2018 10
aiwanjun@sjtu.edu.cn 1201 / 18:00-20:20 213 14:00-17:00
I
II Taylor :
, n R n : x = (x 1, x 2,..., x n ) R; x, x y ; δ( ) ; ; ; ; ; ( ) ; ( / ) ; ; Ů(P 1,δ) P 1 U(P 0,δ) P 0 Ω P 1:
1.1 ( ). Ω R n (n 2), f Ω n, f: Ω R. 1. p, V T : p = CT /V, C = Const., T > 0, V > 0.,.
., : 2.1. a f(x) x 0, ϵ > 0, δ > 0, f(x) a < ϵ, 0 < x x 0 < δ. lim f(x) = a. x x 0
: 2.2. f(x, y) P 0 (x 0, y 0 ) A, ϵ > 0, δ > 0, f(x, y) A < ϵ, (x, y) Ů(P 0, δ). lim f(x, y) = A. (x,y) (x 0,y 0 )
, (x, y) ( ) (x 0, y 0 ), f(x, y) a... lim x x 0 lim f(x, y), y y0 lim y y0 lim f(x, y) x x0 f(x, y).
2.3. : lim f(x, y), lim (x,y) (x 0,y 0 ) lim y y 0 x x 0 lim f(x, y) x x0 lim f(x, y), y y0, ;,, ;,
2. f(x, y) = xy/(x 2 + y 2 ),., lim f(x, y) = 0 = lim f(x, y), x 0 y 0, y = kx, lim f(x, y) = k/(1 + (x,y) (0,0) k2 ),,.
( ) 3. x sin 1 f(x, y) = y + y sin 1, xy 0, x 0, xy = 0 证明., f(x, y) 0 x + y 2 x 2 + y 2 0, lim (x,y) (0,0) f(x, y) = 0. lim sin 1 x 0 x,
2.4. f(t) (a, b), g(x, y) (x 0, y 0 ) Ω: = Ů( (x 0, y 0 ), δ 1 ), g(ω) (a, b). lim g(x, y) = t 0, (x,y) (x 0,y 0 ) lim f(t) = A, t t0 t 0 g(ω) f t 0, lim f(g(x, y)) = A. (x,y) (x 0,y 0 )
证明., ϵ > 0,, ϵ 1 > 0 f(t) A ϵ, 0 < t t 0 < ϵ 1., ϵ 1 > 0, δ > 0, t 0 g(ω), g(x, y) t 0 ϵ 1, (x, y) Ů( (x 0, y 0 ), δ ). 0 < g(x, y) t 0 ϵ 1, (x, y) Ů( (x 0, y 0 ), δ ).
f(g(x, y)) A ϵ, (x, y) Ů( (x 0, y 0 ), δ ). f(t) t 0, f(t) A ϵ, t t 0 < ϵ 1, f(g(x, y)) A ϵ, (x, y) Ů( (x 0, y 0 ), δ ).
4. lim x2 + y 2 1 sin (x,y) (0,0) x2 + y 2.
4. lim x2 + y 2 1 sin (x,y) (0,0) x2 + y 2. 证明. f(t) = t sin 1 t, g(x, y) = x 2 + y 2 lim g(x, y) = 0, lim (x,y) (0,0) g(x, y) = 0 x = y = 0, lim (x,y) (0,0) f(t) = 0. t 0 f(g(x, y)) = lim f(t) = 0. t 0
, P 0 P P 0, P 0.,,.,. 2.5. f Ω, f P 0 (x 0, y 0 ) Ω, lim f(x, y) = f(x 0, y 0 ). (x,y) (x 0,y 0 )., f P Ω, f Ω.
ϵ-δ ϵ-δ,. 2.6. f(x, y) Ω. f P 0 (x 0, y 0 ), ϵ > 0, δ > 0, P = (x, y) U(P 0, δ) Ω, f(x, y) f(x 0, y 0 ) < ϵ.
, : ; ;, ; :..
