t(p) t Za&M,Nf Mp"+D u?D-p, sx ∂f ∂u(p

§1 ^{*u u}

z1 1 ( ^*u ) ^af : D → R ^76M R^{n bt}D ^>1<` (^>G

dp). ^<Bp∈ D ^2uR^{n b*(} u, ^s

t→0lim

f(p + tu) − t(p) t

Za&M,^Nf ^Mp^"+D u^?D-p, ^sx ∂f

∂u(p),^ f ⁺u¹

D-p.

; (1)^D-pj/Gdpϕ(t) = f (p + tu)^Mt= 0^"1-p. ^{3,

, u = e_i = (0, · · · , 0, 1, 0, · · · , 0) (⁴ i ^Q_ 1 ^1*() ^c, ^Ax ∂f

∂u(p)

∂f

∂x_i(p), ^ f ¹⁴ i^QE-p.
⁷⁶,^?

∂f

∂x_i(p) = lim

t→0

f(p₁,· · · , pi−1, pi+ t, p_i+1,· · · , pⁿ) − f(p1,· · · , pⁿ)

t .

(2) ^E-p ∂f

∂xi

Ax f_x⁰_i. ^Za f_x⁰_i = ∂f

∂xi

YVNE-p, ^Nx f_y⁰⁰_ixi =

∂

∂yi

 ∂f

∂xi



, ^ 2 ^E-p. ^s3276 f_x⁰⁰_iyi = ∂

∂xi

 ∂f

∂yi



2uOE

-p. ^Za f ^&MY. k ^1%%E-p, ^N f C^{k dp}. ^Æ7.e=

ZR,1xf:

∂²f

∂x²_i = ∂

∂x_i

 ∂f

∂x_i



, ∂²f

∂x_i∂y_i = ∂

∂x_i

 ∂f

∂y_i

 , · · ·

 1 ^Nf(x, y) = xy ¹1 ^, 2 ^E-p.

f_x⁰(x, y) = ∂f

∂x = ∂

∂x(xy) = y, f_y⁰ = x, f_yx⁰⁰ = 1, f_xy⁰⁰ = 1, f_xx⁰⁰ = f_yy⁰⁰ = 0.

 2 ^Nf(x, y) =px²+ y²+ xy ^M(x₀, y₀) = (1, 2) ^"1E-p.

∂f

∂x(x, y) = x

px²+ y² + y, ∂f

∂y(x, y) = y

px²+ y² + x ⇒

∂f

∂x(1, 2) =

√5

5 + 2, ∂f

∂y(1, 2) = 2 5

√5 + 1.

 3 ^a(x₀, y₀, z₀) ∈ R³,^N

f(x, y, z) =(x − x⁰)²+ (y − y0)²+ (z − z0)²−¹₂

1E-p.

x

r=(x − x⁰)²+ (y − y0)²+ (z − z0)²

1 2

N

∂f

∂x = − 1 r² ·∂r

∂x = −1 r²

x− x0

r = −x− x0

r³ . ,

∂f

∂y = −y− y0

r³ , ∂f

∂z = −z− z0

r³ .

g/Gdp 1j, E-p1&M?V>Gdp1%%!, ^Rj7

E-p\B<dp+{7D1!a.

 4 ^a

f(x, y) =







0, ^, x· y = 0, 1, ^, x· y 6= 0,

N

∂f

∂x(x, 0) = lim

x→0

f(x, 0) − f(0, 0)

x = 0,

∂f

∂y(0, 0) = lim

y→0

f(0, y) − f(0, 0)

y = 0.

V, f ^M(0, 0) ^"%%.

z 1 (^u) ^a f ^Z76 1, x⁰ ∈ D. ^za f_x⁰

i (1 ≤ i ≤ n) ^M x^{0 K}

%%, ^N

(1) f ^M x^{0 "%%};

(2)^Zax_i = xi(t)^Mt₀^"-, x⁰ = (x₁(t₀), · · · , x0(t₀)),^Nf(x(t))^Mt= t₀

"-, ^K

dt dt    t=t0

=

n

X

i=1

∂f

∂xi

(x(t₀)) ·dx_i dt (t₀).

8 (1)"=GbZ7 , ^? f(x) − f(x⁰) =

n

X

i=1

f (x⁰₁,· · · , x⁰i−1, x_i,· · · , xⁿ) − f(x⁰1,· · · , x⁰i−1, x⁰_i, x_i+1,· · · , xⁿ)

=

n

X

i=1

f_x⁰_i(x⁰₁,· · · , x⁰i−1, ξi, x_i+1,· · · , xⁿ) · (xⁱ− x⁰i)

,x→ x^{0 c}, ξi → x⁰i,^>f_x⁰_i ^Mx^{0 "%%W} lim

x→x⁰(f (x) − f(x⁰)) = 0.

(2)^> (1)^1V9, ^? df

dt    t=t⁰

= lim

t→t0

f(x(t)) − f(x(t0)) t− t0

= lim

t→t⁰ n

X

i=1

f_x⁰_i(x⁰₁,· · · , x⁰i−1, ξi, x_i+1,· · · , xⁿ) ·x(t) − xⁱ(t₀) t− t0

=

n

X

i=1

f_x⁰

i(x⁰) · x⁰i(t₀).

