II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, ) (x, y) = (, ) lim h f (h, ) f (, ) h h 3 /h = lim = 1 h h x f x (, ) = 1 y f y (, ) = 3 (6.5) ε (x, y) (, ) ε(x, y;, ) = x3 3y 3 x + y x + 3y = 3x y xy x + y ε(x, y;, ) x + y = 3 cos θ sin θ cos θ sin θ r r 1 (6.5) A, B ε (6.6)
. C 1 C 1 C 1 p.195 y x 1 (1)(4) C 1 () cx + dy = C 1 (3) 1 + x + y = C 1 (5) xy > 1 C 1 (6) x C 1 (7) x + y 1 C 1 (8) x > C 1 C 1 3. 6.5 Z x, y 1 x, y t 1 z t 1 z ax + by + c, x αt + β, y γt + δ = z (aα + bβ)t + (aβ + bδ + c) 1 () a = z x, b = z y, α = dx dt, γ = dy dt dz dt z dx = aα + bβ = x dt + z dy y dt z = f (x(s, t), y(s, t)) y = g(x(s, t)) 1 1 z = f (x(s, t), y(s, t)) [ ] [ ] [ ] x zs z t = zx z s x t y y s y t 15 6 3
II (1 11 ) 1. z = f (x, y) z = f (a, b) + f x (a, b)(x a) + f y (a, b)(y b) (a, b, f (a, b)) x f x (a, b)y f y (a, b) ( f x (a, b), f y (a, b), 1) p13 p.131 p1315 6.16. C n k k n f (x, y) C n C n n 1 C 1 3. 6.7 C n f xy C f xyxy = ( f xy ) xy = ( f xy ) yx = f xyyx f xyxyxx = f xyyxxx 3 4 C n k k + 1 4. 6.7 = f (a + h, b + k) f (a + h, b) f (a, b + k) + f (a, b) φ(x) = f (x, b + k) f (x, b) = φ(a + h) φ(a) f φ(x) = φ (a + θ 1 h)h = ( f x (a + θ 1 h, b + k) f x (a + θ 1 h, b))h f x y x = a + θ 1 h y = f xy (a + θ 1 h, b + θ k)hk
ψ(y) = f (a + h, y) f (a, y) = ψ(b + k) ψ(b) = ψ (b + θ 3 k)k = ( f y (a + h, b + θ 3 k) f ( a, b + θ 3 k))k = f yx (a + θ 4 h, b + θ 3 k)hk hk f xy (a + θ 1 h, b + θ k) = f yx (a + θ 4 h, b + θ 3 k) (h, k) (, ) f xy (a, b) = f yx (a, b) 5. Taylor 1 C n (n 1) p133(6.18) 1 k+l f k!l! k x l y (a, b)(x a)k (y b) l, 1 d m f (a)(x a)m m! dxm 1 Taylor F(t) = f (a + tp, b + tq) Taylor ( F (t) = p f x + q f y = p x + q ) f (a + tp, b + tq) y f p f x + q f y = p x + q y x = a + tp, y = b + tq t 1 t f f x = a + tp, y = b + tq t ( F (m) (t) = m f (a + tp, b + tq) = p x y) + q m f (a + tp, b + tq) m p197 F (m) (t) = m m! m f k!(m k)! k x m k y (a + tp, b + tq)(x a)k (y b) m k k= F(t) Taylor F(1) = F() + F () + F ()! Taylor 6. + + F(n 1) () (n 1)! + R n, R n = F(n) (θ) n! < θ < 1 f (a, b) + f x (a, b)(x a) + f y (a, b)(y b) + 1 f xx(a, b)(x a) + f xy (a, b)(x a)(y b) + 1 f yy(a, b)(y b) 15 6 ()(4)(6)(8) 4(1) z = f (x, y)
