ, B) (u, v) B τ (u, v) > τ (orientation equivalent). 8.2 (φ, E) (ψ, B). (φ, E) (ψ, B).. ( 8.9 ) 8.16 = (φ, E) n F : φ(e) R 3. F. F n dσ := F n dσ := (F φ)(u, v) N φ (u, v) d(u, v). φ(e) E (x, y) E z = f(x, y), 8.16. F n dσ = F (x, y, f(x, y)) ( f x, f y, 1) d(x, y). (8.7) E? F = (φ, E) F n F ( 8.16). n. F n dσ (n ) F (flux)., ( 4.).. 1 (φ, E) (ψ, B),, F (φ(u, v)) N φ (u, v) d(u, v) = F (ψ(s, t)) N ψ (s, t) d(s, t). E B

474 8 F d 8.16 τ ψ(b) φ(e). τ B (φ, E) (ψ, B) B τ <. 8.2 7.14( ) F (ψ(s, t)) N ψ (s, t) d(s, t) B = τ (s, t) (F φ τ)(s, t) (N φ τ)(s, t) d(s, t) B = F (φ(u, v)) N φ (u, v) d(u, v) = τ(b) E F (φ(u, v)) N φ (u, v) d(u, v).,,,. 2 (x, y) [, 1] [, 1] x+y+z = 1 2. n. F (x, y, z) = (xy, x y, z). F n dσ. x + y + z = 1 (1, 1, 1). 1 1 1 F n dσ = (xy, x y, 1 x y) (1, 1, 1) dx dy = 1 4.

8.4 475. 2 = (φ, E) x = φ 1 (u, v), y = φ 2 (u, v), z = φ 3 (u, v). 8.3 1 N φ = ( (y, z) (u, v), (z, x) (u, v) (x, y) ),. (u, v) F = (P, Q, R) : φ(e) R 3 ( (y, z) (x, z) (x, y) ) P + Q + R d(u, v) E (u, v) (u, v) (u, v) = P dy dz + Q dz dx + R dx dy,. dy dz = (y, z) (z, x) (x, y) d(u, v), dz dx = d(u, v), dx dy = d(u, v). (u, v) (u, v) (u, v) ( dy = f (x)dx 2.) Ω R 3 2- ( 2 ) P dydz + Q dzdx + R dxdy. P, Q, R : Ω : R 3. 2- P, Q, R Ω Ω. n 2. P dy dz + Q dz dx + R dx dy = (P, Q, R) n dσ. 1. 2..,.....,

476 8.,. xy- R 2,,, z. R 2 z-., E = {(x, y) : a 2 < x 2 + y 2 < b 2 } {(x, y) : x 2 + y 2 = b 2 } {(x, y) : x 2 + y 2 = a 2 }. 3 z = x 2 + y 2,.. z 4 z = 4 x 2 + y 2 = 4. z-. t [, 2π] φ(t) = (2 sin t, 2 cos t, 4). = j = (φ j, E j ) j.., (φ, E 1 ), (φ, E 2 ), k = 1, 2. φ 1 E k = [π(k 2), π(k 1)] [ 1, 1], k = 1, 2. = j = (φ j, E j ). j n j ±N φ.., = N j=1 j F. N F n dσ = F n j dσ. j j=1 4 F n dσ

8.4 477. x 2 + y 2 = 1 z =, z = 2. n F (x, y, z) = (xy, yz, zx). 1, 2, 3 ( 8.17). E = [, 2π] [, 2] 1 φ(u, v) = (cos u, sin u, v)., N φ = (cos u, sin u, ) F n dσ = 1 2 2π (cos 2 u sin u + v sin 2 u + v 2 cos u) du dv = 2π. 2 n = (,, 1) 8.3 4(a) F n dσ = 2 B 1(,) x d(x, y) = 2π 1, 3. F n dσ = 2π + + = 2π. r 2 cos θ dr dθ =. z (,, 2) 3 1 x (1,, ) 2 (, 1, ) y 8.17 5 F n dσ. F (x, y, z) = (x + z, xy, z) z = x 2 + y 2 z = 1 n. z = x 2 + y 2, z 1 1 x 2 + y 2 1, z = 1 2. 1 φ(u, v) = (u, v, u 2 + v 2 ),

