шамин с сдо

1

z = 1 − x2 − y2 .

D

D = R2

z = sin x · cos y.

ε

|f(x) − A| < ε, x Bδ(ε)(x0) \ {x0}

δ(ε)

Bδ(ε)(x0) \ {x0}

{0}

x2y2

f(x, y) = x2 + y2 .

lim f(x, y)

(x,y)→0

x2y2

lim = 0.

(x,y)→0 x2 + y2

xy

f(x, y) = x2 + y2 ,

(x, y) → 0

f(x)

x0 RN

x0

lim f(x) = f(x0).

x→x0

{xkl }

{xk}

xkn [a, b], n = 1, . . . , N; k = 1, 2, . . . .

N

{xkn}∞k=1 n = 1, . . . , N

xknl , n = 1, . . . , N,

{xkl }

y = f(x1, . . . , xj−1, xj , xj+1, . . . , xn),

xj

x0j

f(x)

x0 Rn

y = f(x1, . . . , xj + j , . . . , xn) − f(x1, . . . , xj , . . . , xn),

f(x)

Q

f(x, y) = x sin y + y cos x.

∂f(x, y) = sin y − y sin x, ∂x

∂f(x, y) = x cos y + cos x. ∂y

f(x)

Q Rn

q

r = x21 + x22 + · · · + x2n.

Q

Q

∂f(x) Ai = ∂xi .

f(x)

n

X

df = Ai xi.

i=1

f(x)

x Rn

f(x)

n

x1 x2

(x1 + x1, x2 + x2)

y = f(x1 + x1, x2 + x2) − f(x1, x2) = = f(x1 + x1, x2 + x2) − f(x1, x2 + x2)+ +f(x1, x2 + x2) − f(x1, x2) =

= fx′ 1 (x1 + θ1 x1, x2 + x2)Δx1 + fx′ 2 (x1, x2 + θ2 x2)Δx2 = = (fx′ 1 (x1, x2) + ε1)Δx1 + (fx′ 2 (x1, x2) + ε2)Δx2 =

= fx′ 1 (x1, x2)Δx1 + fx′ 2 (x1, x2)Δx2 + (ε1 x1 + ε2 x2).

ε1 x1 + ε2 x2 = o(r), r → 0,