Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2

X Y X ×Y

(x, y) X×Y = max{x X , y Y }

−∞) f

lim F (ˆx + h) − F (ˆx)

h→0 h

h → 0

F (ˆx + h) = F (ˆx) + F (ˆx)h + o(h),

j(h) h

c (a, b)

f (b) − f (a) = f (c)(b − a).

f (x)

[a, b] = {x : x = a + t(b − a), 0 t 1},

(a, b)

ϕ(t) = f (a + t(b − a))

|f (b) − f (a) M (b − a)

M = sup |f (x)|

x (a,b)

f (b) − f (a) sup fΓ(c) · b − a .

c (a,b)

f (n)(ˆx) f (ˆx + h) = f (ˆx) + f (ˆx)[h] + (1/2)f (ˆx)[h, h] + . . .

· · · + (1/n!)f (n)[h, . . . , h] + r(h), h → 0

z = f (x)

l

l

y = y(x)

σ XY

f (x) x

x0

v[y(x)] y

y = y(x) y1 = y1(x)

y(x) − y1(x) < ε, . . . , y(k)(x) − y1(k) < ε.

y(x) y1(x)

k

f (x) C(x0) ε > 0 δ = δ(x0) : x : |x − x0| < δ

|f (x) − f (x0)| < ε

v[y(x)] y = y0(x) k

ε > 0 δ = δ(ε) : y(x) Y

y(x) − y0(x) < δ

. . . . . .

y(k)(x) − y0(k)(x) < δ

|v[y(x)] − v[y0(x)]| < ε

ρ(y1, y2) y = y1(x) y = y2(x) (x [x0, x1])

y1 y2 Ck

k

l(x)

l(Cx) = Cl(x) ( C)

l(x1 + x2) = l(x1) + l(x2)

l(x) = kx k = const

L[y(x)]

y = y0(x)

δv = 0

v[y(x)] y = y0(x)

y(x) −y0(x) < ε

v[y(x)] y = y0(x) y = y0(x)

x1

v[y(x)] = F (x, y(x), y (x))dx,

x0

y(x0) = y0 y(x1) = y1 F (x, y, y ) C3

y(x, α) = y(x) + α(¯y(x) − y(x)) = y(x) + α δy, δy = y¯(x) − y(x).