% M7 1 ^1~, ^Za u=

n

X

i=1

u_i· eⁱ = (u₁, u₂,· · · , uⁿ),^N

∂f

∂u(p) = d

dt|^t=0f(p + tu) =

n

X

i=1

∂f

∂x_i(p) · uⁱ.

z 2 (^{us.w -}) ^af : D → R ^@Gdp, (x₀, y₀) ∈ D. ^Z

af_xy^{00 g} f_yx^{00 M}(x₀, y₀) ^"%%,^N

f_xy⁰⁰ (x₀, y₀) = f_yx⁰⁰ (x₀, y₀).

8 <BG1 k6= 0, h 6= 0^G0dp ϕ(y) = f (x₀+ h, y) − f(x0, y), ψ(x) = f (x, y0+ k) − f(x, y⁰),

>Lagrange ^bZ7 ,^?

ϕ(y₀+ k) − ϕ(y0) = ϕ⁰_y(y₀+ θ₁k)k (|θ1| ≤ 1)

= f_y⁰(x₀+ h, y₀+ θk) − fy⁰(x₀, y₁+ θ₁k) k

= f_xy⁰⁰ (x₀+ θ₂· h, y0+ θ₁k) · kh (|θ2| ≤ 1).

,

ψ(x₀+ h) − ψ(x0) = f_yx⁰⁰ (x₀+ θ₃h, y₀+ θ₄k)hk.

3~,

ϕ(y₀+ k) − ϕ(y0) = ψ(x₀+ h) − ψ(x0),

Y

f_xy⁰⁰ (x₀+ θ₂h, y₀+ θ₁h) = f_yx⁰⁰ (x₀+ θ₃h, y₀+ θ₄k).

.k, h→ 0,^>f_xy⁰⁰ , f_yx^{00 M}(x₀, y₀) "%%v0FV2g.

% >Gdp1RE-pZa%%, ^NGZDN-$$[.

 5

f(x, y) =







xyx²− y²

x²+ y², (x, y) 6= (0, 0), 0, (x, y) = (0, 0).

Nf_xy⁰⁰ (0, 0) = 1, f_yx⁰⁰ (0, 0) = −1,^Rq97 2 b%%!zajT^1.

§2 ^)

aσ : [α, β] → Rⁿ R^{n b/~%%P},^x σ(t) = (x1(t), · · · , xⁿ(t)). ^Za x_i(t)(1 ≤ i ≤ n)^M t= t₀ ^"-, ^Nσ ^M t₀ ^"-,^x

σ⁰(t₀) = dσ

dt(t₀) = dσ dt    t=t⁰

= (x⁰₁(t₀), · · · , x⁰n(t₀))

σ⁰(t₀) σ ^M t₀ ^"1J(.

, σ⁰(t₀) 6= 0 ^c, ^ {σ(t0) + σ⁰(t₀)u|u ∈ R} σ ^M t₀ ^"1J, ^GD

P− σ(t0) = u · σ⁰(t₀)

q

x− x0

x⁰₁(t₀) = y− y0

x⁰₂(t₀) = z− z0

x⁰₃(t₀).

Æbσ(t₀)KDJU
1F8 A8, ^GD

(q − σ(t0)) · σ⁰(t₀) = 0.

 1 ^af ^/Gdp, ^.

σ(t) = (t, f (t))

Nσ⁰(t₀) = (1, f⁰(t₀)), σ ^M t₀ ^"JD

x− t0

1 = y− f(t0) f⁰(t₀)

v

y= f (t₀) + f⁰(t₀) · (x − t0).

 2 ^N3&

σ(t) = (a cos t, a sin t, t), t ∈ R

1JgA8D.

Mt= t₀ ^",

σ⁰(t₀) = (−a sin t0, acos t₀,1)

YJD

x− a cos t⁰

−a sin t0

= y− a sin t⁰

acos t₀ = z− t⁰ 1 ,

A8D

−(x − a cos t0)a sin t₀+ (y − a sin t0)a cos t₀+ (z − t0) · 1 = 0.

aD R^{m bt},^Æ7%%<` Σ : D → Rⁿ(n > m) R^{n b1/Q}

pP8. ^a u⁰ = (u⁰₁,· · · , u⁰m) ∈ D, ^N

u7→ Σ(u⁰1,· · · , u⁰i−1, u, u⁰_i+1,· · · , u⁰m)

R^{m bP},^ Σ ^]1 ui ^P. ^Zaui ^PMu⁰_i ^"-,^Nx ∂Σ

∂u_i(u⁰) = Σ⁰_ui(u⁰),^yjLPM u^{0 "1J(}. ^Za {Σ⁰ui(u⁰)|1 ≤ i ≤ m}^![ (^#

cu⁰ Σ^1UN5), N>RJ(P1 Æb Σ(u⁰) ^{1i| J}

|,J|1U
 A|, A|b1Gt A(.