: 3.1. f(x) x 0. a f(x) x 0, f f(x 0 + x) f(x 0 ) (x 0 ): = lim x 0 x. ϵ-δ, : ϵ > 0, δ > 0, x δ, f(x 0 + x) f(x 0 ) f (x 0 ) x ϵ x.
,. f(t) t, f (t) t,.,,, f (t),..
, 100km/h 28m/s, 5, f(t) = 25(10t t 2 )/9. 2:, V km/s T
f(x, y), f (x, y 0 ), f (x 0, y), f(x, y), f x (x, y 0 ), f x (x, y 0)., ( ϵ-δ )., f x (x, y 0 ) f(x, y 0 ).. f x, f x ( ); ;
I,. 5. z = ln(1 + arctan x y ).
II 证明., z D = {(x, y) : y 0}, {x = 0},. z D = {(x, y) : x 0, y 0}, z x = (1 + arctan(x/y) ) 1 1 sgn(x/y) 1 + (x/y) 1/y, 2 sgn(x/y)y = (x 2 + y 2 )(1 + arctan(x/y) ), z y = (1 + arctan(x/y) ) 1 1 sgn(x/y) 1 + (x/y) x 2 y 2 sgn(x/y)x = (x 2 + y 2 )(1 + arctan(x/y) ).
, f(x) ϵ-δ : ϵ > 0, δ, x U(x 0, δ), f(x 0 + x) f(x 0 ) f (x 0 ) x ϵ x., o( x ) f(x 0 + x) f(x 0 ) = f (x 0 ) x + o( x ), o( x ) lim x 0 x = 0.
, P 0 (x 0, y 0 ) f(x, y),. 3.2. A, B, f(x 0 + x, y 0 + y) f(x 0, y 0 ) = A x + B y + o(ρ), ρ = x 2 + y 2. Adx + Bdy f(x, y) P 0, df P0. f P 0., f P 0, f P 0., f P 0 f x (P 0 ) = A, f y (P 0 ) = B., f x (P 0 )dx + f y (P 0 )dy.
I 6. f(x, y) P 0 (0, 0), f(x, y) + ax by lim (x,y) (0,0) ln(1 + x 2 + y 2 ) = 1, a, b, f x (P 0 ) + f y (P 0 )
II 证明. f P 0, f(0, 0) = 0. f(x, y) f(0, 0) ( ax + by) lim (x,y) (0,0) x2 + y 2 f(x, y + ax by) = lim ln(1 + x2 + y 2 ) (x,y) (0,0) ln(1 + x 2 + y 2 ) x2 + y 2 = 1 0 = 0, f x (P 0 ) = a, f y (P 0 ) = b, f x (P 0 ) + f y (P 0 ) = a + b.
,,.,.
3.3. f(x, y) P 0 (x 0, y 0 ), f P 0. 证明. f(x 0 + x, y 0 + y) f(x 0, y 0 ) = f(x 0 + x, y 0 + y) f(x 0, y 0 + y) + f(x 0, y 0 + y) f(x 0, y 0 ). f(x 0, y 0 + y) f(x 0, y 0 ) = f y (x 0, y 0 ) y + o( y ), f(x 0 + x, y 0 + y) f(x 0, y 0 + y) = f x (x 0, y 0 + y) x + o( x ).
, f(x 0 + x, y 0 + y) f(x 0, y 0 ) = f x (x 0, y 0 + y) x + f y (x 0, y 0 ) y + o( x ) + o( y ), f x P 0, f x (x 0, y 0 + y) f x (x 0, y 0 ) = o( y ). f(x 0 + x, y 0 + y) f(x 0, y 0 ) = f x (x 0, y 0 ) x + f y (x 0, y 0 ) y + o( y ) x + o( x ) + o( y ),, f P 0.
I : u = u(x) x 0, f = f(u) u 0 = u(x 0 ), z = f(u(x)) x 0, z = f (u(x 0 )) u (x 0 ).