478 8 (u, v) B 1 (, ). N φ = ( 2u, 2v, 1). 1 F n dσ = ( 2u 2 2u(u 2 + v 2 ) 2uv 2 + (u 2 + v 2 )) d(u, v) 1 = B 1(,) 1 2π =. (2r 2 cos 2 θ + 2r 3 cos θ + 2r 2 cos θ sin 2 θ r 2 )r dθ dr 2 n = (,, 1) 2 F n = z = 1, 8.3 4(a) F n dσ = 2 B 1(,) d(x, y) = B 1 (, ) = π. F n dσ = + π = π. z (,, 3) 1 (1,, ) (, 1, ) 2 x y 3 (,, 1) 8.18 6 F n dσ. F (x, y, z) = (x, y, z) x2 +y 2 z 2 = 1 z = 1, z = 3 n. 1, 2, 3 ( 8.18). 1 n = (,, 1) F n dσ = 3 d(x, y) = 4 3π. 1 B 2(,)

8.4 479, F n dσ = 2π. 3 2 F n z = u x 2 + y 2 = 1 + u 2. φ(u, v) = ((1 + u 2 ) cos u, (1 + u 2 ) sin v, u), (u, v) [ 3, 3] [, 2π] 2. N φ = ( (1 + u 2 ) cos v, (1 + u 2 ) sin v, 2u(1 + u 2 )) F N φ = ((1 + u 2 ) cos v, (1 + u 2 ) sin v, u) ( (1 + u 2 ) cos v, (1 + u 2 ) sin v, 2u(1 + u 2 )) = (1 + u 2 ) 2 + 2u 2 (1 + u 2 ) = u 4 1 2 F n dσ = 3 2π 1 3 (u 4 1) dv du = 2π (1 u 4 ) du = 8π 1 5 (1 3). F n dσ = 4 3 + 2π + 8π 5 (1 3) = 6π 5 (3 + 2 3). (8.4) 1., ( ) F T ds. (a) y = 9 x 2 z 2, y, F (x, y, z) = (x 2 y, y 2 x, x + y + z). (b) x, y, z x + 2y + z = 1. F (x, y, z) = (x y, y x, xz 2 ).

48 8 (c) z = x 2 + y 2, 1 z 4 F (x, y, z) = (5y + cos z, 4x sin z, 3x cos z + 2y sin z). 2. F n dσ. (a) z = x 2 + y 2, z 1 F (x, y, z) = (x, y, z). (b) n z = 4 y 2, x 1 F (x, y, z) = (x 2 + y 2, yz, z 2 ). (c) 8.3 4 n F (x, y, z) = (y, x, z). (d) x 2 + y 2 = 1 z = x 2 n. F (x, y, z) = (y 2 z, cos(2 + log(2 x 2 y 2 )), x 2 z) 3., ω. (a), [, 1] [, 1] z = x 4 + y 2. ω = x dydz + y dzdx + z dxdy. (b), z = a 2 x 2 y 2, ω = x dydz + y dzdx. (c), x 2 + y 2 = b 2, < b < a z = a 2 x 2 y 2. ω = xz dydz + dzdx + z dxdy. (d) z-, z = 2 x 2 + y 2, z 2. ω = x dydz + y dzdx + z 2 dxdy. 4. (ψ, B) (φ, E) C p, Z B, (ψ, B) Z τ : B R 2 B E C 1. τ B o \ Z 1-1, τ > ψ = φ τ F (φ(u, v)) N φ (u, v) d(u, v) = F (ψ(s, t)) N ψ (s, t) d(s, t) E B. F : φ(e) R 3.