,m= n − 1^c,^Σ ^P8, ^#cA|p 1. ^{3,^<BR^{3 b}

1(^) ^P8 Σ,

Σ(u, v) = (x(u, v), y(u, v), z(u, v))

Σ⁰_u(u₀, v₀) = (x⁰_u(u₀, v₀), y⁰_u(u₀, v₀), z_u⁰(u₀, v₀) Σ⁰_v(u₀, v₀) = (x⁰_v(u₀, v₀), y_v⁰(u₀, v₀), z_v⁰(u₀, v₀))

ZaΣ⁰_u(u₀, v₀), Σ⁰_v(u₀, v₀) ^![, ^N

~n = Σ⁰_u(u₀, v₀) × Σ⁰v(u₀, v₀)

= (y_u⁰ · z⁰v− zu⁰ · y⁰v, z_u⁰ · x⁰v− x⁰u· zv⁰, x⁰_u· yv⁰ − y⁰u· x⁰v) 6= 0

~

n ^A(, ^%?Σ ^1JF8D

(P − Σ(u0, v₀)) · ~n = 0

qM

x− x(u0, v₀) y − y(u0, v₀) z − z(u0, v₀) x⁰_u(u₀, v₀) y⁰_u(u₀, v₀) z⁰_u(u₀, v₀) x⁰_v(u0, v₀) z⁰_v(u0, v₀) z_v⁰(u0, v₀)

= 0.

 3 ^NM8 S² = {(x, y, z) ∈ R³| x²+ y²+ z² = 1}^1J8.

M8pP8

x= sin θ cos ϕ, y = sin θ sin ϕ, z = cos θ, 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π,

GA(

~

n= (cos θ cos ϕ, cos θ sin ϕ, − sin θ) × (− sin θ sin ϕ, sin θ cos ϕ, 0) = sin θ · (x, y, z)

YM(x0, y₀, z₀) "JF8D

(x − x0) · x0+ (y − y0) · y0+ (z − z0) · z0 = 0.

§3 ^4w&

Æ7p5/, <B/Gdp?*, j[Ldp2!dp/

. <B>Gdp, Æ7.2b! 76!.

aD ⊂ R^{n t}, ^Æ7<` f : D → Rⁿ  >G(Zdp, ^G

(g

f(x₁,· · · , xⁿ) = (f₁(x₁,· · · , xⁿ), · · · , f^m(x₁,· · · , xⁿ))

D H~, 2Ck|b1(2*(h.

z1 1 (^&
) ^af ^Z], x⁰ = (x⁰₁,· · · , x⁰n)^T ∈ D. ^Za&M m× n^1

SA= (a_ij)_m×n,^e0<Bx^{0 K 15} x,^?

kf(x) − [f(x⁰) + A · (x − x⁰)]k = o(kx − x⁰k), x → x0,

Nf ^M x^{0 "}, ^!<`

df(x⁰) : Rⁿ→ R^m v7→ A · v

 f ^M x^{0 "1G}.

" 1 (^& ⇒ ^u) ^Za f : D → R^{m M} x^{0 "}, ^NGG( f_i (1 ≤ i ≤ n)^M x⁰ "&MD-p, ^K

A= ∂f_i

∂x_j(x⁰)



m×n

.

8 %G1762 , ^Zaf ^Mx^{0 "},^Nf ^Mx^{0 "%%}. ^

82m= 1 #q9D-p1&M!.

#, ^Q*( u,^>76,^Æ7?

f(x⁰+ tu) − f(x⁰) = A(x⁰+ tu − x⁰) + o(kx⁰+ tu − x⁰k)

= t · Au + o(|t|)

Rq9

∂f

∂u(x⁰) = A · u, vD-p&M. ^{3,

∂f

∂xi

(x⁰) = A · eⁱ ⇒ A= ∂f

∂x₁(x⁰), · · · , ∂f

∂xⁿ(x⁰)

 .

 1 ^a

f(x, y) =





 x²y

x²+ y², (x, y) 6= (0, 0), 0, (x, y) = (0, 0).

7 |f(x, y)| ≤ 1

2|x|, ^Y f ^M (0, 0) ^"%%, ^K f_x⁰(0, 0) = f_y⁰(0, 0) = 0. ^Za u= (u₁, u₂) ^*(,^N

∂f

∂u(0, 0) = lim

t→0

f(tu₁, tu₂)

t = lim

t→0

t³u²₁u₂

(u²₁+ u²₂)t³ = u²₁u₂ u²₁+ u²₂.

V?f ^M (0,0) ^" (why?).

Za f_i (1 ≤ i ≤ m) ^1E-p&M, ^Nx J f = 

∂fi

∂xj



m×n, ^ f ¹ Jacobian. Jf M6/51ZW/Q<` J f : D → R^m·n,^R!Æ7m× n^

Sl R^{m·n b15}.

z 1 (&wr
#) ^Za J f ^M D ^b&M, ^{Kyn <`M} x^{0 "}

%%,^Nf ^M x^{0 "}.

