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Московский государственный индустриальный университет

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Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2

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ζ ξ

y =

= M (ζ/ξ = x) ξ ζ	x = M (ξ/ζ = y)
ξ ζ

u =

x − aξ

v =

y − aζ

σξ

σζ

ϕ(x; y) =

−

e−2(1 − rξζ2 )

u2 + v2

ξζ

2πσξ σζ "

1 − rξζ

ϕξ

(x) =

√

e−

2 .

2π

−

− rξζ u)2

ϕ(x; y)

2(1

r2 )

ϕ(y/ξ = x) =

· e

− ξζ

ϕξ (x)

√

2πσξ "1 − rξζ2

y − aζ

−

ξζ

x − aξ

σζ

σξ

−

2 σξ "1

− rξζ2

= 2

· e

√

σξ

1 − rξζ2

2π

y − aζ + rξζ

σζ

(x − aξ)

σξ

−

· e

2 σξ "1 − rξζ2

√2π σξ "1 − rξζ

ζ		ξ
M (ζ/ξ = x) = aζ + rξζ	σζ	(x − aξ )
	σξ
σζ2(1 − rξζ2 )

		ξ ζ
M (ξ/ζ = y) = aξ + rξζ	σξ	(y − aζ ).

	σζ

(ξ, ζ)

ξ\ζ

	ξ ζ
ξ	ζ = 7
ζ	ξ = 5, 1

M (ξ) = 3, 4 · 0, 31 + 5, 1 · 0, 69 = 4, 573,

M (ζ) = 4 · 0, 37 + 7 · 0, 24 + 8 · 0, 39 = 6, 280.

P (ζ = 7) = 0, 11 + 0, 13 = 0, 24.

P (ξ = 3, 4|ζ = 7) = 0, 11/0, 24 = 11/24, P (ξ = 5, 1|ζ = 7) = 0, 13/0, 24 = 13/24.

ξ ξ

P (ξ = xi|ζ = 7)

M (ξ|ζ = 7) = 3, 4 · 11/24 + 5, 1 · 13/24 ≈ 4, 321.

P (ξ = 5, 1) = 0, 32 + 0, 13 + 0, 24 = 0, 63

P (ζ = 4|ξ = 5, 1) = 0, 32/0, 69 = 32/69,

P (ζ = 7|ξ = 5, 1) = 0, 13/0, 69 = 13/69,

P (ζ = 8|ξ = 5, 1) = 0, 24/0, 69 = 24/69.

P (ζ = yi|ξ = 5, 1)

M (ζ|ξ = 5, 1) = 4 · 32/69 + 7 · 13/69 + 8 · 24/69 ≈ 5, 957.

F (x, y) =

cos x · cos y 0 x π/2, 0 y π/2, 0

(ξ, ζ) x = π/4, x = π/2, y = 0, y = π/4

P (x1 < ξ < x2, y1 < ζ < y2) = (F (x2, y2) − F (x2, y1))−(F (x1, y2) −F (x1, y1)).

x1 = π/4, x2 = π/2, y1 = 0, y2

= π/4

√

2 ·

4 −

−

4 −

P = (cos

cos

cos 0)

(cos

cos

cos 0) =

− 1

F (x, y) =	0	−	−	ax)(1	−	−	by) x > 0, y > 0,
	(1		e	ax)(1		e	by) x > 0, y > 0,

(ξ, ζ)

∂F (x, y) = b(1 − e−ax)e−by. ∂y

				x
			ϕ(x, y) = abe−ax−by
x > 0, y > 0	0	−	−	ϕ(x, y) = 0 x < 0	y < 0
ϕ(x, y) =	0	−	−	by x > 0, y > 0,
	abe		ax	by x > 0, y > 0,
π/4, −π/4 y π/4		ϕ(x, y) = (1/2) cos(x + y)			−π/4 x
π/4, −π/4 y π/4				ϕ(x, y) = 0

(ξ, ζ)

				x	y
				F (x, y) = −π/4	−π/4			ϕ(x, y)dxdy.
			−π4 x π4 −π4 y π4
	1		x	y		1
F (x, y) =	1		−π/4 dx	−π/4 cos(x+y)dy =		1	(cos(x−π/4)+cos(y−π/4)−cos(x+y)).

	2					2
		x < −π4		y < −π4
				x				y
				F (x, y) = −∞ dx		−∞ 0dy = 0.
		x > π4 −π4 y π4
	1		π/4	y			1
F (x, y) =	1		−π/4 dx	−π/4 cos(x + y)dy =			1	(1 + cos(y − π/4) − cos(y + π/4)).

