. () ; () ; (3) ; (4).. () : P.4 3.4; P. A (3). () : P. A (5)(6); B. (3) : P.33 A (9),. (4) : P. B 5, 7(). (5) : P.8 3.3; P ; P.89 A 7. (6) : P.

() * 3 6 6 3 9 4 3 5 8 6 : 3. () ; () ; (3) (); (4) ; ; (5) ; ; (6) ; (7) (); (8) (, ); (9) ; () ; * Email: huangzh@whu.edu.cn

. () ; () ; (3) ; (4).. () : P.4 3.4; P. A (3). () : P. A (5)(6); B. (3) : P.33 A (9),. (4) : P. B 5, 7(). (5) : P.8 3.3; P.84 3.5; P.89 A 7. (6) : P.3 5.4; P.33 A 4()(3). (7) : P.3 6.6; P.38 A (3). (8) : P.343 A 5(); P.348 A 3. (9) : P.39 A 8, 9; B 3. () : P.5 A (3), 8; B. () : P.4 A 9; B 5. : ; ; ; : ;. () a, b a b =, a x b x + a y b y + a z b z =. a, b a b (â, b) = arccos a b = arccos(ê a, e b ), e a, e b a, b. () a b a b. a b k(a b). a b a b =, a x b x = a y b y = a z b z, b x, b y, b z,.

. (3) a, b, c [a, b, c]. a, b, c [a, b, c]. 6 a, b, c, a, b, c [a, b, c] =.. A(x x ) + B(y y ) + C(z z ) =. A, B, C}, (x, y, z ). Ax + By + Cz + =. A, B, C}. x a y b z c a a b b c c =. (a i, b i, c i ), i =,, 3. a 3 a b 3 b c 3 c x a + y b + z c =. a, b, c. A x + B y + C z + =,. A x + B y + C z + =. x x m = y y n x = x + mt, = z z. (m, n, p). m, n, p p,. y = y + nt, t. z = z + pt. x a = y b = z c. (a i, b i, c i ), i =,. a a b b c c l : f(x, y) =, z = x f (x, ± ) y + z =, y f ( ± ) x + z, y =.. 3

. x a, l : = y + = z π : 3x + 6y + 3z + 5 =, a l π : x y + z = l. a =. l s =,, a}, π n = 3, 6, 3}. s n, l. 3 = 6 = a 3,. l π π, π π. π n, n s, n n. i j k s n = = 3i 3k, n =,, }, l P (,, ), P π, π : (x ) + (y + ) + ( ) (z ) =, x z =. l x z =, x y + z =.. l x = y+, x y 4 =, y+ = z, y z + 4 =. l x y 4 + λ (y z + 4) =, x + (λ )y λz + (4λ 4) =. π π, λ (λ ) λ =, λ =. l x z =, x y + z =.. l M(,, ), π N. M π x = y = z, x = y = z, x y + z =, N ( 4,, 4 3 3 3). l π, l π Q(, 4, ), 3 3 3 Q, N l : x 4 3 = y 3 4 = z 4 3.

. l : x 9 4 = y + 3 = z l : x = y + 7 = z 9 l. (). l. l l π, l l π, π π l. l l s = 4, 3, },s =, 9, }. i j k s s = 4 3 = 5i j + 3k, 9 l s = 3,, 6}. π l l, n = s s = P (9,, ), π. π i j k 4 3 3 6 = 6i + 7j + 7k. l 6(x 9) + 7(y + ) + 7(z ) =, 6x + 7y + 7z 9 =. l l π : 58x + 6y + 3z =. π π l, 6x + 7y + 7z 9 =, 58x + 6y + 3z =. ()., l π, l l. x = y+7 = z, 9 6x + 7y + 7z 9 =, (,, 4), l : x+ 3 = y = z 4 6. () l l, l M(x, y, z ), N(x, y, z ), M, N l, l (), x = 9 + 4t, y = 3t, z = t, x = λ, y = 7 + 9λ, z = + λ. MN = λ 4t 9, 9λ + 3t 5, λ t + }. MN l, MN l, 4( λ 4t 9) 3(9λ + 3t 5) + (λ t + ) =, 33λ + 6t + 9 =, 89λ + 33t 3 =, ( λ 4t 9) + 9(9λ + 3t 5) + (λ t + ) =. t =, λ =. MN = 3,, 6}. l x = y 4 = z + 3 6. M, N M(, 4, ), N(,, 4), 5