8 Y2 m = 1 ^#V9. ^>~, f_x⁰_i ^M x^{0 "%%}, i = 1, 2, · · · , n.

TGbZ7 , ^? f(x) − f(x⁰) =

n

X

i=1

f (x⁰₁,· · · , x⁰i−1, xi, x_i+1,· · · , xⁿ) − f(x⁰1,· · · , x⁰i, x_i+1,· · · , xⁿ)

=

n

X

i=1

f_x⁰

i(x⁰₁,· · · , x⁰i−1, x⁰_i + θ · (xⁱ− x⁰i), xi+1,· · · , xⁿ) · (xⁱ− x⁰i)

=

n

X

i=1

f_x⁰_i(x⁰) · (xⁱ− x⁰i) +

n

X

i=1

αi· (xⁱ− x⁰i)

Gb

α_i = f_x⁰_i(x⁰₁,· · · , x⁰i−1, x⁰_i+θ(x_i−x⁰i), x_i+1,· · · , xn)−fx⁰i(x⁰₁,· · · , x⁰n) → 0, (x_i → x⁰i)

%?

f(x) −

"

f(x⁰) +

n

X

i=1

f_x⁰_i(x⁰) · (xi− x⁰i)

#

≤

n

X

i=1

α²_i

!¹₂

· kxi− x⁰ik

= o(kx − x⁰k)

vf ^Mx^{0 "}.

ZaÆ7 m× n ^1Sl R^{mn b15}, NS.76jV1Cp.

v,^ZaA= (aij)_m×n,^NGCp76 kAk =





 X

1≤i≤m 1≤j≤n

a²_ij







1 2

.

>Schwarz ^2g,^?

kA · vk ≤ kAk · kvk, ∀v ∈ Rⁿ.

z 2(^u) ^a ∆ R^{l bt}, D R^{m bt}, g : ∆ → D ^u f : D → R^{n <`}. ^Za g ^M u⁰ ∈ ∆ ^", f ^M x⁰ = g(u⁰) ^", ^NIi

<`h= f ◦ g : ∆ → R^{n M}u^{0 "}, ^K

J h(u⁰) = Jf (x⁰) · Jg(u⁰).

8 7 g ^M u^{0 "},^Y

g(u) − g(u⁰) = Jg(u⁰) · (u − u⁰) + Rg(u, u⁰) (1)

GbR_g(u, u⁰) = o(ku − u⁰k). ,⁷ f ^Mx⁰ = g(u⁰)^",^Y

f(x) − f(x⁰) = Jf (x⁰) · (x − x⁰) + Rf(x, x⁰) (2)

GbR_f(x, x⁰) = 0(kx − x⁰k).

>(1) ^W, ^,u→ u⁰) ^c, g(u) → g(u⁰) = x⁰. ²x= g(u)^)[(2), ⁰ f◦ g(u) − f ◦ g(u⁰) = Jf (x⁰)(g(u) − g(u⁰)) + Rf(g(u), g(u⁰))

= Jf (x⁰) · Jg(u⁰) · (u − u⁰) + R_f◦g(u, u⁰)

(3)

Gb

R_f◦g(u, u⁰) = Jf (x⁰) · R^g(u, u⁰) + R_f(g(u), g(u⁰))

%??ZXw

kRf◦g(u, u⁰)k ≤ kJf(x⁰) · Rg(u, u⁰)k + kRf(g(u), g(u⁰)k

≤ kJf(x⁰)k · kRg(u, u⁰)k + o(kg(u) − g(u⁰)k)

= o(ku − u⁰k) + o(O(ku − u⁰k)) = o(ku − u⁰k).

%?>(3) ^uG176W f◦ g ^Mu^{0 "}, ^KJ(◦g)(u⁰) = Jf (x⁰) · J^g(u⁰).

Za f, g ^GhG(g

y_i = fi(x₁,· · · , xⁿ), i= 1, · · · , m, x_j = gj(u₁,· · · , u^l), j= 1, · · · , n.

NJ(f ◦ g)(u⁰) = Jf (x⁰) · J^g(u⁰) ^M















∂y₁

∂u₁(u⁰) · · · ∂y₁

∂u_l(u⁰)

· · ·

∂y_n

∂u₁(u⁰) · · · ∂y_n

∂u_l(u⁰)















n×l

=















∂y₁

∂x₁(x⁰) · · · ∂y₁

∂x_m(x⁰)

· · ·

∂y_n

∂x₁(x⁰) · · · ∂y_n

∂x^m(x⁰)















n×m

·















∂x₁

∂u₁(u⁰) · · · ∂x₁

∂u_l(u⁰)

· · ·

∂x_m

∂u₁(u⁰) · · · ∂x_m

∂u^l (u⁰)















m×l

v

∂y_i

∂uj

(u⁰) =

n

X

s=1

∂y_i

∂xs

(g(u⁰)) ·∂x_s

∂uj

(u⁰).

R.jx1&_N.

 2 ^af(x, y)^, ϕ(x) ^,^N u= f (x, ϕ(x))^[B x^1-p.

>&_N

u⁰_x = f_x⁰(x, ϕ(x)) · x⁰x+ f_y⁰(x, ϕ(x)) · ϕ⁰(x)

= f_x⁰(x, ϕ(x)) + f_y⁰(x, ϕ(x)) · ϕ⁰(x).

