Клевчихин - Матан II семестр

inf

ρ ϕ

ϕi 1

ϕ

ϕi

ϕi 1

ϕ

ϕi

21 mi2

ϕi

1

n

mi2 ϕi

2 i

1

Mi

sup

ρ ϕ ϕi

1

ϕ

ϕi

ϕi 1 ϕ

ϕi

1 n

Mi2

ϕi

2 i

1

β

1

ρ2 ϕ dϕ

ρ2 ϕ

2

R3

z

R3

y

Π

x

Π

x, y, z : a x b; c y

d; e z

f ,

a b c d e f

Π

Π

Π

b a d

c f

e ,

γ : y y t

a t b,

~r t

x t ~ı y t ~ z t ~κ

b

b

ℓ γ

x2 t

y2 t

z2 t dt

~r t dt .

a

a

y f x x a; b

f

b

ℓ

1 f x 2 dx.

a

ρ ρ ϕ ϕ α; β

ρ

β

ℓ

ρ ϕ 2 ρ ϕ 2 dϕ.

α

x

ρ ϕ cos ϕ,

y

ρ ϕ sin ϕ .

x

ϕ

ρ

ϕ

cos ϕ

ρ ϕ sin ϕ,

y

ϕ

ρ

ϕ

sin ϕ

ρ ϕ cos ϕ.

f

a; b

x

a; b

f x

0

L

τ

a

x0

x1

xn

b

a; b

Ox

S f, τ

y

y = f (x)

0

x

S

π R

r l,

a

b

R

r

l

f

xk r f xk 1

R f xk

l

l

xk

xk 1 2

f xk

f xk 1

2

xk 2

f xk

2.

n

2.

S f, τ

π f xk

f xk 1

xk 2

f xk

k

1

λ τ

S f, τ

λ τ

0

k

ξk

f xk

f xk f xk 1

f ξk xk

xk 1

f ξk xk.

n

xk 2

2

S f, τ

π f xk

f xk 1

f xk

k

1

n

π f x

k

f x

k

1

x

k

2

f ξ

x

k

2

k

k

1

n

π

f x

k

f x

k

1

1 f ξ

2 x

.

k

k

k

1

b

n

2πf x

1

f

x 2 dx

lim

π

f ξ

k

f ξ

1 f ξ

2

x

.

λ τ

0

k

k

k

a

1

k

f

a; b

f x

0

x

b

S f

2πf x

1

f

x 2 dx

a

f

Ox

M

sup

f x

x

a;b

n

n

π

f x

k

f x

k

1

1 f ξ

2 x

k

2πf

ξ

1 f ξ

2 x

k

k

k

k

k 1

k

1

n

π f x

k

f ξ

f x

k

1

f ξ

k

1 f ξ

2

x

k

k

k

k

1

n

π 1 M 2

f x

k

f ξ

k

f x

k

1

f ξ

x

k

k

k 1

n

π 1 M 2

2ωk f xk.

k

1

L

~rL ~rL t t

a; b

τ

a

t0 t1

tn

b

k

tk 1; tk

~rL t

Ak

~rL tk

ℓ L

n

ℓ L

A0A1 A1A2

An 1An

~r tk

~r tk 1

k

1

Ak 1

~r tk 1

Ak

~r tk

~r tk

~r tk 1

L

~r

~r t

t

α; β

tk

Ak

tk 1

tk

z

A6

z

A6

A5

A5

0

A4 y

0

A3 y

A1

A1

x

A2

A3

x

A2

A4

~r

~r t

t

a; b

ℓ

sup ℓ L

: L

~r ~r t .

~r t

~

tk 1; tk

~

~a b t t

~a b

~r t

t

a, b

τ

a; b

a

τ

τ

a; b

~r a

~r

τ

τ

s τ .

~r t

~r

~r t

a; b

~r

t

M :

~r

t

M,

~r

~r

t

a; b

~r

t

~r

~r

t

ds

~r

t

dt

~r t

x t ~ı

y t ~

z t ~κ

ds

x t 2

y t 2 z t 2 dt.

τ

a

t0

t1

tn b

a; b

Pk

~r tk

L

~r t

Pk

n

ℓ L

P0P1

P1P2

Pn 1Pn

~r tk

~r tk 1 .

k

1

~r

tk

~r tk

1

~r tk

~r tk 1

~r ξ

tk

tk 1

M tk

tk 1 .

n

n

ℓ L

~r tk

~r tk 1

M tk

tk 1

M b a .

k

1

k

1