II (1 11 ) f (x, y) = sin(xy) f xy (x, y) = xy sin(xy) f xy (x, y) = cos(xy) xy sin(xy) 4.4(1) z = f (x, y) (x, y, z) = (a/, b/ 3, c/ 6) z > z = f (x, y) = c x/a f x = c, 1 x y a b 1 x a y b y/b f y = c 1 x y a b f (a/, b/ 3) = c/ 6, f x (a/, b/ 3) = c a 3, fy (a/, b/ 3) = c b z = c 6 c 3 a (x a ) c (y b ) 3 b 1 x a + y b 3 + z c 6 = 1 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) = x y x 1 x a + y b + z = 1 z x, y f (x, y) x c x a + z c f x(x, y) = f x (x, y) = c x a z z = c 1 x y f a b x
x c x a z y = x y = x y = x(x a) + a 1. Taylor ( F (k) (t) = p x y) + q k f (a + tp, b + tq) (A) f (x, y) p f x + q f y p x + q y (B) 1 F(t) F (t) d dt (C) 1 f (x, y) f (a + tp, b + tq) (A)(C) (C)(B) (A)(A)(C) (C)(B) (A)(C)(B) (C)(B) (C)(B)(B) (A) (C) 1 (C) 1 (B). ( 6.9) C 1 F(x, y) = c z = F(x, y) z = c (a, b) F(x, y) = c F(a, b) = c z = F(x, y) (a, b, F(a, b)) = (a, b, c) z = F(a, b) + F x (a, b)(x a) + F y (a, b)(y b) = c + F x (a, b)(x a) + F y (a, b)(y b) (F x (a, b), F y (a, b)) (, ) z = c F x (a, b)(x a) + F y (a, b)(y b) = z = F(x, y) z = c F(x, y) = c (a, b) F(x, y) (a, b) C 1 (F x (a, b), F y (a, b)) (, ) F(x, y) = c x = a F x (a, b)(x a) + F y (a, b)(y b) =
F y (a, b) y x = a y = y(x) F(x, y(x)) = c F(x, y) = c y = y(x) 6.9 3. 6.4 6.18 y = ±x 1 + x 6.9 ( 1, ) (, ) ( 1, ) ( 1, ) x = 1x 1 (, ) 1 4. ( 6.1) (x, y) x = x(s, t), y = y(s, t) (s, t) x = r cos θ, y = r sin θ 6.1 (s, t ) [ xs (s, t ) ] x t (s, t ) y s (s, t ) y t (s, t ) (s, t ) (s, t) (x, y) 1 1 1 1 (s, t) (x, y) (x, y) (s, t) x = x(s, t ), y = y(s, t ) 1 [ ] x x y y [ ] [ ] xs (s, t ) x t (s, t ) s s y s (s, t ) y t (s, t ) t t 1 (x, y) (s, t) (x, y) (s, t) 1 1 5. p139 (6.7) 1 [ ] [ ] 1 sx s y xs x = t t x t y y s y t s x = x s p14 [ ] [ ] 1 rx r y xr x = θ = θ x θ y y r y θ [ cos θ r sin θ sin θ r cos θ ] 1 = 1 r [ r cos θ ] r sin θ sin θ cos θ
r = x + y, θ = tan 1 (y/x) 1(6) 6. 6.5 = [ ] [ ] y f x f y x = [ ] [ ] cos θ sin θ f r f θ = [ ] [ ] f (sin θ)/r (cos θ)/r r f θ = f 1 θ 5 6.9
II (1 5 ) f (x, y) = x /3 + y /3 x = x y = y x y 4 (±a, ), (, ±a) p.14 4.4 1 a 4 1. p17(6.13) f r = cos θ f x + sin θ f y, f θ = r sin θ f x + r cos θ f y (1) ( f r ) + 1 r ( f θ) = ( f x ) + ( f y ) f rr = ( cos θ f x + sin θ f y )r = cos θ( f x) r + sin θ( f y ) r ( f x ) r f (x, y) x f x (x, y) f x (r cos θ, r sin θ) r f x (1) ( f x ) r = cos θ f xx + sin θ f xy, ( f y ) r = cos θ f yx + sin θ f yy f rr = cos θ f xx + cos θ sin θ f xy + sin θ f yy () f θθ f θθ = r cos θ f x r sin θ f y r sin θ( f x ) θ + r cos θ( f y ) θ ( f x ) θ, ( f y ) θ (1) ( f x ) θ = r sin θ f xx + r cos θ f xy f xx + f yy = f rr 1 r f r + 1 r f θθ p.1 1. 1 6.11