8.5 481 5. M 1 M 2. M 1 M 2 M 1 M 2. 6. E x =, y =, z =, x + y + z = 1. E T = E, C 1 P, Q, R : E R P dydz + Q dzdx + R dxdy = (P x + Q y + R z ) dv. E E 7. T, 6. T. x =, y =, z =.. C 1 P, Q, R : R P dx + Q dy + R dz = (R y Q z ) dy dz + (P z R x ) dz dx + (Q x P y ) dx dy. 8.5 f C 1 f(b) f(a) = b a f (t) dt. [a, b] f [a, b] {a, b} f. f : [a, b] R F : Ω R m. Ω m-, m = 2 m = 3., Ω F Ω F. 2.

482 8 8.3 ( ) E E 2. P, Q : E R C 1 F = (P, Q) ( Q F T ds = x P ) da. y E E [ ] E I- II-. P dx + Q dy = E I 1. E I- E P dx + Q dy = I 1 + I 2. E E = {(x, y) R 2 : a x b, f(x) y g(x)} f, g : [a, b] R. E y = g(x), y = f(x), ( 8.19). x b a y = g(x) x a b y = f(x). dx = I 1. 8.9 b a I 1 = P dx = P (x, f(x)) dx + P (x, g(x)) dx E = = a b (P (x, g(x)) P (x, f(x)) dx a b g(x) a f(x) b P P (x, y) dy dx = y E y da. 8.19

8.5 483 E II- I 2 = E I 1 I 2. Q dy = E Q x da. E I- II-. I- II- 2., E 8.2. E II- E I- II E 1 E 2. 8.3 8.2 E ( Q x P ) da = y = = ( Q E 1 x P ) ( Q da + y E 2 x P ) da y F T ds + F T ds E 1 E 2 F T ds + F T ds + F T ds. E C E 1 C E 2, C E 1 E 2. E 1 E 2 C E 1 C E 2. C. E F T ds.. 1 E = [, 1] [, 1], E, F (x, y) = (xy, x 2 + y 2 ) F T ds. E

484 8 E., 1 1 F T ds = (2x x) dy dx = 1 2. E. 2 E = B 1 (, ), E, F = (xy 2, arctan(log(y 2 + 3)) x 3 ) F T ds. E F., F T ds = ( 3x 2 2xy) dx dy E = B 1(,) 2π 1 ( : E.) (3r 2 sin 2 θ + 2r 2 cos θ sin θ)r dr dθ = 3 4 π. R 2 2 Q x P y. Ω R 3 3. 8.17 E R 3, F = (P, Q, R) : E R 3 E C 1. F F. curl F = ( R y Q z, P z R x, Q x P ) ; y div F = P x + Q y + R z F = (P, Q, ) curl F = (,, Q x P y ). (Nabla ) ( = x, y, ) z

8.5 485 i j k curl F = F = x y z P Q R. div F = F = ( x, y, ) (P, Q, R) z E 3 E E o. E n. E. 3 a > E = {x : a x b} n {x : x = b}, {x : x = a}. Ω 3. (divergence theorem). 8.4 ( ) E E 3. F : E R 3 E C 1 F n dσ = div F dv. E [ ] E I, II, III-. F = (P, Q, R) F n dσ = P dy dz + E E = I 1 + I 2 + I 3. E E Q dz dx + R dx dy E I 3. E I-, B R 2 E = {(x, y, z) R 3 : (x, y) B, f(x, y) z g(x, y)} f, g : B R. E z = g(x, y), z = f(x, y) ( 8.21). E xy-.