II,, 3.3. u = u(x, y), v = v(x, y) P 0 (x 0, y 0 ). z = f(u, v) Q 0 = (u(x 0, y 0 ), v(x 0, y 0 )), z = f(u(x, y), v(x, y)) P 0 ( ), z x (P 0 ) = f u (Q 0 ) u x (P 0 ) + f v (Q 0 ) v x (P 0 ), z y (P 0 ) = f u (Q 0 ) u y (P 0 ) + f v (Q 0 ) v y (P 0 ).
III., P 0., u, v P 0. [1,. 6.13].,, dz = f u du + f v dv, du = u x dx + u y dy, dv = v x dx + v y dy. dz = (f u u x + f v v x )dx + (f u u y + f v v y )dy, z = f(u(x, y), v(x, y)) ( )
I 7. z = z(x, y), z = z xx + z yy z. x = r cos θ, y = r sin θ, f(r, θ) = z(r cos θ, r sin θ) ( z ).
II 证明. f rr = (z x x r + z y y r ) r = z xx x 2 r + 2z xy x r y r + z yy y 2 r + z x x rr + +z y y rr = z xx cos θ 2 + 2z xy cos θ sin θ + z yy sin θ 2, f θθ = z xx x 2 θ + 2z xy x θ y θ + z yy y 2 θ + z x x θθ + z y y θθ = z xx ( r sin θ) 2 2z xy (r 2 sin θ cos θ) + z yy (r cos θ) 2 + z x ( r cos θ) + z y ( r sin θ) = r 2 ( z xx sin 2 θ 2z xy sin θ cos θ + z yy cos 2 θ ) rf r,
III, z = z xx + z yy = f rr + 1 r f r + 1 r 2f θθ., f(r, θ) = ln r, r > 0, f = 0, z(x, y) = ln x 2 + y 2 ( z = 0).
I : D = { (x, y) R 2 : x 2 + y 2 = 1 }. F (x, y) = x 2 + y 2 1, D F (x, y) xy-. y = y(x), x. ( 1, 0) (1, 0) P U(P ; δ), U(P ; δ) y = y(x)., y = ± 1 x 2 = y (x) = x 1 x 2.
II ( 1, 0) (0, 1) y (x)., y (x) x = x 0, F x + F y y = 0 = y = F x /F y = F y 0. F y, F y (x 0, y(x 0 ))., x, f(y): = F (x, y) y., x y F (x, y) = 0. y = y(x), F (x, y(x)) = 0.
. 3.4. F (x, y) P 0 (x 0, y 0 ) U(P 0 ; δ) F y, F (x 0, y 0 ) = 0, F y (x 0, y 0 ) 0, F (x, y) = 0 P 0 U(P 0 ; δ ) y = y(x), y F (x, y(x)) = 0, y 0 = y(x 0 )
. 3.5. F (x, y, z) P 0 (x 0, y 0, z 0 ) U(P 0 ; δ) F z, F (x 0, y 0, z 0 ) = 0, F z (x 0, y 0, z 0 ) 0, F (x, y, z) = 0 P 0 U(P 0 ; δ ) z = z(x, y), F (x, y, z(x, y)) = 0, z 0 = z(x 0, y 0 ) z(x, y) z x = F x (x, y, z)/f z (x, y, z), z y = F y (x, y, z)/f z (x, y, z).
I,.,,., (x 0, y 0 ) z x z y, z = z(x, y) (x 0, y 0 ). 8. F (x, y, z) = x 2 + y 2 + z 2 1 = 0 z = z(x, y).
II 证明. F x + F z z x = 0 = z x = F x /F z = 2x/2z = x/z, (x 0, y 0, z 0 ), z 0 0, z = z(x, y), x z x = x/z., (x 0, y 0, z 0 ), z 0 > 0, z = 1 x 2 y 2, z x = x/z = x/ 1 x 2 y 2.,.
I,. :,?,?