f (x, y) (a, b) (a, b) f x (a, b) = f y (a, b) = F(x) = f (x, b) F(x) a x = a F (a) = f x (a, b) = y 3. f (x, y) C f x (a, b) = f y (a, b) = f (x, y) f (a, b) + 1 ( fxx (a, b)(x a) + f xy (a, b)(x a)(y b) + f yy (a, b)(y b) ) (3) f (x, y) f (a, b) f (x, y) (x, y) = (a, b) Taylor 6.1 4. f xx (a, b) = A, f xy (a, b) = B, f yy (a, b) = C x a = p, y b = q () Ap + Bpq + Cq 3 A Ap + Bpq + Cq = A (p + B ) A q + AC B q A AC B >, A > (, ) AC B >, A < AC B < p = r cos θ, q = r sin θ Ap + Bpq + Cq = r ( A cos θ + B cos θ sin θ + C sin θ ) = r ( A + C Ap + Bpq + Cq = r A + C (A C) + 4 r m = A + C (A C) 4 + B, M = A + C + A C + B sin(θ + α) (A C) + + B 4 ) cos θ + B sin θ
r = p + q m(p + q ) Ap + Bpq + Cq M(p + q ) m > M < m < < M Mm = AC B Mm = AC B > AC > B A, C, M, m p.146 m, M 3 5. f (x, y) = x 4 + y 4 x + 4xy y f x (x, y) = 4x 3 4x + 4y =, f y (x, y) = 4y 3 + 4x 4y = 4x 3 + 4y 3 = y = x x (, ) (±, ) 3 f xx = 1x 4, f xy = 4, f yy = 1y 4 A = f xx (a, b) B = f xy (a, b) C = f yy (a, b) AC B (, ) 4 4 4 (±, ) 4 (, ) 6
II (11 8 ) 6(1) f (x, y) = x xy + 4y + x + y f x = x y + 1, f y = x + 8y + f x = f y = 1 ( /3, 1/3) f xx =, f xy = 1, f yy = 8 f xx f yy ( f xy ) = 15 >, f xx > f (x, y) ( /3, 1/3) f ( /3, 1/3) = /3 6() f (x, y) = x 3xy + y 3 f x = f y = f x = x 3y = x = 3 y f y = 3x + 6y = 9 ( y + 6y = 6y y 3 ) = 4 y = y = 3 4 y = x = y = 3 4 x = 3 3 4 = 9 8 (, ) (9/8, 3/4) f xx =, f xy = 3, f yy = 1y A = f xx B = f xy C = f yy AC B (, ) 3 9 (9/8, 3/4) 3 9 9 (9/8, 3/4) 7/64 y =(x ) x =(y ) f x + f y = f x =, f y = f x + f y = f x + f y = 1. f (x, y) = xy(x + y 4) (1) f x = f y = () f xx, f xy, f yy 6.1 f x = f y = y = x x = y
p(x, y)q(x, y) = p(x, y) = q(x, y) = f x = 3x y + y 3 4y = y(3x + y 4) = y = 3x + y 4 = f y = x 3 + 3xy 4x = x(x + 3y 4) = x = x + 3y 4 = f x = f y = 4 9 y = x = (, ) y = x + 3y 4 = (±, ) 3x + y 4 = x = (, ±) 3x + y 4 = x + 3y 4 = (±1, ±1), (±1, 1) 6.1 f xx = 6xy, f xy = 3x + 3y 4, f yy = 6xy f xx f yy ( f xy ) AC B A = f xx B = f xy C = f yy AC B (, ) 4 16 (±, ) 8 64 (, ±) 8 64 (±1, ±1) 6 6 3 (±1, 1) 6 6 3. f (x, y) = x 4 + y 4 x + 4xy y (, ) AC B = f (, ) = (, ) (, ) f (x, y) f (x, y) f (x, y) = (x 4 + y 4 ) (x y) x 4 + y 4 (x y) 4 x = y (x y) = f (x, ) = x 4 x < ( < x < 1/), f (x, x) = x 4 > (x ) 3. φ(x, y) = f (x, y) Lagrange 6.13 z = f (x, y) f (x, y) = f (a, b) (a, b) φ(x, y) =