 3 ^au= f (x, y) ^, x = r cos θ, y = r sin θ,^V9

 ∂u

∂x

2

+ ∂u

∂y

2

= ∂u

∂r

2

+ 1 r²

 ∂u

∂θ

2

.

8 >&_N,

∂u

∂r = ∂u

∂x·∂x

∂r +∂u

∂y ·∂y

∂r = ∂u

∂xcos θ + ∂u

∂y · sin θ

∂u

∂θ = ∂u

∂x·∂x

∂θ +∂u

∂y ·∂y

∂θ = −r ·∂u

∂x· sin θ + r∂u

∂ycos θ

Rq9

 ∂u

∂r

2

+ 1 r²

 ∂u

∂θ

2

=  ∂u

∂xcos θ + ∂u

∂ysin θ

2

+



−∂u

∂xsin θ +∂u

∂ycos θ

2

=  ∂u

∂x

2

+ ∂u

∂y

2

.

 4 ^az= f (u, v, w), v = ϕ(u, s), s = ψ(u, w), ^N ∂z

∂u, ∂z

∂w.

z= f (u, v, w) = f (u, ϕ(u, s), w) = f (u, ϕ(u, ψ(u, w)), w).

>&_N,

∂z

∂u = ∂f

∂u +∂f

∂v ·∂v

∂u = ∂f

∂u +∂f

∂v · ∂ϕ

∂u +∂ϕ

∂s · ∂s

∂u



= ∂f

∂u +∂f

∂v ·∂ϕ

∂u + ∂f

∂v ·∂ϕ

∂s ·∂ψ

∂u

∂z

∂w = ∂f

∂w +∂f

∂v · ∂v

∂w = ∂f

∂w + ∂f

∂v

 ∂ϕ

∂s · ∂s

∂w



= ∂f

∂w +∂f

∂v ·∂ϕ

∂s ·∂ψ

∂w.

lk, ^{Æ7}*3 _SG} (^gG) ^1NB. ^a f : D → R ^1

>Gdp, ^>76, f ^Mx ^"1G df(x)^j/Q!<`

df(x) : R^m → R v7→

m

X

i=1

∂f

∂x_i · vi

Æ7<` x7→ df (x)^ f ^1SG,^x df. ^>B d(λf + µg)(x) = λdf (x) + µdg(x), λ, µ ∈ R,

7#,SGX|.276yAgpKu, ^MRQ46,^? df =

n

X

i=1

∂f

∂x_i · dxⁱ (∗) d(λf + µg) = λdf + µdg,

d(f, g) = f · dg + gdf, d(f

g) = gdf − f dg

g² (g 6= 0).

Za (*) ^Sg:

df = Jf (dx₁,· · · , dxn)^T

NIi<`1&_N

d(f ◦ g) = J(f ◦ g) · (du1,· · · , du^m)^T,

= Jf (x) · Jg(u)(du1,· · · , du^m)^T (x = g(u))

= Jf (d(u)) · (dg1,· · · , dgⁿ)^T

RQ2g SG1gÆ!.

 5 u = log z²

px²+ y²,^N du^uu ^1E-p.

du= d log z²

x²+ y² = d logz²− ln(x²+ y²)

= 2

zdz− 1

x²+ y²d(x²+ y²)

= −2x

x²+ y²dx− 2y

x²+ y²dy+2 zdz.

Rq9

∂u

∂x = − ∂x

x²+ y², ∂u

∂y = − 2y

x²+ y², ∂u

∂z = 2 z.

§4 ^:95 Taylor ^

ap, q∈ Rⁿ,^.

σ(t) = (1 − t) · p + t · q, t∈ R.

Æ7 σ : [0, 1] → Rⁿ R^{n b%} p, q ^1Y;. ^aA R^{n b1it}, ^Za

∀a1, a₂∈ A, ^%a₁, a₂ ^1Y;YcB A,^NA ^t. ^{3,^t9j

//%1. Æ71t E.

 1 R^{n bM} B_r(x) ^9jt.

z 1 (^&
:9z) ^a D R^{n bE}, f : D → R^, ^N ∀ a, b ∈ D, ∃ ξ ∈ D,^e0

f(b) − f(a) = Jf(ξ) · (b − a),

Gbξ = a + θ(b − a), θ ∈ (0, 1), ξ ^B%a, b ^1Y;].

8 aσ: [0, 1] → D ^% a, b^1Y;, ^NIidp ϕ(t) = f ◦ σ(t)^j

1/Gdp. >/Gdp1GbZ7 , ∃ θ ∈ (0, 1) ^e0 ϕ(1) − ϕ(0) = ϕ⁰(θ) · (1 − 0).

]gv

f(b) − f(a) = Jf(ξ) · (b − a).

Gbξ = σ(θ) = a + θ(b − a).

% 1 ^aD R^nbOE(^//%t), f : D → R^. ^ZaJ f ≡ 0,

Nf ^p.

8 ZaD ^E, ^NFV2%7 1 ^$v0.. ^/3,^XQp, q∈ D,^aτ : [0, 1] → D ^j% p, q ^1%P. ^.

T = sup{t |^, 0 ≤ s ≤ t ^c, f◦ τ(s) = f(p)}.