3 l : x + = y = z x = 4t +, 3 l : y = t + 3, z = t 4,?, ;, d. l P (,, ), s =,, 3}; l Q(, 3, 4), s = 4,, }. l, l. d. [ ] 3 4 P Q, s, s = 3 =, 4. l π l, l π, d. i j k s s = 3 = i 6j 6k, 4 π n =, 6, 6}. π l P (,, ), π (x + ) + 6 (y ) + 6 (z ) =, x + 6y + 6z 5 =. l Q(, 3, 4) π : d = + 6 3 + 6 ( 4) 5 + 6 + 6 = 93.. s, l, l s,. i j k s s = 3 = i 6j 6k, 4 s =, 6, 6}. P, Q l, l, P Q = 3,, 4}, P Q s d = Prj sp Q = s = 3 + 6 4 6 =. + 6 + 6 93 : ; ; ; : (); ; ; ; 6

....,. (). (),,. (3),. 4 z = f(x, u, v), u = g(x, y), v = h(x, y, u), f, g, h, z x, z. ( 4.9) y. (: u,,.) z.. x u v x y x y u x y x 4, y 3. z x = f x + f u g x + f v h x + f v h u g x, z y = f u g y + f v h y + f v h u g y.. :. 7

.3.3 (). Γ x = ϕ(t), y = ψ(t), z = ω(t), t [α, β]. T = ϕ(t ), ψ(t ), ω(t ) }. (). Γ F (x, y, z) =, () G(x, y, z) =. () T = i j k F x F y F z G x G y G z, P Fx (P )(x x ) + F y (P )(y y ) + F z (P )(z z ) =, (3) G x (P )(x x ) + G y (P )(y y ) + G z (P )(z z ) =. (4) (3) (4) () ().. :,. x x y y z z F x (P ) F y (P ) F z (P ) =. G x (P ) G y (P ) G z (P ) () (). Σ: F (x, y, z) =, n = F x, F y, F z } P. : F x (P )(x x ) + F y (P )(y y ) + F z (P )(z z ) =. : x x F x (P ) = y y F y (P ) = z z F z (P ). 8

.4 3 (). Σ: z = f(x, y), n = f x, f y, } P. : z z = f x (P )(x x ) + f y (P )(y y ). : x x f x (P ) = y y f y (P ) = z z. (3). Σ : x = x(u, v), y = y(u, v), z = z(u, v), n = i j k x u y u z u x v y v z v. P.4 : gradf(x, y ) = f x (x, y ), f y (x, y ) }. (),., f l = f x(x, y ) cos θ + f y (x, y ) sin θ = f x (x, y ), f y (x, y ) } cos θ, sin θ } = gradf(x, y ) e l. (),.. f l = gradf(x, y ) e l = gradf(x, y ) e l cos gradf(x, y ), e l = gradf(x, y ) cos gradf(x, y ), e l e l gradf(x, y ),, gradf(x, y ). : 3 : ; 9

3. 3 3. X Y. ϕ (x) y ϕ (x), X : a x b, Y : f(x, y) dσ = b a ( ϕ (x) ψ (y) x ψ (y), c y d, f(x, y) dσ = d c ϕ (x) ( ψ (y) ψ (y) ) f(x, y) dy dx. ) f(x, y) dx dy. x y. sin y 5 I = y, y = x x = y. sin y X, y, dy, y y x y, Y. : y, I = = = ( y y sin y y ) dx dy sin y y (y y ) dy = [ cos y ] + [ y cos y sin y 3 6 I = dy e x dx. 3y e x dx, X, y. (sin y y sin y) dy ] = sin. : 3y x 3, y, X : y x 3, x 3, I = = 3 3 x 3 dx e x dy x 3 ex dx = 6 3 = [e x] 3 = 6 6 (e9 ). e x d(x )