f x (a, b)(x a) + f y (a, b)(y b) =, φ x (a, b)(x a) + φ y (a, b)(y b) = {[ ] [ ]} fx (a, b) φx (a, b), 1, f x (a, b) φ x (a, b) f y (a, b) φ y (a, b) f y (a, b) φ y (a, b) = 6.13 φ x = φ y = 1 [ ] [ ] fx (a, b) φx (a, b) + λ = f y (a, b) φ y (a, b) λ 3 1 φ x = φ y = φ(x, y) =, f x φ y f y φ x = f (x, y) = (x + y ) a (x y ), (a > )
II (11 15 ) f (x, y) = (x + y ) a (x y ), (a > ) f x = f y = f x = 4x 3 + 4y x 4a x = 4x(x + y a ) =, f y = 4x y + 4y 3 + 4a y = 4y(x + y + a ) = a > x + y + a a > f y = y = f x = x(x a ) = (, ) (±a, ) f xx = 1x + 4y 4a, f xy = 8xy, f yy = 4x + 1y + 4a A = f xx B = f xy C = f yy AC B (, ) 4a 4a 16a 4 (±a, ) 8a 8a 64a 4 a f y = y = x + y + a = 1.. f (x, y) R R f (x, y) C 1 f f x = f y = C 1 φ(x, y) = f φ = f x φ y f y φ y = f (x, y) C 1 C 1 φ(x, y) = f (x, y) 3. 6.6p.15 f x = f y = (, )
φ(x, y) = x a + y b 1 =, f x f y φ x φ y = y x/a x y/b = y b x a = x a = y b = 1 (x, y) = (± a, ± b ), (x, y) = (± a, b ) f (x, y), φ(x, y) C 1 f (x, y) f (, ) =, f (± a, ± b ) = ab (±, f a, b ) = ab ab ab p.15 4. ax + by + c = (l, m) φ(x, y) = ax + by + c = f (x, y) = (x l) + (y m) 1 φ = ax + by + c =, f x φ y f y φ x = b(x l) a(y m) = 1 5. f (x, y) : a x b, c y d xy z = f (x, y) y = S (y) = a b f (x, y)dx y x V = d c S (y)dy = d c ( b a ) f (x, y)dx dy d ( b ) d b f (x, y)dx dy = dy f (x, y)dx c a 6 7 (3) f (x, y) = e x y (x + y ) (x + y ) e x +y (x + y ) c a
II (11 ) C 1 f x = f y = (1) f x = x + y =, f y = x + 4y =, (x, y) = (, ) (, ) φ(x, y) = x + 3y 1 =, f x φ x = x + y x x + 4y 6y = x + 4xy + 6y = (x 3y)(x + y) = f y x = y 4y = 1x = 3y 1y = 1 φ y ( (x, y) = 1 ) 3, ±1, (x, y) = ±, ± 1 3 f (, ) =, ( f 1 ), ±1 = 1 3 ±, f, ± 1 = 7 3 6 7 6 () f x = 3x =, f y = 3y = (, ) φ(x, y) = x + y 1 =, f x f y φ x φ y = 3x 3y x 4y = 1x y 6xy = 6xy(x y) = x = y = 1 y = x = 1 x = y 9x = 1 (, ± 1 ), (±1, ), ( ± 1 ) 3, ± 3 f (x, y) f (, ) =, f (, ± 1 ) 1 1 = ± 1 (, f (±1, ) = ±1, f ± 1 ) 3, ± = ± 1 3 3 (3) f (x, y) = x + y e x +y x + y e x +y x + y * 1 f x = x(1 x y )e x y =, f y = y( x y )e x y = *1
f x = f y = x = y( y ) = x + y = 1 y = (, ), (, ±1), (±1, ) 5 f (x, y) f (, ) =, f (, ±1) = e, f (±1, ) = 1 e e (3) f (x, y) = x + y x + y e x +y e x +y u = x + y g(u) = ue u g(u) /e u = x + y = 1 f (x, y) e x + y = (x, y) = (, ) x + y = x + y x + y = 1 (, ±1) 1. p.161 7.3. y = φ 1 (x) y = φ (x) a x b x y y x 7. f (x, y) = e x 7.3 7. e x 3.