7 D ^t, ^Y ∃ ε > 0 ^e0 B_ε0(p) ⊂ D. ^> τ ^1%%!W, ∃ δ > 0, ^e0 τ([0, δ)) ⊂ Bε0(p). ^>BB_ε0(p) ^E, f ^MB_ε0(p)^{b Zdp},^Y T ≥ δ > 0.

8V9T = 1 (^BVA). ^ZaV,^NMτ(T ) ∈ D^"ng]8Mτ(0) = p

"/,1z2W, ∃ δ⁰ >0 ^e0

[0, T + δ⁰) ⊂ {s ∈ [0, 1] | f(s) = f(p)}.

Rg T ^1765=! ^%? T = 1. ^>%%!$W f(q) = f (1) = f (T ) = f (p).

7 p, q ^jXQ1, ^Yf ^p.

 2 ^0(Zdp f : R → R², f (t) = (t², t³),^N J f(t) =





 2t 3t²





.

Za∃ θ ∈ (0, 1) ^e0

f(1) − f(0) = Jf(θ) · (1 − 0)

N





 1 1





=





 2t 3t²





. (^)

RQ#iq97 1<(ZdpL$.

z 2 (&
Æ9z) ^a D R^{n bE}, f : D → R^{m }, ^N

∀ a, b ∈ D, ∃ ξ ∈ D^e0

kf(b) − f(a)k ≤ kJf(ξ)k · kb − ak.

Gbξ = a + θ · (b − a), θ ∈ (0, 1).

8 xσ : [0, 1] → D ^% a, b^1Y;.^0Iidp ϕ: [0, 1] → R

ϕ(t) = hf(b) − f(a), f ◦ σ(t)i

Nϕ ^/Gdp, ^K

ϕ⁰(t) = hf(b) − f(a), Jf(σ(t)) · (b − a)i

>/Gdp1GbZ7 , ∃ θ ∈ (0, 1) ^e0

|ϕ(1) − ϕ(0)| = |ϕ⁰(θ)| ≤ kf(b) − f(a)k · kJf(σ(θ)) · (b − a)k

≤ kf(b) − f(a)k · kJf(ξ)k · kb − ak

7

|ϕ(1) − ϕ(0)| = |hf(b) − f(a), f(b)i − hf(b) − f(a), f(a)i|

= kf(b) − f(a)k²

i'Q2g0.)7 1V9.

% 2 ^a D R^{n bOE}, f : D → R^{m }. ^Za J f ≡ 0, ^N f ^Z

<`.

8 g21 ^1V9s, ¹.

T76, Za/Q>Gdp, Ny2=!dp , ^"=R/5

Æ72m swu.

 3 ^N 1.03²

√0.98√³ 1.06

1 sZ.

0dpf(x, y, z) = x²

√y√³z, x₀= (1, 1, 1), h = (0.03, −0.02, 0.06), ^N 1.03²

√0.98√³

1.06 = f (1.03, 0.98, 1.06)

= f (x₀+ h) ≈ f(x0) + Jf (x₀) · h

= 1 + (2, −1 2,−1

3)











 0.03

−0.02 0.06













= 1.05.

)Ue3N , Æ7-=>G>g >Gdp. ^#8/

xf. ^aα_i∈ Z(1 ≤ i ≤ n), ^xα= (α1,· · · , αⁿ), ^{ >c[}. ^x

|α| =

n

X

i=1

α_i, α! = α₁! · α2! · · · αn!.

Zax= (x₁,· · · , xⁿ) ∈ Rⁿ,^Nx

x^α= x^α₁¹ · x^α2²· · · x^αnⁿ.

<B>Gdp f,^n=81xfh |α| ^E-p: D^αf(x⁰) = ∂^|α|f

∂x^α_x1¹· · · ∂^αxnn(x₀).

z 3 (Taylor ^) ^a D R^{n bE}, f ∈ C^m+1(D)(^v f ^? m+ 1

%%E-p), a = (a₁,· · · , a^m) ∈ D. ^N ∀ x ∈ D, ∃ θ ∈ (0, 1)^e0 f(x) =

m

X

k=0

X

|α|=k

D^αf(a)

α! · (x − a)^α+ X

|α|=m+1

D^αf(a + θ(x − a))

α! · (x − a)^α.

8 0/Gdp ϕ(t) = f (a + t · (x − a)), t ∈ [0, 1]. ϕ ^? m+ 1^%

%-p, ^Y>/Gdp1 Taylor^Vg,^? (∗) ϕ(1) = ϕ(0)+ϕ⁰(0)+1

2!ϕ⁰⁰(0)+· · ·+ 1

m!ϕ^(m)(0)+ 1

(m + 1)!ϕ^(m+1)(θ), θ ∈ (0, 1)

"=`;A<V9

ϕ^(k)(t) = X

|α|=k

k!

α!D^αf(a + t(x − a)) · (x − a)^α

{3, t = 0 ^c,^?

ϕ^(k)(0) = X

|α|=k

k!

