):
4.1 
; L: ABC,
A(1;3), B(2;4), C(2;3)
4.2 
L: ABC,
A(0;0), B(1;1), C(1;0)
4.3 
; L: OAB,
O(0;0), A(0;1), B(1;1)
4.4 
; L: x 
+ y = R
4.5 
; L: 
+ 
= 1
4.6 
; L:
A(1;1), B(2;2), C(1; 3)
4.7 
; L: ABC,
A(1;2), B(-1;3), C(0;4)
4.8 
; L: (x 1) + (y - 1) 
= 1
4.9 ; L: x + y =25
4.10 
; L: x + y = R
2
1.
f(x,y)
L,
y = 
(x), a 
x b
1.1 
; L:
A(0,1), B(3,0),
C(3,2), D(0,2)
1.2 
; L: x + y + z = a
x + y + z = 0
1.3 
; L:
A(0,0), B(1,0), C(0,1)
1.4 
; L: x 
+ y = a , x 
0, y 0
1.5 
; L: x + y = x - y ; x 0, y 0
1.6 
; L:
x = a cos t, y = a sin t, z = bt, 0 t 2 
1.7 
; L: (x + y ) = xy
1.8 
; L:
A(0,1), B(2,0), C(0,2)
1.9 
; L: x + y = a , x 0, y 0
1.10 
; L: x + y = z , y = ax
2.
f (x,y)
L,
:
2.1 f (x,y) = y ; L: x = a cos t, y = a sin t, 0 t 
2.2 f (x,y) = xy; L: x = a cos t, y = b sin t, 0 t
2.3 f (x, y) = y ; L: x = a(tsin t), y = a(1-cos t),
0 t
2.4 f(x, y) = 
; L: x = a(cos t + t sin t),
y = a(sin t t cost), 0 t 2
2.5 f(x,y) = 3x -y ; L: x = a(cos t + t sin t),
y = a(sin t t cost), 0 t 2
2.6 f(x, y) = x ; L: x = a(t t sin t),
y = a(1 cos t), 0 t 2
2.7 f(x,y) = xy L: x = ch t, y = ash t, 0 t t 
2.8 f(x,y) = 
; L: x = a(t - t sin t),
y = a(1-cos t), 0 t
2.9 f(x,y) = x 
+ y ; L: x = a cos 
t, y = a sin t,
0 t 2
2.10 f(x,y) = 
+ 
; L: x = a cos t, y = a sin t,
0 t 2
3.
.
3.1 P = x , Q = -yz, R = z; L:
(1;2;-1) B(3;3;2)
3.2 P = yz, Q = xz, R = xy; L: x = t,
y = t ; z = t ; 0 t 1
3.3 P = z , Q = x + y + z,R = x +y ; L:
A(2;1;0) B(4;3;1)
3.4 P = x , Q = y, R = z; L: x = t,
y = t , z = t 
, 0 z 1
3.5 P = z , Q = yz, R = x y; L:
A(1;0;2), B(2;-1;0)
3.6 P = x + z, Q = y + z, R = x + y; L: x = t ,
y = t 
, z = t 
, 0 t 1
3.7 P = x, Q = y, R = x + y; L:
A(0;1;1), B(2;4;6)
3.8 P = x + z, Q = x , R = xy; L: x = sin t
y = sin t, z = sin t, 0 z
3.9 P = z, Q = xy, R = x + y ; L:
x = a cos t, y = a sin t, z = bt, 0 t
|
|
|
3.10 P = 2yz, Q = y - z , R = -x ; L: x = t,
y = t , z = t , 0 t 1
4. ,
U(x,y)
U
.