II, xy- v x-, y- α,β. v = (cos α, cos β). y cosβ O β α v cosα x 3:
III f(x, y) v 3.6. f(x, y) P 0 (x 0, y 0 ), v = (cos α, cos β) (, cos 2 α + cos 2 β = 1). f(x 0 + t cos α, y 0 + t cos β) f(x 0, y 0 ) lim t 0 t, f P 0 v. f. (x0,y 0 ) v
IV v = (1, 0) v = (0, 1), f v f x, f y. f v v.,. 3.7. f(x, y) P 0 (x 0, y 0 ), v = (cos α, cos β). f P 0, f P 0 v, f v = f x (x 0, y 0 ) cos α + f y (x 0, y 0 ) cos β. x0,y 0 p. 116, 40(3)
f(x, y) P 0 (x 0, y 0 ), v = (cos α, cos β) f v = (f x, f y ) P0 v. (x0,y 0 ) (f x, f y ) P0 f P 0, f P0 f(p 0 )., f(p 0 ) v = f(p 0 ) v cos ( f(p 0 ), v)., :,, ;,,.
f/ f = 1 (f x, f y ), fx 2 + fy 2. (x, y, f(x, y)) 1 (f x, f y, f (fx 2 + fy 2 )(1 + fx 2 + fy 2 x 2 + fy 2 ) ). ( ).
, f(x, y) = sin(x 2 + y 2 ) x 2 + y 2 = 1. 4:
I 9. x + y + z = a 2az + a 2 = x 2 + y 2 γ P 0 (0, a, 0), a > 0. γ : x(t) = t, y(t) = a (2a t)t z(t) = ± (2a t)t t,
II, t [0, 2a], γ, γ(0) = P 0 = (0, a, 0): γ (0) = ( 1, t a a t, 1) (2a t)t (2a t)t t=0 = (1, 1 0, 1 0 1),.?
III F (x, y, z) = x + y + z a, G(x, y, z) = x 2 + y 2 + 2az a 2. P 0 y = y(x), z = z(x), F x + F y y + F z z = 0, G x + G y y + G z z = 0. (y, z ) Jacobi ( ) (F, G) (y, z) = det Fy G y 0. F z G z
IV, P 0, (F, G) (y, z) = det ( ) 1 2y = 0 1 2a, P 0, x y = y(x), z = z(x)., y, x = x(y), z = z(y). ( ) ( ) x 1 ( ) ( Fx F z = z Fy 0 = G x G z G y 1)
V, P 0 γ (0, 1, 1). P 0 x 0 0 = y a 1 = z 0 1. x(t) = 0 y(t) = a + t z(t) = t
VI P 0 ( ) (x 0), (y a), (z 0) (0, 1, 1) = 0, y a z = 0., : F, G, P 0, dx + dy + dz = 0, 2ady + 2adz = 0,
VII dx = 0, dz = dy. γ (x(t), y(t), z(t), ( ( ) ), (x (t), y (t), z (t)) = (0, y (t), y (t)), (0, 1, 1). p. 116, 44(3).
I S F (x, y, z) = 0, P 0 (x 0, y 0, z 0 ), F. γ(t) = (x(t), y(t), z(t)) S P 0, F (x(t), y(t), z(t)) = 0, γ(t 0 ) = P 0, γ (t) = 0. F x (P 0 )x (t 0 ) + F y (P 0 )y (t 0 ) + F z (P 0 )z (t 0 ) = 0, F (P 0 ) γ (t 0 ) = 0,
II F (P 0 ) S P 0 ( )., S P 0 F (P 0 ) ((x x 0 ), (y y 0 ), (z z 0 ) ) = 0, F x (P 0 )(x x 0 ) + F y (P 0 )(y y 0 ) + F z (P 0 )(z z 0 ) = 0., x x 0 F x (P 0 ) = y y 0 F y (P 0 ) = z z 0 F z (P 0 ).
III, z = f(x, y)( ), F (x, y, z) = f(x, y) z F = (f x, f y, 1). z = f(x, y). x = x 0, z(y) = f(x 0, y) (0, 1, f y ), y = y 0, z(x) = f(x, y 0 ) (1, 0, f x ), (0, 1, f y ) (1, 0, f x ) = (f x, f y, 1),.,,.