α!D^αf(a) · (x − a)^α,

]g)[ (∗) ^v0FVVg.

x

R_m= f (x) −

m

X

k=0

X

|α|=k

D^αf(a)

α! · (x − a)^α,

>7 3, f ∈ C^m+1(D)^c

R_m= X

|α|=m+1

D^αf(a + θ(x − a))

α! · (x − a)^α. (Lagrange^C)

% 3 ^M7 3 ^1~,^, kx − ak^Gc, R_m= O(kx − ak^m+1).

8 Qδ >0,^e0B¯_δ(a) ⊂ D. ^>Bf ∈ C^m+1(D), ¯B_δ(a)^{^},^Y∃ M > 0

e0

|D^αf(x)| ≤ M, ∀x ∈ ¯B_δ(a), |α| ≤ m + 1.

7#

|R^m| ≤ M · X

|α|=m+1

|(x − a)^α|

≤ M · X

|α|=m+1

kx − ak^|α| = M · n^m+1· kx − ak^m+1.

; (1) ^Za f ∈ C^m(D), ^NTaylor ^Vg

f(x) =

m−1

X

k=0

X

|α|=k

D^αf(a)

α! · (x − a)^α+ X

|α|=m

D^αf(a + θ(x − a))

α! · (x − a)^α

=

m

X

k=0

X

|α|=k

D^αf(a)

α! (x − a)^α+ Rm

Gb

R_m = X

|α|=m

1

α![D^αf(a + θ(x − a) − D^αf(a)] · (x − a)^α

=23 1V9DA0ZXw:

R_m = o(kx − ak^m), x → a. (Peano^C).

(2)^>GdpTaylor ^O1I\

f(x) = f (a) + Jf (a) · (x − a) +1 2

n

X

i,j=1

∂²f

∂x_i∂x_j(a) · (xⁱ− aⁱ) · (x^j− a^j) + · · ·

xHess(f ) =

 ∂²f

∂x_i∂x_j(a)



n×n

,^ f ^M a^"1Hessian.

(3) Taylor ^Og1p> f ^/U7.

z1 1 (^$ ) ^a D R^{n b1E}, f : D → R ^>Gdp. ^Za

∀ x, y ∈ D, ^?

f(tx + (1 − t)y) ≤ t · f(x) + (1 − t)f(y), ∀ t ∈ [0, 1]

Nf D^]1dp. ^]gb “≤” ^o “<”^c,^{ )Pdp}.

z 4 (1)^Zaf ^ME D^]?%%/E-p, ^Nf ^dp⇔

f(y) ≥ f(x) + Jf(x) · (y − x), ∀ x, y ∈ D.

(2)^Zaf ^?%%@E-p, ^Nf ^dp ⇔ Hess(f) ≥ 0 (^U7S).

8 (1) “⇒” ∀ x, y ∈ D, t ∈ [0, 1], ^?

t· (f(y) − f(x)) ≥ f(x + t(y − x)) − f(x)

= Jf (x) · t(y − x) + o(tky − xk)

]g'!2 t,^Vk.t→ 0^v0

f(y) ≥ f(x) + Jf(x) · (y − x).

“⇐” ∀ x, y ∈ D, t ∈ [0, 1], ^xz= tx + (1 − t) · y, ^N f(x) ≥ f(z) + Jf(z)(x − z) f(y) ≥ f(z) + Jf(z)(y − z)

Rq9

t· f(x) + (1 − t)f(y) ≥ f(z) + Jf(z) [t(x − z) + (1 − t)(y − z)] = f(z).

(2) “⇐” ^a f ^?%%1@E-p, ^KHess(f ) ^U7,^N> Taylor^Vg,

∀ x, y ∈ D, ∃ ξ = x + θ · (y − x), θ ∈ (0, 1) ^e0 f(y) = f (x) + Jf (x) · (y − x) + 1

2(y − x)^T · Hess(f)(ξ) · (y − x)

≥ f(x) + Jf(x) · (y − x).

>(1) ^Wf ^dp.

“⇒” (^BVA). ^ZaHess(f )^jU71,^N∃ x ∈ D,^2uh= (h₁,· · · , hⁿ) ∈ R^ne0

(h₁,· · · , hⁿ) · Hess(f)(x) · (h1,· · · , hⁿ)^T <0.

>Taylor ^VguGfx,^, ε→ 0^c,^? f(x + ε · h) = f(x) + ε · Jf(x) · h +1

2ε²· h^T · Hess(f) · h + o(kεhk²)

= f (x) + ε · Jf(x) · h + ε² 1

2h^T · Hess(f)(x) · h + o(1)



,ε6= 0 ^Gc,^]g4\ <0. ^#c

f(x + εh) < f (x) + Jf (x) · εh.

RDf ^dp5= (^=.(1)).

§5 4z34z

p5/: ^<B/Gdp, ZayK-p""F,, ^NLdpAKG

AY. ^R!, !g-pF,V)dpM]DA!dp?e 1 , ^7?.j (^) ^A1.8Æ70>G(Zdps1 |.