4.1 (10xy +12x + 6)dx + (15x y- 5)ydy
4.2 (cos x cos y + 6x +3)dx + (18y - sin x sin y)dy
1. (2cos 2x cos 3y - 
)dx + (
- 2sin 2x sin 3y)dy
2. (e 
- 
)dx + (sin 3y - 
)dy
3. (xye 
+ cos 2x + x )dx + (
+ y)dy
4. (
- 1)dx + (
- 10)dy
5. (arcsin x x ln y)dx (arcsin y + 
)dy
6. (2x 3xy +2y)dx + (2x 3x y + 2y)dy
7. (x - 2xy + 3)dx + (y - 2x y + 3)dy
4.10 (y + ln(x + 1))dx + (x + 1 - e 
)dy
3.
1.
f(x,y,z)
L:
4.7 f(x,y,z) = x + y + z; L: x = cos t, y = sin t, z = t,
0 t
4.8 f(x,y,z) = x + y; L: x = t, y = t, z = 
,
0 t 
4.9 f(x,y,z) = z; L: x = t cos t, y = t sin t, z = t,
0 t t
4.10 f(x,y,z) = z; L: x = t, y = 
, z = 
,
0 (0,0,0) B(
, , 
)
4.11 f(x,y,z) = x + z; L: x = t, y = 
, z = t ,
0 t 1
4.12 f(x,y,z) = 
; L: x = a cos t, y = a sin t,
z = bt, 0 t 2
4.13 f(x,y,z) = z - ; L: x = a cos t, y = a sin t,
z = bt, 0 t 2
4.14 f(x,y,z) = 
; L: x = t, y = 
, z = ,
0 t 1
4.15 f(x,y,x) = ax; L: x = a cos t, y = a sin t,
z = 
, 0 t 2
4.16 f(x,y,z) = x + y + z ; L: x = cos t, y = sin t, z = t
0 t 2
2.
(L
).
2.1 
L: x + y = ax
1.6 
;
L: x + y = R
2.3 
; L: (x 1) + (y - 1) = 1
2.4 
; L: 
+ 
= 1
2.5 
L:
, y = sin x, y = 0,
0 x
2.6 
;
L: x + y = ax
2.7 
; L:x + y = R
2.8 
; L: x + y = R
y = 0, y 0
a. ;
L: 
+ 
= 1
b. ;
L: x = a cos t, y = b sin t
3. ,
= P(x,y) 
+ Q(x,y) 
L:
3.1 = { x , xy }; L:
A(0;1) B(1;2)
3.2 = { x + y, x + y }; L: ABC,
A(1;1), B(3;1), C(3;5)
3.3 = { x , 
}; L: xy = 1,
A(1;1) B(4; 
)
3.4 = { y, x }; L: x = a cos t,
y = asin t M 
(t )
M 
(t ), t = 0, t = 
3.5 = { x y, 2x + y }; L:
A(1;1), B(3;3), C(3;-1)
3.6 = { x , x }; L: y = x
A (1;1) B(3;9)
3.7 = { cos x, y}; L: y = sin x, 0 x
3.8 = { cos x, 
}; L: y = tg x, x 
3.9 = { x + y, y x }; L: 5x - 6xy + 5y = 6 
3.10 = { - x, -y }; L: x = a cos t, y = b sin t
0 t 2
4.
.
4.1 
4.2 
2.7 
4.4 
4.5 
4.6 
dx + 2x yz dy + 3x y z dz
4.7 
4.8 
4.9 
4.10 
.
3
1.
1.
|
|
|
:
1.1 : x + 2y +3z = 6, 
f(x,y,z) = 6x + 4y + 3z
1.2 : y = 
, x = 0,
x = a; f(x,y, z) = x + 3y + z + 5
1.3 : x + y + z =a, f(x,y,z) = 1
1.4 : z = 
, y = 0, y = 5 f(x,y,z) =
1.5 : 6x + 4y + 3z = 12,
, f(x,y,z) = z + 2x + 
1.6 : z = , z =3;
f(x,y,z) = xyz
1.7 : x + y + z =1,
, f(x,y,z) = 2x + y - 
1.8 : z 1; f(x,y,z) =x + y
1.9 : 
+ 
+ 
= 1, f(x,y,z) = x + y + z
1.10: 6x + 4y + 3z = 12,
f(x,y,z) = z + 2x +
2.