4. p.183 (4) 7 1
II (11 9 ) () 1 (1) a b a dx x + xy + y dy = x b + xb x dx x + xy + y dy = x 1 dx [x y + xy + y3 3 + b3 3 dx = a3 b 3 + a b 4 + ab3 3 ] x 1 = x 11 6 x3 x 4 x5 x6 3 dx = 17 84 (4) π/ (3) dx a 1 x dy 1 y (1 x) 3 (1 y) dx = sin(x + y)dy = (5) π/ 1 1 ] 1 (1 1 x)4 dy [ (1 y) (1 y) 6 = dy = 1 4 y 4 8 dx [ cos(x + y) ] π/ x = cos x cos xdx = x 1 dx x y π dy = 4 x dx = π 1 [ sin x a x dx a 1/4 πa /4 ] π/ sin x = 1 (1) (4) () 1 dx 1 dy 1 dx x x dy x x y x y x y x x (3) (5) x x y dy 1 x 1 y = x sin θ a a x dx y = a x x x a 1/4 1 1.
= [a, b] [c, d] i j = [x i 1, x i ] [y j 1, y j ] (p i j, q i j ) i j S = f (p i j, q i j )(x i x i 1 )(y j y j 1 ) = f (p i j, q i j ) i j i j 156. (p i j, q i j ) (p i, q j ), x i 1 < p i < x i, y j 1 < q j < y j i j p.161 7.9 f (p i, q j )(x i x i 1 )(y j y j 1 ) = i j j=1 i j m n f (p i, q j )(x i x i 1 ) (y j y j 1 ) 1 f (x, q j ) x b f (x, q a j)dx F(y) = b f (x, y)dx x [a, b] a m F(q j )(y j y j 1 ) j=1 F(y) y [c, d] b F(y)dy a 3. p.159 7. 3 4. f (x, y)dxdy x = x(s, t), y = y(s, t) st Ω xy Ω Ω Ω i j Ω i j i j i j Ω i j (u i j, v i j ) (p i j, q i j ) i j p i j = x(u i j, v i j ), q i j = y(u i j, v i j ) i=1
f (x, y)dxdy { i j} f (x, y)dxdy f (p i j, q i j ) i j = 5. i j Ω i j i j i j f (x(u i j, v i j ), y(u i j, v i j ) i j Ω i j Ω i j x(s, t) y(s, t) C 1 x, y s, t 1 Ω i j i j p.169 p.17 [ ] x(s + h, t + k) y(s + h, t + k) [ ] [ ] [ ] [ ] x(s, t) hxs + kx t xs x = t h y(s, t) hy s + ky t y s y t k h st k xy x s y s x t y t x s y s x t y t h k p.138 6.1 Ω i j 6. i j Ω i j (x, y) (s, t) (u i j, v i j ) (x, y) (s, t) i j 4 5 f (x, y)dxdyrisingdotseq f (x(u i j, v i j ), y(u i j, v i j ) (x, y) (s, t) (u i j, v i j )Ω i j i j f (x(s, t), y(s, t)) (x, y) (s, t) (s, t) p.167 7.7 p.183
II (1 6 ) 1. dxdy = (x, y) (s, t) dsdt. 7.4 p.171 x = r cos θ, y = r sin θ r θ x r dxdy = rdrdθ r 3 cos θ sin θdrdθ, Ω : r a, θ π Ω f (r)g(θ) f (x)g(y)dxdy = b a f (x)dx d c g(y)dy : a x b, c y d Ω r 3 cos θ sin θdrdθ = a r 3 dr π/ cos θ sin θdθ = a4 1 4 f (x)g(y)dxdy = = b a b a d dx c f (x)g(y)dy = ( d ) g(y)dy f (x)dx = c b a d c ( dx f (x) g(y)dy d c b a ) g(y)dy f (x)dx y f (x) d g(y)dy c 3. x + y dxdy, : x a + y b 1