486 8 z z g(x, y) z f (x, y) x B y 8.21 dx dy E. I 3 E.. 3.13 I 3 = E R dx dy = = (R(x, y, g(x, y)) R(x, y, f(x, y))) d(x, y) B g(x,y) B f(x,y) R R (x, y, z) dz d(x, y) = z E z dv., E II- I 2 = E III- I 1 + I 2 + I 3. I 1 = E E Q y dv, P x dv. E I, II, III-. I, II, III- 3. E = E 1 E 2 E 1 E 2 div F dv = div F dv + div F dv E E 1 E 2 = F n dσ + F n dσ + F n dσ. E E 1 E 2

8.5 487 E 1 E 2 E 1 E 2. 4 E = {(x, y, z) : x 2 + y 2 z 1}, n. F (x, y, z) = (2x+sin z 2, cos x 5 +log z 7, cos(x 2 )+ tan(y 3 ) z 2 ) 8.4 F n dσ. div F = 2 2z, F ndσ = (2 2z)dV = 2 2π 1 1 E r 2 (1 z)r dz dr dθ = π 3. 5 Q [, 1] [, 1] [, 1], n, F (x, y, z) = (2x z, x 2 y 2, x + z 2 ) F n dσ. Q Q. F n dσ = 1 1 1 Q (2 + 2x 2 y 2z) dx dy dz = 4 3.. F a curl F (a) a. div F (a) a ( 7). 6 F (x, y, z) = (x, y, z). curl F = div F = 3. G(x, y, z) = (y, x, ). curl G = (,, 1) div G =. curl G..

488 8 2 E R 2, C F T ds T C (8.2 5 )., E E. F 3 E R 3 = E, F n ( 8.14 ). = E E. (8.5) 1., F T ds. C (a) C, x =, y =, y = 4 x 2 1 2 F (s, y) = (sin( x 3 x 2 ), xy). (b) C, (, ), (2, ), (, 3), (2, 3) F (x, y) = (e y, log(x + 1)). (c) C = C 1 C 2, C 1 = B 1 (, ), C 2 = B 2 (, ). F (x, y) = (f(x 2 + y 2 ), xy 2 ), f [1, 2] C 1. 2. ω. C (a) C [a, b] [c, d]. f : [, 1] R ω = (f(x) + y) dx + xy dy. (b) C y = x 2 y = x 2. ω = yf(x) dx + (x 2 + y 2 ) dy, f : [, 1] R 1 xf(x) dx = 1 x2 f(x) dx. (c) C 2 E. ω = e x sin y dy e x cos y dx.

8.5 489 3. F n dσ. n. (a) [, 1] [, 2] [, 3] F (x, y, z) = (x + e z, y + e z, e z ). (b) x 2 + y 2 1, z =, 1 x 2 + y 2 1, z 1 F (x, y, z) = (x 2, y 2, z 2 ). (c) E. E R 3 z = 2 x 2, z = x 2, y =, z = y. F (x, y, z) = (x + f(y, z), y + g(x, z), z + h(x, y)), f, g, h : R 2 R. (d) x 2 /a 2 + y 2 /b 2 + z 2 /c 2 = 1, F (x, y, z) = (x y, y z, z x ). 4., n ω. (a) y = x 2, z =, z = 1, y = 4 3 ω = xyz dydz + (x 2 + y 2 + z 2 ) dzdx + (x + y + z) dxdy. (b) x 2 + z 2 1, y = x 2 + z 2 2, y = 1 x 2 y 2 + z 2 = 1, y 1 ω = xy z dydz + x 2 z dzdx + (x 3 + y 3 ) dxdy. (c) x 2 + y + z 2 = 4 4x + y + 2z = 5 E R 3. ω = (x+y 2 +z 2 ) dydz +(x 2 +y +z 2 ) dzdx+(x 2 +y 2 +z) dxdy. 5. (a) E. (b) Folium φ(t) = Area(E) = 1 2. E x dy y dx ( 3t 1 + t 3, 3t 2 ) 1 + t 3, t [, ) (c) (a) R 3 E. (d) a > b ( ) (8.3 4 ).