2.1 
2x + 3y + z = 6
2.2 
, x = 0,
y = 0, z = 0, x =1, y =1, z =1
1.7 
, x = 0, y =0
z = 0, x + y + z = 1
1.8 
,
x = 0, y =0, z =0, z = h
x + y =R
1.9 
z = 
,
y = 0, y =2
1.10 
, x = 0,
y = 0, z = 0, x =1, y =1, z =1
1.11 
x = 4y,
y = 4, z = 0, z = 3
1.12 
, x =0,
y = 0, z = 0, x =3, y = 3, z = 3
1.13 
x + y + z = a,
4.7 
,
x + y + z =1 c
3.
3.1 a z = 2xy,
x = 2, y =4
3.2 z = ,
x = 0, y =0, x + y =2
3.3 x + y + z = R ,
x + y = Rx
3.4 x + y = Rx,
x + y + z = R
2.7 2x + 2y + z = 8a,
y + z =0, z = 0
2.8 x + y = R z = 0,
y + z = 0
2.9 z + y = R , x + y = R
2.10 x + y = 6z, x + y = 27
3.9 x + y + z = 3a,
x + y = 2az
3.10 z = 2 
,
XOY
2.
1.
1.1 : z = 
; f(x,y,z) = x
1.2 :
z = 
(x + y ), z =0,
z = 2; f(x,y,z) = x + y
1.3 : z = x + y ,
z = 0, z = 1,
f(x,y,z) = x + y
1.6 :
z = 1- x - y , z =0,
z =1; f(x,y,z) = 
1.7 : x + y = z ,
0 z 1; f(x,y,z)=
1.8 : z = , z = 0, z =1; f(x,y,z) = 3x + 3y + 5z
1.9 : x + y + z = 4,
x + z = 4; f(x,y,z) = x + y + 2x z + z
1.10 : z = 
;
f(x,y,z) = x + y + z
1.11 : y = 
, x = 0, x = a; f(x,y,z) = y(x + z)
1.12 : z = 
; f(x,y,z) = x
2.
.
1.6 
x + y R
1.7 
x + y = z , 0 z 1
1.8 
|
|
|
x + y = R
1.9 z = 1 - x , 0 y 1
1.10 
y = y =0, y =1
1.11 
z = 1 - 
- 
, z = 0
1.12 
x = 
, z = 0, z = 2
1.13 
x = a - y - z , YOZ
2.9 
x = 
,
x = 0, x = a
2.10 
3.
1. z = , z = 0, z =1
2. z = , y = 0, y = 2;
3. 2z = 2 - x - y , XOY;
4. y = 
,
x + y = 2x; 
5. x = 
, x = 0,
x = 2; 
6. x + y + z = a (a > 0),
x + y = R , 
3.7 y = 
, 
3.8 x = 
,
3.9 2az = x - y , x + y = a ;
, k > 0
3.10 0 x 1; 0 y 1; 0 z 1; 
3.
1.
1.1 : 2z = 
,
z = 0; f(x,y,z) = x + y + z 2
1.2 : x + z = 2az,
z = ; f(x,y,z) = z
1.3 : x + y = 9 - z ,
f(x,y,z) = x + y + z
1.4 : z = ,
x + y = 2ax; f(x,y,z) = xy + yz + zx
1.5 : x + y + z = 1; f(x,y,z) = x + y + z
1.6 : z = 
,
x + y = 8x;
f(x,y,z) = xy + yz + zx
1.7 : x + y + z = a,
; f(x,y,z) = x + y + z
1.8 : ,
z = x + y = 4x;
f(x,y,z) = zy + xy + xz
1.9 : z = ;
f(x,y,z) = 
1.10 : ,
z = k x + y - 2ax = 0
f(x,y,z) = y z + z x + x y
2.
:
2.1 
,
x + y + z = R , z = 0
2.2 
,
x + y + z = a
2.3 
,
z = 
, x 0, y 0, z H
2.4 
dydz,
x =