3. 3 f(x, y) dσ = f(ρ cos θ, ρ sin θ)ρ dρ dθ. : ; ( P.63.) : dσ = ρ dρ dθ. 3. :,.. z (x, y) z z (x, y), : (x, y),. ( z (x,y) f(x, y, z) dv = z (x,y) ) f(x, y, z) dz dxdy,.,., ρ φ z,. (,.) ρ = a() ( z ).,.,,. :,.. (x, y) z, : a z b, f(x, y, z) dv = b a ( z z = z. 7 I = z x dxdydz, : + y + z. a b c z ) f(x, y, z) dxdy dz.

3. 3 : (x, y) z : x + y a b c z c, I = = c c c c z c, ( ) c z dxdy dz = (z c z z πab ( z c ) dz = 4 5 πabc3. ( ) dxdy z πab z c. ) dxdy dz z 8 : x a + y b + z c. : () (x + y 5z ) dxdydz. (x + y + z ) dxdydz; () ( x a + y b + z ) dxdydz; (3) c () (x + y + z ) dxdydz, I z = z dxdydz. I z = z dxdydz = 4 5 πabc3, I x = y dxdydz = 4 5 πab3 c. x dxdydz = 4 5 πa3 bc, I y = () z c dxdydz = c (x + y + z ) dxdydz = 4 5 πabc( a + b + c ). z dxdydz = 4 πabc. 5 ( x a + y b + z c ) dxdydz = 4 5 πabc. (3) (x + y 5z ) dxdydz = 4 5 πabc( a + b 5c ). () : z, x, y;, z. (x, y) z, h(z) dxdydz, : a z b. z, (z). h(z) dxdydz = z. b a ( ) h(z) dxdy dz = b a h(z)(z) dz, z

4 r, θ, φ. r = a() ( a, ); θ = α() ( α); φ = β() ( x β).,. r, θ, φ.... (, r = a(), θ = α().) 9 z = x + y z = x y.. : V = dv = r, φ π 4, θ π. π π 4 dθ dφ r sin φ dr [ ] π 4 = π cos φ 3 = π 3 ( ). 4 : ; : ;. (). 4., f(x, y) ds, ds, ds = ( dx) + ( dy).. x = x(t), (x : (α t β), ds = (t) ) ( + y (t) ) dt, y = y(t),. β f(x, y) ds = f ( x(t), y(t) ) ( x (t) ) ( + y (t) ) dt. : ; α 3

4. 4 : ; : ds = ( dx) + ( dy). (i) : y = y(x), a x b. f(x, y) ds = (ii) : x = x(y), c y d, 4. f(x, y) ds = b a d c x = x, y = y(x), f ( x, y(x) ) + ( y (x) ) dx. f ( x(y), y ) ( x (y) ) + dy., P (x, y) dx + Q(x, y) dy.. x = x(t), AB : y = y(t),. t = a, t = b A B. AB : ; : ; P (x, y) dx + Q(x, y) dy = b a t : a b, (a x b), ( P ( x(t), y(t) ) x (t) + Q ( x(t), y(t) ) ) y (t) dt. : a, b. a b. P (x, y) dx + Q(x, y) dy = ( P (x, y) cos α + Q(x, y) cos β ) ds, (5) cos α, cos β} e τ (x, y) (x, y). (5) : F (x, y) = P (x, y)i + Q(x, y)j W. s. [s, s + ds], (x, y). e τ (x, y) (x, y)., F [s, s + ds], F (x, y), e τ (x, y) ds ( ds, e τ (x, y)). F [s, s + ds] dw = F (x, y) e τ (x, y) ds. 4

4.3 4 W = dw = = F (x, y) e τ (x, y) ds ( P (x, y) cos α + Q(x, y) cos β ) ds.. P dx + Q dy = P dx + Q dy, () P (x, y) dx =. Q(x, y) dy =. (:, x = C(), dx =. ) () :,.. 4.3, P (x, y), Q(x, y), P dx + Q dy = ( Q x P ) dxdy, y. () x P dx + Q dy = P Q dxdy. y () P, Q. (3) P dx = P y dxdy, Q dy = Q dxdy,. x d ( u(x, y) ) = P (x, y) dx + Q(x, y) dy, P dx + Q dy = (x, y ), (x, y ). [ ] (x,y ) u(x, y) = u(x, y ) u(x, y ), (x,y ) 5