49 8 6. (a) P, Q E ( : P = y/(x 2 + y 2 ), Q = x/(x 2 + y 2 ), E = B 1 (, ) ). (b) F E. 7. 8.6. V R 3 F : V R 3 C 1 x V n B r (x ). 1 div F (x ) = lim F n dσ. r Vol(B r (x )) B r(x ) 8. F, G : R 3 R 3 f : R 3 R.. (a) (F + G) = ( F ) + ( G) (b) (ff ) = f( F ) + ( f F ) (c) (ff ) = f F + f ( F ) (d) (F + G) = F + G (e) (F G) = ( F ) G ( G) F 9. 8.6. E R 2. C 1 f : E R grad f := f := (f x, f y ). (a) E F f : E R C 2. E F = grad f. F T ds =. E (b) f F x C 2 curl grad f(x ) = div curlf (x ) =.

8.6 491 (c) E F f : E R C 2. E F = grad f. ff n dσ = F F dv. E 1. E R m. u : E R E 2 m 2 u u := x 2. j j=1 (a) u E C 2 E u = ( u). (b) ( 1 ) E R 3 C 1 F C 2 u, v : E R (u v + u v) dv = u v n dσ E. (c) ( 2 ) E R 3 C 1 F C 2 u, v : E R (u v v u) dv = (u v v u) n dσ E. (d) u : E R E u C 2 x E u(x) =. E C 1 F R 3. u E E, E u = E u =. (e) V R 2 u V C 2 u V. u V C 1 F = (P, Q) 2 E V. E E E E (u x dy u y dx) =

492 8 8.6 R 3. 8.5 ( ) n R 3 C 2. F : R 3 C 1 F T ds = curl F n dσ. [ ] C 1 F 2 E C 2. F = (P, Q, R) F T ds = P dx + Q dy + R dz. z = f(x, y), (x, y) E. f : E R C 2. n = N/ N, N = ( f x, f y, 1). (ψ(t), σ(t)), t [a, b] E. φ(t) = (ψ(t), σ(t), f(ψ(t), σ(t))), t [a, b], ( 8.22). x = ψ(t), y = σ(t) z = f(ψ(t), σ(t)), dx = ψ (t)dt, dy = σ (t)dt,, P dx + Q dy + R dz = dz = z z dx + x y dy. E ( P + R z ) ( dx + Q + R z ) dy. (8.8) x y. ( Q + R z ) = Q x y x + Q z z x + R z x y + R z z z x y + R 2 z x y

8.6 493 8.22 ( P + R z ) = P y x y + P z z y + R y z x + R z z y z = f(x, y) C 2 2. ( Q + R z ) ( P + R z ) x y y x = ( R x Q z = curl F N. )( z ) + x ( P z R x )( z x + R 2 z y x. z ) ( Q + y x P ) y (8.8),, (8.7) F T ds = curl F N d(x, y) = curl F n dσ. E.. 1 C y- x 2 +y 2 = 1, y =. F (x, y, z) = (x 2 z + x 3 + x 2 + 2, xy, xy + z 3 + z 2 + 2) F T ds. C curl F = (x, x 2 y, y) F T ds. x 2 + z 2 1, y =. = C. C y-., n = (, 1, ). curl F n = x 2 y = x 2 F T ds = x 2 da = 2π 1 C r 3 cos 2 θ dr dθ = π 4.

494 8 1 C.. 2 9x 2 + 4y 2 + 36z 2 = 36, z, n. F (x, y, z) = (cos x sin z + xy, x 3, e x2 +z 2 e y2 +z 2 + tan(xy)). curl F n dσ. C =. curl F n dσ C F T ds., C F T ds E = C C 2 E curl F ndσ. E 2 9x 2 +4y 2 36. E n = (,, 1)., curl F x (x3 ) y (cos x sin z + xy) = 3x2 x. curl F n dσ = (3x 2 x) d(x, y). E x = 2r cos θ y = 3r sin θ. 2π 1 (3x 2 x) d(x, y) = (12r 2 cos 2 θ 2r cos θ)6r dr dθ = 18π. E. 3 z = x 2 + y 2, z 1 x 2 + y 2 = 1, 1 z 3. n F (x, y, z) = (x + z 2,, z 3) F n dσ,. x 2 + y 2 = 1, z = 3. curl G = F