x = ar cos θ, y = br θ < r 1 x = ra cos θ, y = rb sin θ r θ π 1 rθ Ω Ω : r 1, θ π (x, y) (r, θ = x r y r x θ y θ = a cos θ b sin θ ar sin θ br cos θ = abr abr dxdy = abrdrdθ Ω abr(a r cos θ + b r sin θ)drdθ = ab 1 r 3 dr π/ 4. a cos θ + b sin θdθ 1 x y 1 + x + y dxdy, : x + y 1, x, y Ω 1 r 1 + r rdrdθ, Ω : r 1, θ π f (x)g(y) u = 1 r π 1 r 1 r 1 + r dr 1 + r r 1 u 1 1 r = u r 1 + r r = 1 u 1 + u 4u rdr = (1 + u ) du 1 r 1 r 1 + r dr = u 1 (1 + u ) du = [ ] 1 1 1 = u + 1 + u 1 u u (1 + u ) du 1 1 + u du = 1 + [ tan 1 u ] 1 = 1 + π 4 1 dr du
x = x(t) dx = x (t)dt 3p.183(1)()(3) (5) (4) x + y = s, x y = t
II (1 6 ) 1 1. f (x)g(y) = {(x, y) a x b, c y d} f (x)g(y)dxdy =. b a f (x)dx d c g(y)dy 3(4) x + y = s, x y = t x, y s, t dxdy = (x, y) (s, t) dsdt 3. 1 (1 + x + y ) dxdy, : (x + y ) x y, x 3 p.14 r 4 r (cos θ sin θ), r cos θ r θ x r r cos θ cos θ cos θ, r cos θ
cos θ, cos θ θ π 4 θ π 4 Ω : π/4 θ π/4, r cos θ 1 (1 + x + y ) = 1 (1 + r ) dxdy = rdrdθ = π/4 π/4 cos θ r dθ (1 + r ) dr = 1 (1 + x + y ) dxdy = r Ω (1 + r ) drdθ π/4 π/4 [ dθ 1 1 1 + r ] cos θ 1 + cos θ = cos θ π/4 1 = 1 [θ cos θ dθ = 1 ] π/4 tan θ = π 4 1 = 1 π/4 4. π/4 1 1 1 + cos θ dθ 1 e x y dxdy R : x + y R, x, y π/ R x y e dxdy = dθ re r dr = π [ 1 ] R = π R e r 4 (1 e R ) E R : x R, y R e x e y x y e E R dxdy = R e x dx R ( R e y dy = ) e x dx R E R R ( π R ) 4 (1 ) < e e R x dx < π 4 (1 ) e R p.173 7.5 e x y dxdy = π 4 1 17
II (1 ) 1. 4(3) : x + y ax r ar cos θ r r a cos θ cos θ θ θ π π θ π cos θ π/ θ π/ y a θ π/ θ π/ a cos θ θ r r a cos θ xy a x y dxdy Ω : π/ θ π/, r a cos θ a x y = a x Ω dxdy = rdrdθ. π/ a r rdrdθ = dθ = a3 3 = 4a3 3 = 4a3 3 π/ π/ π/ π/ a cos θ r a r dr = 1 sin 3 θ dθ = 4a3 3 ( π 3 4 + 1 1 : x + y ax π/ π/ π/ [ 13 (a r ) 3/ ] a cos θ 1 sin 3 θdθ 1 3 4 sin θ + 1 4a3 sin 3θdθ = [θ + 34 4 3 cos θ 11 ] π/ cos 3θ ) ( = 4a3 π 3 ) 3 C 1 z = f (x, y) xy