4.4 4,,,,. [ ] [ I = φ(y) cos x πy dx + φ (y) sin x π ] dy, AMB A(π, ) AMB B(3π, 4) AB, AB. φ(y) cos x dx + φ (y) sin x dy = φ(y) d(sin x) + sin x d(φ(y)) = d ( φ(y) sin x ), AMB φ(y) cos x dx + φ (y) sin x dy = I = π y dx + dy. y dx + dy. AMB AMB x π BA:, x = πy π, y: 4. = y 3π π 4 AMB BA, AMB y dx + dy = 4.4 BA y dx + dy = AMB BA y dx + dy = ( ) y dx + dy AMB BA [ ] (3π,4) φ(y) sin x = φ(4) sin 3π φ() sin π =. (π,) 4 (, dxdy =. (y π + ) dy = 6π, BA f(x, y, z) d, ) y dx + dy = 6π, I = 6π. d.. z = z(x, y), xy xoy, (. ) f(x, y, z) d = x, y, z(x, y) + zx + zy dxdy. : ; : ; : d = + zx + zy dxdy. xy f 4.5,.. 6

4.5 4 : Φ = v e n. : Φ = v e n d. v = P, Q, R}, e n = cos α, cos β, cos γ}, Φ = (P cos α + Q cos β + R cos γ) d (6) P dydz + Q dzdx + R dxdy. (7) dydz cos α d, dzdx cos β d, dxdy cos γ d, d.,. ( Prj a b = cos θ b,,.) 4 : (), (), (3), (4). : z = z(x, y). P dydz + Q dzdx + R dxdy = v e n d = v n d + z x + zy = v n dσ xy [ P ( zx ) + Q ( z y ) + R ] dσ, n = z x, z y, }, = xy [ P zx + Q z y + R ( ) ] dσ, n = z x, z y, }. xy n. I = P dydz + Q dzdx + R dxdy, () I = v e n d; () I = v n dσ. xy. : e n n. :, 7

5 : z = z(x, y),, n = z x, z y, };, n = z x, z y, }. : y = y(z, x),, n = y x,, y z };, n = y x,, y z }. : x = x(y, z),, n =, x y, x z };, n =, x y, x z }., : z = z(x, y),. n = z x, z y, } n = z x, z y, }., n z γ, cos γ >. n = z x, z y, }. ( cos γ = z >.) x +zy + xoy, R cos γ d =.) R(x, y, z) dxdy =. (: cos γ =,,., : x = C, Q dzdx =, R dxdy =. ( dx =.), P dydz + Q dzdx + R dxdy = P dydz + R dxdy, P dydz, Q dzdx, R dxdy. R dxdy = Q dzdx + P (x, y, z), Q(x, y, z), R(x, y, z) V, P x, Q y, R z, P dydz + Q dzdx + R dxdy = V V V. V ( P x + Q y + R ) dxdydz, z : () V. () V..,. 5 : ; ;. : ; ; ;. u n = u + u + + u n +,.,. 8

5. 5, u n n s n }. u n. s n }., s n u + u + + u n., u n,. ( ). :. 5. u n = lim n u n =. : lim u n n =,. n :,,. :, ;,.,. lim v n =.) n u n u lim n =, n v n v n. (: lim n u n =, () p, ( p ), () n p aq n = aq + aq + aq 3 + + aq n + = aq n = a + aq + aq + + aq n + = n=,. () ρ <, ; () ρ >, ( u n ); (3) ρ =,. u n+ 3 u n }, lim n u n u n+ lim = ρ, lim n u n n = ρ, lim n n un = ρ, n= aq n. aq, ( q < ) q a. ( q < ) q n un = ρ.. 9