8.6 495, R y Q z P z R x Q x P y = x + z 2, (8.9) = (8.1) = z 3 (8.11) G = (P, Q, R) : R 3. (8.9). Q z = x, R y = z2. (8.12) (8.12) g : R 2 R Q = xz + g(x, y). (8.12) h : R 2 R R = z 2 y +h(x, z). Q x = z + g x g = P y = 3,, σ : R 2 R P = 3y + σ(x, z) (8.11)., σ = h = P z R x = σ z h x (8.1). P = 3y, Q = xz, R = yz 2, G = (3y, xz, yz 2 ). φ(t) = (sin t, cos t, 3), t [, 2π] (G φ) φ = (3 cos t, 3 sin t, 9 cos t) (cos t, sin t, ) = 3 cos 2 t + 3 sin 2 t = 3., F n dσ = curl G n dσ = G T ds = 2π 3 dt = 6π. 3 curl G = F G.. (8.12) Q z = z2,. R y = x G(x, y, z) = (zy, (3x + z 3 /3), xy). G T curl G = F G.

496 8 curl G = F G.. G E G curl G = F C 2. C 2 F (8.5 9(b)), E div F =. div curlf = (8.13) 8.6 F G div F =.. curl G = F (8.14) 4 Ω = B 1 (,, ) \ {(,, )} w = w(x, y, z) = x 2 + y 2 + z 2 F (x, y, z) = ( x w 3/2, y w 3/2, z ) w 3/2. Ω div F = curl G = F G. div F = 2x2 + y 2 + z 2 w 5/2 + x2 2y 2 + z 2 w 5/2 + x2 + y 2 2z 2 w 5/2 =. B 1 (,, ). curl G = F G. F = (x, y, z) = n F n = x 2 = y 2 + z 2 = 1 F n dσ = 1 da = σ() = 4π. (8.15) 1 2 F n dσ = F n dσ + F n dσ (8.16) 1 2 = G T 1 ds + G T 2 ds =. (8.17) 1 2 1 = 2 T 1 = T 2. (8.15) (8.16) curl G = T G.

8.6 497 4 (8.14) G. E 8.6. 8.7 Ω (,, ) (,, ) F : Ω R 3 Ω C 1.. (i) Ω curl G = F C 2 G : Ω R 3. (ii) F, E, = E E Ω F n dσ = (8.18) (iii) Ω div F =. (i) G 1, div F = div(curl G) =. (8.13). (ii) 8.5 7 x Ω o 1 div F (x ) = lim r Vol(B r (x )) 1 = lim r Vol(B r (x )) B r(x ) B r(x ) div F dv F n dσ = div F Ω, Ω div F =. (iii). F = (p, q, r) G = (, Q, R). curl F = G R y Q x = p, R x = q, Q x = r. (8.19), g, h : R 2 R R = x q(u, y, z) du + g(y, z), Q = x r(u, y, z) du + h(y, z). ( : (, y, z) (x, y, z) Ω.) ( 6.3) (iii),

498 8 p = R y Q z = = x x (q y (u, y, z) + r z (u, y, z)) du + g y h z p x (u, y, z) du + g y h z = p(x, y, z) p(, y, z) + g y h z. (8.19) g y = p(, y, z) h =., Q = x r(u, y, z) du, R = y p(, v, z) dv x q(u, y, z) du. 8.6 (x, y, z) Ω, L((, y, z); (x, y, z)) L((,, z); (, y, z)) Ω 3 Ω ( 9). (8.6) 1., F T ds. C (a) C x 2 + y 2 = 1 z = x, z- F (x, y, z) = (xy 2,, xyz) (b) C z = y 3 x 2 + y 2 = 3, z- F (x, y, z) = (e x + z, xy, ze y ) 2. curl F n dσ. (a) y = x 2, z = 1 y z, n F (x, y, z) = (x sin z 3, y cos z 3, x 3 + y 3 + z 3 ). (b) z = 3 x 2 y 2, z, n F (x, y, z) = (y, xyz, y).