j j S j S j S j (p j, q j ) j j T j θ j T j j 1/ cos θ j 1/ cos θ j 1 = S j T j = j cos θ j j j ax + by + cz = d (a, b, c) *1 (a, x) = d θ j ( f x (p j, q j ), f y (p j.q j ), 1) (,, 1) 1 = 1 + ( f x (p j, q j )) cos θ + ( f y (p j, q j )) j j j 1 + ( f x (p j, q j )) + ( f y (p j, q j )) j 1 + ( f x ) + ( f y ) = 1 + ( f x ) + ( f y ) dxdy p.1 4.13 3. z = a x y 1 + ( f x ) + ( f y ) a dxdy = a x y dxdy, : x + y a π a dθ ar a r dr = 4πa [ a x ] a = 4πa 4 (3) 1 14 ()1 *1
II (1 1 ) 1. 3 n 3 ρ(x, y, z) j j (p j, q j, r j ) j ρ(p j, q j, r j )v( j ), v( j ) j ρ(p j, q j, r j )v( j ) j ρ(x, y, z) Riemann 3 ρ(x, y, z)dxdydz. 3 p179 p18 3. 3 p18 6 4. 7. 3 r, θ, φ r sin θ 5. R M = R M = dr π dθ π ρ( x + y + z )dxdydz ρ(r)r sin θdφ = 4π R r ρ(r)dr ρ(r) M = 4π 3 R3 ρ 4π 3 R3 6. (,, a), a > R P m j j P (p j, q j, r j ) j Gmρ(p j, q j, r j )v( j ) p j + q j + (a r j)
G z Gmρ(p j, q j, r j )v( j ) j p j + q j + (a r j) Gmρ(p j, q j, r j )v( j ) a r j p j + q j + (a r j) p j + q j + (a r j) a r j p j + qj + (a r j) = j Gm (a r j)ρ(p j, q j, r j ) ( p j + qj + (a r j) ) 3 v( j ) (a z)ρ(x, y, z) Gm( x + y + (a z) ) 3 Riemann (a z)ρ(x, y, z) Gm ( x + y + (a z) ) dxdydz 3 P Gm R dr π dθ π (a r cos θ)ρ(r) ( r ar cos θ + a ) 3 r sin θdφ φ π πgm R dr π (a r cos θ)ρ(r) ( r ar cos θ + a ) 3 r sin θdθ r ρ(r) 1 I = π (a r cos θ) sin θ ( r ar cos θ + a ) 3 dθ r ar cos θ + a = u θ : π u : a r a + r r +a u a r cos θ a a ( r ar cos θ + a ) = = a r + u 3 u 3 au 3 I = 1 a r ar sin θdθ = udu sin θdθ = udu ar a+r a r a r + u u du = 1 a r I r 4π Gm a [u a r R u ] a+r a r = r ρ(r)dr = GmM a a + r (a r) (a r) + (a + r) a r M 3 = a