5. 5 : f(n) : + + dx, n x :. f(x) dx. + [ ] + x dx = ln x = +. 5.. u n u n. u n = u n., ( ) n u n,,.,.,. : u n, ( ) n u n. : ( ) n n = + 3 4 +. ( ln.), : (i) p >, ; ( ) n, n p (ii) < p, ; (iii) p,. ( ) n n. 5.3 : () nx n, () x n. n xn n. :. n= x n = x ( < x < ). x n = :,,,. () x x ( < x < ), nx n = x ( nx n ) = x ( x n) ( x ) x = x = ( < x < ) x ( x)

5.4 5 x = ±, nx n. () s(x) = x n n, nx n = s (x) = ( x n n x. ( < x < ) ( x) ) = x, s(x) = x s (x) dx = s(x) s() = s(x), s(x) = x x n =, ( < x < ) x dx = ln( x), ( < x < ) x x = x n n = ; x = n x n n = ( ) n n, x n n = ln( x). ( x < ) () nx n, () x n n () nx n nx n+, () x n n : n n +, n. x n+, n (n + )x n. x n n+. nx n+ = x ( x n), (n + )x n = ( x n+), x n+ n = x( x n ), n x n n + = ( x n+ ), x n +. 5.4 f(x) ;, f(x). f(x),.

5.5 5 5.5 f(x) ;, f(x). f(x) a + ( an cos nx + b n sin nx ) s(x), s(x) = f(x), x f(x) ; ) ( f(x ) + f(x + ), x f(x). f(x),.. π f(x), f(x) : f(x) = a + ( an cos nx + b n sin nx ), (8) f(x). a n = π b n = π π π π π. f(x) cos nx dx, n =,,, 3, f(x) sin nx dx, n =,, 3, () f(x) ( π, π) (), a n = (n =,,, ), b n = π : π f(x) sin nx dx (n =,, 3, ). b n sin nx. () f(x) ( π, π) (), a n = π π f(x) cos nx dx (n =,,, 3, ), b n = (n =,, ). : a + a n cos nx.

6 : l f(x), f(x) = a + ( an cos nπx l f(x).. a n, b n a n = l b n = l l l l () l = π, π. () f(x), b n = l l (3) f(x), f(x) sin nπx l l f(x) = f(x) cos nπx l f(x) sin nπx l dx, (n =,, ). + b n sin nπx ), l dx, (n =,,, ), dx, (n =,, ). b n sin nπx, (x f(x) ) l f(x) = a + a n cos nπx, (x f(x) ) l a n = l l f(x) cos nπx l dx, (n =,,, ). 6 :,,. : () ; () ().,,., ;,.. 6.,,.,. x ds, x + y =. x y : x, y,. x ds = y ds. 3

6. 6 : x ds = y ds = ( ) x + y ds (9) = ds ( x + y = ) = 4 =., : () a x ds; () x + y =. ( a x + b y ) ds. (x : x + y R, I = a + y ) dx dy =. b (x, y, ), : x dx dy = y dx dy = (x + y ) dx dy, () (x I = a + y ) (x dx dy = b a + x ) dx dy b = ( a + ) b = ( a + b ) π x dx dy = dθ R ( a + ) b (x + y ) dx dy ρ ρ dρ = πr4 ( 4 a + ). b = (x, y) x + y 4, x, y }, f(x), a, b, a f(x) + b f(y) dσ = f(x) + f(y) (A) abπ. (B) ab a + b π. (C) (a + b)π. () π., f(x) dσ = f(x) + f(y) f(y) dσ = f(x) + f(y) dσ = f(x) + f(y) f(x) + f(y) dσ, () a f(x) + b f(y) f(x) + f(y) dσ = a f(x) dσ + b f(x) + f(y) f(y) f(x) + f(y) dσ 4

6. 6 : = (a + b) f(x) dσ = (a + b) f(x) + f(y) dσ = a + b π. (). 4, (9), (), (),.,, (): ( x, y,, : x + y R ), : f(x, y) dx dy = f(y, x) dx dy = [ ] f(x, y) + f(y, x) dx dy. () : () (9), (), (),. 3 (x + y + z) ds, Γ Γ x + y + z = R, x + y + z =. Γ x, y, z,, Γ. x ds = y ds = z ds = Γ Γ Γ 3 Γ(x + y + z ) ds = R ds = π 3 Γ 3 R3. x ds = y ds = z ds = Γ 3 (x + y + z) ds = 3 Γ Γ Γ Γ Γ (x + y + z) ds =. (x + y + z ) ds + (x + y + z) ds = 4 3 3 πr3. Γ 6.,.,. : x,, f(x, y) y ; f(x, y) dσ = f(x, y) dσ, f(x, y) y. x. 5