	2						2

−π4 x π4 y > π4

F (x, y) = 12 (1 + cos(x − π/4) − cos(x + π/4)). x > π4 y > π4

F (x, y) = 1.

ϕ(x, y) =

= a/(x2 + y2 + 2)4	(ξ, ζ)
a

		+∞		+∞
		−∞	−∞		ϕ(x, y)dxdy = 1.
+∞	+∞	a			2π	∞	1
−∞	−∞			dxdy = a 0		dϕ 0		rdr = 1.
−∞	−∞	(x2 + y2 + 2)4		dxdy = a 0		dϕ 0	(r2 + 2)4	rdr = 1.
a = 24/π						ϕ r		πa/24 = 1
a = 24/π
a = 24/π
						(ξ, ζ)
	ϕ(x, y) = a/((1 + x2)(4 + y2))							a
				F (x, y)

(ξ, ζ) G : x [0, π/4], y 0, π/2]

ξ ζ

+∞

a −∞

−∞

= 1,

a · [arctgx]−∞+∞

· [

arctg

]−∞+∞ = 1,

1 + x2

4 + y2

π2

a · (

) ·

(

) = 1, a ·

= 1, a =

π2

ϕ(ξ, ζ) = π2(1 + x2)(4 + y2) .

−∞

· (arctgx +

· (

F (x, y) =

)

arctg

π2

1 + x2

4 + y2

π2

P ((ξ, ζ) G) = F (

) − F (0,

) − (F (

, 0)

− f (0, 0)) =

{(arctg

) ·

(

arctg

) −

(

arctg

)−

π2

π π

−((arctg

)

−

)} =

π2

+∞	2	+∞ dy			1
ϕξ(x) = −∞ ϕ(x, y)dy =		−∞		=		.
	π2(1 + x2)		4 + y2		π(1 + x2)

ϕζ (y) = π(4 + y2) .

ϕ(x, y) = ϕξ(x) ·ϕζ (y) ξ ζ

ϕ(x, y) = a(x2 + xy) 0 x 2, 0 y ≤ 2 ϕ(x, y) = 0 a

2 2

ϕ(x, y)dxdy = 1.

2 2

adx (x2 + xy)dy = 1

		2			8
		a 0	(2x +		8		)dx = 1.

						3				a·28/3 = 1
		a = 3/28								a·28/3 = 1
		a = 3/28
			ϕ(x, y) = (3/28)(x2 + xy)
		ξ	ζ
2	2				3			2		2	10
M (ξ) = 0	0	xϕ(x, y)dxdy =			3			0	xdx 0	(x2 + xy)dy =	10			,

				28							7
2		2			3			2	2		8
M (ζ) = 0	0	yϕ(x, y)dxdy =			3			0	dx 0	y(x2 + xy)dy =	8		.

					28							7

(ξ, ζ)

ϕ(x, y) = π1 e−(x2+2xy+2y2).

+∞

ϕξ(x) = −∞ ϕ(x, y)dy =

e−x

−∞ e−2(y+x/2)

dy =

√

e−x

/2.

2π

+∞

−∞ e−t

dt = √π

t = √2(y + x/2).

+∞

ϕζ (y) = −∞

ϕ(x, y)dy =

e−y

−∞ e−(x+y)

dx =

√

e−y

ξ = x

ϕ(x, y)

ϕ(y/ξ = x) =

e−(1/2)(x+2y)

ϕξ(x)

ζ = y

ϕ(x, y)

ϕ(x/ζ = y) =

√

e−(x+y)

ϕζ (y)

(ξ, ζ)

O(0, 0), A(0, 6), B(6, 0)

	(x, y)				AB y = 6 − x
ϕ(x, y) = a					AB y = 6 − x
a
0	6 dx 0	6−x ady = 1,	a 0	6	(6 − x)dx = 1,

18a = 1, a = 1/18, ϕ(x, y) = 1/18

6−x 1				3		1
ϕξ(x) = 0			dy =		−			x (0 < x < 6),
		16		8		16
6−y 1				3		1
ϕζ (y) = 0			dx =		−			y (0 < y < 6).
		16		8			16
ϕ(x, y) =	ln2 4		4−x−y				x ≥ 0, y ≥ 0,
	0		· x < 0					y < 0.

M (ξ) = 0	∞ 0	∞ x · ϕ(x, y)dxdy = 0			∞ 0	∞ x · ln2 4 · 4−x−ydxdy =
			∞			1
		= x ln 4 0		x · 4−xdx =				.
		= x ln 4 0		x · 4−xdx =			ln 4	.
		x
		∞	∞						1
	M (ζ) = 0		0	y · ϕ(x, y)dxdy =						.
	M (ζ) = 0		0	y · ϕ(x, y)dxdy =					ln 4	.