II (1 4 ) 4(4) 5(3) 4 5 45 4(1) z = x/3 y/3 x, y, (z =) x/3 y/3 x 3 y 3 6 y 3 dxdy = dy x 3 y 3 [ ] 6 y 6 y 3 dx = dy x x 3 6 x= 3 (6 y) ] 3 (6 y)3 = dy = [ = 6 6 36 4() : x + y a x + y z = h h a x + y h a π h πh a hdxdy = dr (a r)rdθ = (a r)rdr = πh [ ] ar a a a a a r3 = a πh 3 3 4(3) 1 1 4(4) z 3 z = ± c ( ) x + y a xy z : c ( x + y a ) ( ( c x + y a) dxdy, : c x + y a) (r a) c a c r a + c = a+c a c dr π a+c c (r a) rdθ = 4π r c (r a) dr r = a + c sin u π/ u π/ = 4π π/ = 4ac π π/ π/ (a + c sin u)(c cos u)(c cos u)du = 4π cos u + 1du = ac π a c π/ π/ ac cos u + c 3 cos u sin udu cos u sin u
5(1) z = 1 x y : x + y 1 1 + (z x ) + (z y ) dxdy = 1 + 4x + 4y dxdy 5() z x = 1 π = dr r 1 1 + 4r dθ = π r [ ] 1 1 1 + 4r dr = π 1 (1 + 4r ) 3/ = π ( ) 5 5 1 6 x a x y, z y y = a x y 1 + (z x ) + (z y ) a dxdy = a x y dxdy, : x + y ax 4(3) = = π/ π/ π/ π/ a cos θ dθ ar π/ a r dr = [ a a r ] a cos θ π/ π/ a a sin θ dθ = 4a 1 sin θdθ = a (π ) a a cos θ = a sin θ 5(3) z = ±c 1 x y 1 + (z x ) + (z y ) 1 + (c 1)(x + y ) dxdy = dxdy, : x + y 1 1 x y = 1 dr π r 1 + (c 1)r 1 r dθ = 4π 1 r 1 + (c 1)r 1 r dr u = 1 + (c 1)r 1 r r 1 u r = u 1 u + c 1, rdr = c u (u + c 1) du = 4πc u du = 1 u + c πc 1 [ = πc u ] + πc u + c 1 1 = π + π c c 1 1 1 ( 1 u u + c 1 ) du 1 du = π + πc u + c 1 πc 1 c 1 tan 1 c 1 = π + [ tan 1 c 1 πc c 1 tan 1 c 1 ] u c 1 1 1 x > tan 1 x + tan 1 1 x = π 38.4(3)
1 31
II (9 7 ) 6 1. z = f (x, y) (x, y, f (x, y)) 1 z = ax + by + c a y x b y c z z z = ax + by xz z = ax yz z = by z = c ax + by = c a, b a, b 1 xy z = 1 x y. 6.1 6.1 118 3 3. 117 6.1
xy lim (x,y) (,) x + y (x, y) (, ) (x, y) (, ) r = x + y xy x + y = r cos θ sin θ = cos θ sin θ r r cos θ sin θ θ cos θ sin θ (x, y) (, ) xy lim (x,y) (,) x + y 4 xy x + y 4 = r 3 cos θ sin θ r cos θ + r 4 sin 4 θ = r cos θ sin θ cos θ + r sin 4 θ cos θ cos θ = x = y (x, y) = (t, t) 1/ (x, y) (, ) 116 6.6 x 3 + y 3 lim (x,y) (,) x + y 4. r x3 + y 3 x + y = r(cos3 θ + sin 3 θ) r (x, y) (, ) θ θ 119 6. x y 1 f 11 6. f (x, y) = e x y 5. xy (x, y) (, ) f (x, y) = x + y (x, y) = (, ) (x, y) (, ) (, ) x f (h, ) f (, ) f x (, ) = lim = lim = x h x h
y f y (, ) = f (x, y) (x, y) (, ) f (x, y) f (x, y) 1/ 1 1/ 1 x y f x (.) x f (x, ) x 15 6 1