6. 6 : : yoz,, f(x, y, z) x ; f(x, y, z) dv = f(x, y, z) dv, f(x, y, z) x. yoz. : y, x. : Σ yoz,, f(x, y) x ; f(x, y) ds = f(x, y) ds, f(x, y) x., f(x, y, z) x ; f(x, y, z) d = f(x, y, z) d, f(x, y, z) x. Σ Σ Σ Σ yoz. 5, :, ;., x, y ;, xoy, z.. π π sin x ( 4 M = + x π cos4 x dx, N = sin 3 x + cos 4 x ) π ( dx, P = x sin 3 x cos 4 x ) dx, π π (A) N < P < M. (B) M < P < N. (C) N < M < P. () P < M < N., M =, N = (). 5 l x 4 + y 3 π π π cos 4 x dx >, P = cos 4 x dx <, π =, a, (xy + 3x + 4y ) ds =. l. l x, xy y, (xy + 3x + 4y ) ds = xy ds + (3x + 4y ) ds = + (3x + 4y ) ds = ds = a. l l l l l 6

6. 6 : 6 = (x, y) x + y, x }, I = x, π + xy dx dy = dx dy = + x + y + x + y π xy y, + x + y dθ + xy dx dy. + x + y xy dx dy =. + x + y + ρ ρ dρ = π [ ln( + ρ ) ] = π ln. 7 (, ), (, ) (, ),, (xy + cos x sin y) dx dy (A) cos x sin y dx dy. (B) xy dx dy. (C) 4 (xy + cos x sin y) dx dy. (). x, y,,. BO, 3 : AOB, 3 COB. xy dσ = xy dσ + xy dσ =. 3 y, xy x, xy dσ = ; 3 x, xy y, xy dσ =. 3, y, cos x sin y x ; 3 x, cos x sin y y, (A). cos x sin y dσ = cos x sin y dσ + cos x sin y dσ = cos x sin y dσ. 3 8 Σ : x + y + z = a (z ), Σ Σ, (A) x d = 4 x d. (B) y d = 4 x d. Σ Σ Σ Σ (C) z d = 4 x d. () xyz d = 4 xyz d. Σ Σ Σ Σ B C y. O A x 7

6. 6 : Σ yoz zox, f(x, y, z) = z x, y, z d = z d (f(x, y, z) = z x ) Σ = 4 Σ x} Σ z d, (f(x, y, z) = z y ) Σ ( Σ : x + y + z = a, x, y, z.), Σ Σ z d = y d = x d, (C).,,. (A), yoz, x x, x d =. x d >, (A)., y d =, Σ Σ (B). yoz, xyz x, xyz d =, () Σ Σ. Σ.,,. I = P (x, y) dx, : () y,, P (x, y) x ; P (x, y) dx = P (x, y) dx, P (x, y) x. x. () x,, P (x, y) y ; P (x, y) dx = P (x, y) dx, P (x, y) y. y.. I = f(x, y, z) dy : Σ 8

6. 6 : Σ yoz,, f(x, y, z) x ; f(x, y, z) dy = f(x, y, z) dy, f(x, y, z) x. Σ Σ Σ Σ yoz. 6,.. 9 x y dx, y = x (, ) (, ). x, x y y, x y dx =. x I = r dy + y 3 r dx + z 3 r dx dy, r = x + y + z, Σ x + y + z = R 3 Σ. z Σ x, y, z, I = 3 dx dy. Σ xoy, r3 Σ z r z, z z dx dy = dx dy. 3 r3 r3 Σ Σ I = 6 Σ z r 3 dx dy = 6 R 3 x +y R R x y dx dy = 6 R 3 π dθ a R r r dr = 4π. 9