Taylor, f(x) x 0 n, f(x) = f(x 0 ) + f (x 0 )(x x 0 ) + f (x 0 ) (x x 0 ) 2 + 2! + f (n) (x 0 ) (x x 0 ) n + R n (x), n!, R n (x) = { o ( (x x0 ) n), Peano f (n+1) (ξ) (n+1)! (x x 0 ) n+1, ξ x 0 x f x 0 n + 1
Taylor I Taylor f(x 0 + x) = f(x 0 ) + x x f(x 0 ) + 1 2! ( x)2 2 xf(x 0 )+ + 1 n! ( x)n n xf(x 0 ) + R n (x), { Rn (x) = o( x n ), Peano ( x) n+1 x n+1 (x 0 +θ x) (n+1)!,
Taylor II Taylor f(x, y) (x 0, y 0 ) n + 1, f(x 0 + x, y 0 + y) = n k=0 1 k! ( x x + y y ) k f(x 0, y 0 ) + R n (x, y), R n (x, y) = { o(ρ n ), 1 (n+1)! ( x x + y y ) (n+1) f(x 0 + θ x, y 0 + θ y), Peano.
f(x) x 0 x 0 f (x 0 ) = 0. f(x) x 0, x 0 f (x 0 ) > 0( ), f (x 0 ) < 0( ).
Taylor, f (x 0 ) f(x 0, y 0 )( = (f x, f y )) x0,y 0, f (x 0 ) fxx f Jacobi(f) = xy., f(x, y) f yx f yy P 0 (x 0, y 0 ) f(x, y) P 0, P 0 f f x (P 0 ) = f y (P 0 ) = 0 f(p 0 ) = 0. f(x, y) P 0, P 0 f xx (P 0 ) > 0& det Jacobi(f)(P 0 ) > 0( ) f xx (P 0 ) < 0& det Jacobi(f)(P 0 ) > 0( ).
10. f(x, y) = x 2 y 2 + 2 D = { (x, y) : x 2 + y 2 /4 1 } 证明. f = (2x, 2y) = 0 = x = 0 = y, det Jacobi(f)(0, 0) = 4 < 0 (0, 0) f D = {θ [0, 2π] : x = cos θ, y = 2 sin θ} f f(x, y) D = cos 2 θ 4 sin 2 θ + 2 = 2 + 5 cos 2 θ. f(0, ±2) = 2, f(±1, 0) = 3.
u = f(x, y), ϕ(x, y) = 0 f(x, y) D := {(x, y) : ϕ(x, y) = 0} F (x, y, λ) = f(x, y) + λϕ(x, y)..
I 11.,, A B y x z C
II 证明. x, y, z, ABC S(x, y, z) = 1 (sin x + sin y + sin z) 2 ϕ(x, y, z) = x + y + z 2π = 0. F (x, y, z, λ) = S(x, y, z) + λϕ(x, y, z),
III F = 0, S x + λϕ x = 0, S y + λϕ y = 0, S z + λϕ z = 0, ϕ = 0 = { cos x = cos y = cos z = 2λ, x + y + z = 2π. x, y, z (0, π), x = y = z = 2π/3, ABC
I p. 110: 3(2)( ); 4(3); p. 111: 5( ) ( ): f: R n \ {0} R. f k (, t > 0, x = (x 1,..., x n ) R n \ {0}, f(tx) = t k f(x)), x f(x) = kf(x)., g(t, x) = f(tx), t g(t, x) = kg(t, x)/t ODE. p. 111: 7(1)( ) : { x sin 1 y f(x, y) = + y sin 1 x, xy 0, 0, xy = 0,
II O(0, 0). (,,,.,.) p. 113: 25 p. 116: 40(3)( ), u = x arctan(y/z) P 0 (1, 2, 2) v, v M 0 (5, 5, 15). p. 117: 46(1); 47 p. 118: 56 :
III 1. f = f xx + f yy + f zz, f(x, y, z) = f(r) (r, φ, θ) 2. 2 f(x, y) 3. f(x, y, z) = sin(x 2 + y 2 + z) P 0 (0, 0, 0) Taylor
I [1],. II: [M]., :, 2008.