8.6 499 (c) z = 1 x 2 y 2, n F (x, y, z) = (x, x, x 2 y 3 log(z + 1)). (d) x =, y =, x + 2y + 3z = 1, z z, n F (x, y, z) = (xy, yz, xz). 3. curl F n dσ. (a) x 2 +y 2 +z 2 = 1, n F (x, y, z) = (xz 2, x 2 y z 3, 2xy+ y 2 z) (b) B 1 () z = y, n F (x, y, z) = (xy, xz, yz) (c) y = 2 x 2 + z 2, 2 y 4, n F (x, y, z) = (x, 2y, z) (d) z = 4 x 2 y 2, z 4 z = x 2 +y 2 4, 4 z, n F (x, y, z) = (x + y 2 + sin z, x + y 2 + cos z, cos x + sin y + z). (e) z = x 2 + y 2 ( z 2), 2 = x 2 + y 2 (2 z 5), z = 7 x 2 y 2 (5 z 6), n F (x, y, z) = (2y, 2z, 1) 4. ω. (a) y 2 + z 2 9, x 2 ω = xy dydz + (x 2 z 2 ) dzdx + xz dxdy. (b) x 2 + z 2 = 8, y 1 ω = (x 2z) dydz y dzdx. (c) R = [, π/2] [, 1] [, 3] ω = e y cos x dydz + x 2 z dzdx + (x + y + z) dxdy. (d) 2x 2 + z 2 1 x = y x-. ω = x dydz y dzdx + sin y dxdy.

5 8 5.. 6. Π n R 3, x Π. r > r r x Π., r = B r (x ) Π. F : B 1 (x ) R C 1 r n 1 curl F (x ) n = lim F T ds. r σ( r ) r. 7. n C 1 F. (a) F : R 3 \ {} C 1, T n. T (x ) F (x ) x curl F n dσ = x T (x ) F (x ). (b) F, F k : R 3 C 1 F k F curl F k n dσ = curl F n dσ lim k 8. E (x, y) E (, ) (x, ) (x, ) (x, y) E 2. F : E R 2 C 1. (a) f : E R E F = f. (b) F = (P, Q) (exact), E Q x = P y. (c) C = Ω C F T ds =. Ω 2 Ω E. 9. Ω 3 F : Ω R 3 Ω C 1. (x, y, z) Ω L((x, y, ); (x, y, z)) L((x,, ); (x, y, )) Ω..

8.6 51 (a) Ω curl G = F C 2 G : Ω R 3. (b) F, E, = E E Ω F n dσ =. (c) Ω div F =. 1. F C 1 R 2 \ {(, )}, P y = Q x. (a) C 1 C 2. E E C 1 C 2 2. (E C j.) (, ) / E. F T ds = F T ds. C 1 C 2 (b) E (, ) E o 2. E F (x, y) =, F T ds. E ( y x 2 + y 2, x ) x 2 + y 2 (c) F : R 3 \ {(,, )} 3, (a). 11.. (a) C y2 dx + z 2 dy + x 2 dz, C : φ(θ) = (cos θ, sin θ, 1), θ 2π (b) C (y2 + z 2 ) dx + (x 2 + z 2 ) dy + (x 2 + y 2 ) dz, C x 2 + y 2 + z 2 = 4x, z > x 2 + y 2 = 2x (c) C 2y dx 2x dy + z2 x dz, C : φ(θ) = (cos θ, sin θ, 5), θ 2π (d) C yz2 dx + (xz 2 2y) dy + 2xyz dz, C : φ(θ) = (cos θ, sin θ, cos θ), θ 2π 12. C, a n dσ = 1 a φ dφ. 2 C, a C : φ = (x(t), y(t), z(t)), a t b.

52 8 13. u v 2, n u v dσ = u v dφ. C C : φ = (x(t), y(t), z(t)), a t b.