 1 ^{Æ70!<`</sup>. <sup>>!)p</sup>, <sup>!<`} A : Rⁿ → R^{n A}

⇔ detA 6= 0 ⇔ A ^{*`</sup>. <sup>Æ7?Z\</sup>: <sup>Za</sup> B : R<sup>n</sup> → R<sup>n !<`}, kBk < 1, ^NIn− B ^A.

id], ^a(In− B)v = 0,^N

kvk = kIⁿvk = kBvk ≤ kBk · kvk

Rq9v= 0.

v, ^{j <`R,/QA<`n/Q1=82kYV A<`}.

/3, XhA<`MWY A<`.

z 1 (^4z) ^aW R^{n bt}, f : W → Rⁿ C^k(k ≥ 1)^<`, x⁰ ∈ W . ^Za detJf (x⁰) 6= 0, ^N&M x^{0 1+E} U ⊂ W ^2u y⁰ = f (x⁰) ^1

+EV ⊂ Rⁿ,^e0 f|^U : U → V ^{jA<`</sup>,<sup>KGAY</sup> C<sup>k <`}.

8 b/!, ^a x⁰ = 0, y⁰ = f (x⁰) = 0. ²A ^x f ^Mx⁰ = 0^"1

G,^NA ^A,^Kf◦ A^{−1 M}0^{"G j <`}. ZaFV12< f◦ A⁻¹

$,^N<f ^.$. ^7#,^E%/fza J f(x⁰) = In.

76<`

g: W → Rⁿ x7→ f(x) − x

Ng C^{k <`},^K J g(0) = 0. ^7#, ∃ ε0>0^e0 kJg(x)k ≤ 1

2, ∀x ∈ ¯B_ε0(0) ⊂ W.

>@GFZ7 ,

kg(x1) − g(x2)k ≤ kJg(ξ)k · kx1− x2k

≤ 1

2· kx1− x2k, ∀ x1, x₂∈ ¯Bε0(0).

ay∈ B^ε0₂ (0), ^Æ7D

f(x) = y, x∈ B^ε0(0). (5.1)

R2{BM Bε0(0)^b'Qgy(x) = x + y − f(x) = y − g(x) ^185. ^Æ7"=

(w<H QR,185. ^m,

kg^y(x)k = ky − g(x)k ≤ kyk + kg(x)k

< ε₀ 2 +1

2kxk ≤ ε₀ 2 +ε₀

2 = ε₀, ∀x ∈ ¯B_ε0(0)

(5.2)

G$, gy : ¯B_ε0(0) → B^ε0(0) ⊂ ¯B_ε0(0) ^j(w<`: kg^y(x1) − g^y(x2)k = kg(x²) − g(x¹)k ≤ 1

2kx¹− x²k, ∀x¹, x₂ ∈ ¯B_ε0(0).

%?(5.1) ^M B¯_ε0(0)^b?/, ^x x_y. ^>(5.2), xy ∈ B^ε0(0). ^x U = f⁻¹(B^ε0

2 (0)) ∩ B^ε0(0), V = B^ε0

2 (0),

NÆ71ÆV9) f|^U : U → V ^j//1 C^{k <`</sup>, <sup>GA<`} h(y) = xy ^4k

y− g(h(y)) = h(y). (5.3)

(1) h : V → U ^j%%<`: ^,y₁, y₂ ∈ V ^c

kh(y1) − h(y2)k ≤ ky1− y2k + kg(h(y1)) − g(h(y2))k

≤ ky1− y2k + 1

2kh(y1) − h(y2)k

Rq9

kh(y1) − h(y2)k ≤ 2 · ky1− y2k, y1, y₂∈ V.

(2) h : V → U ^j<`: ^ay₀ ∈ V ,^N ∀ y ∈ V ,^? h(y) − h(y0) = (y − y0) − [g(h(y)) − g(h(y0))]

= (y − y0) − Jg(h(y0)) · (h(y − h(y0)) + o(kh(y) − h(y0)k)

]gM

[In+ Jg(h(y₀))] · (h(y) − h(y0)) = (y − y0) + o(ky − y0)k).

7?

h(y) − h(y0) = [In+ (Jg)(h(y₀))]⁻¹· (y − y0) + o(ky − y0k).

vh ^M y₀ ^".

(3) h : V → U C^{k <`}.

>(2) ^1V9W

J h(y) = [In+ Jg(h(y))]⁻¹ = [Jf (h(y))]⁻¹, ∀y ∈ V. (5.4)

>f ∈ C^k⇒ Jf ∈ C^k−1. ^>(2) ^u(5.4) ⇒ Jh ∈ C⁰,^v h∈ C¹. ^L>J f ∈ C^k−1, h∈ C^{1 u}(5.4) ⇒ Jh ∈ C^{1 v}h∈ C². ^0$,^lkÆ70. h∈ C^k.

; %V92 , ^Zaf : W → R^{n 1} Jacobian^Fm, ^N f(W ) ^

t.

 2 ^0 f : R² → R², f (x, y) = (e^x· cos y, e^x· sin y). ^V, f ^j*`,

+

detJf (x, y) = det







e^xcos y −e^xsin y e^xsin y e^xcos y





= e^2x 6= 0.

Rq97 1 ^12\?3$.