Αθ.Κεχαγιας - users.auth.gr · Αθ.Κεχαγιας Κεφάλαιο 1 Οριο και...
TRANSCRIPT
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v. 0.95
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2010
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1 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 82.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 213.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 334.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5 Taylor 425.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6 646.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7 787.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
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8 948.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
9 1059.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
10 12410.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12410.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12610.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
11 13611.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13611.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13811.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
12 16912.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16912.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17312.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
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. v.0.9( 2010).
(x, y), (x, y, z), (x1, x2, ..., xN) . . - , .. F (t) = ix (t)+jy (t)+kz (t) (. ), F (x, y, z) = iP (x, y, z) + jQ (x, y, z) + kR (x, y, z) (. ) .
.
1. : u =ai+bj+ck u = (a, b,c).
2. i, j,k x, y, z.
3. , . u = (a, b,c) a, b, c . , ..,
M1M2
M1 (0, 0, 0) M2 (1, 2, 3) N1N2
N1 (1,1, 1) N2 (2, 1, 4) , u = (1, 1, 1).
4. u =ai + bj + ck, v =di + ej + fk
u v = (ai + bj + ck) (di + ej + fk) = ad+ be+ cf.
5. u =ai + bj + ck, v =di + ej + fk
u v= (ai + bj + ck) (di + ej + fk)
=
i j ka b cd e f
= (bf ce) i + (dc af) j + (ae db) k.
iv
-
.
v
6. u =ai + bj + ck
|u| =a2 + b2 + c2.
7. , N- N- : a = (a1, a2, ...,aN) . N- - () ( ).
8. N- a = (a1, a2, ...,aN) ||a|| =a21 + ...+ a
2N
a b a b =N
n=1 anbn.
R2 . ..
D ={
(x, y) : x2 + y2 1}
. (0, 0) 1. R3 . ..
D ={
(x, y, z) : x2 + y2 + z2 1}
. (0, 0, 0) 1.
. , . . !
, 2010
-
. 1
1.1
1.1.1. (x, y). - (x, y), (x, y) (x0, y0), o" ( lim(x,y)(x0,y0) (x, y) = 0" )
> 0 > 0 : 0 0 > 0 .
1.1.3. :
lim(x,y)(x0,y0)
(x, y) 6= limx,x0
(limyy0
(x, y)
)
limx,x0
(limyy0
(x, y)
)6= lim
y,y0
(limx,x0
(x, y)
).
1.1.4. (x, y) (x0, y0)
lim(x,y)(x0,y0)
(x, y) = (x0, y0) .
1.1.5. (x, y) D R2 (x0, y0) D.
1.1.6.
(x, y) = a00 + a10x+ a01y + a11xy + a20x2 + a02y
2 + ...+ amnxmyn
1
-
.
1. 2
R2, . (x0, y0) R2
lim(x,y)(x0,y0)
(x, y) = (x0, y0) .
1.1.7.
(x, y) =f (x, y)
g (x, y)
f (x, y), g (x, y) , R2, . (x0, y0) R2 P
lim(x,y)(x0,y0)
(x, y) = (x0, y0)
P g (x, y) :
P = {(x, y) : g (x, y) = 0} .
1.1.8. f (x, y) , g (x, y) D R2
f (x, y) + g (x, y) , f (x, y) g (x, y) , f (x, y) g (x, y) , f (x, y)g (x, y)
.
D R2 ( f(x,y)g(x,y)
D g (x, y)).
1.1.9. f (x) R g (x, y) R2 f (g (x, y)) R2.
1.1.10. -. .. lim(x,y,z)(x0,y0,z0) (x, y, z) = 0
> 0 > 0 : 0 0
() > 0
0 0 xx (x0, y0) < 0.2.
xx (x0, y0) xy (x0, y0)yx (x0, y0) yy (x0, y0) > 0 xx (x0, y0) > 0.
3. xx (x0, y0) xy (x0, y0)yx (x0, y0) yy (x0, y0)
< 0.4. (
) xx (x0, y0) xy (x0, y0)yx (x0, y0) yy (x0, y0)
= 0.5.1.5. (x, y, z) N (x1, x2, ..., xN). - , ( , )
x (x0, y0, z0) = y (x0, y0, z0) = z (x0, y0, z0) = 0
-
.
5. TAY LOR 44
x1 = x2 = ... = xN = 0
.
5.1.6. (x, y, z). - (x, y, z)
(x, y, z) = 0
(x, y, z, ) = (x, y, z) + (x, y, z)
/ .
5.1.7. , (x1, x2, ..., xN). (x, y, z)
1 (x1, x2, ..., xN) = 0, ..., M (x1, x2, ..., xN) = 0
(x1, x2, ..., xN , 1, ..., M) = (x1, x2, ..., xN) +Mm=1
mm (x1, x2, ..., xN)
/ .
5.2
5.2.1. Taylor F (x, y) = e2x+3y (0, 0). 3 .
.
ez = 1 + z +1
2!z2 +
1
3!z3 + ....
z = 2x+ 3y
e2x+3y = 1 + (2x+ 3y) +1
2!(2x+ 3y)2 +
1
3!(2x+ 3y)3 + ... .
Taylor F (x, y). - xmyn. - (2x+ 3y)k:
e2x+3y = 1 + (2x+ 3y) +1
2!
(4x2 + 12xy + 9y2
)+
1
3!
(8x3 + 36x2y + 54xy2 + 27y3
)+ ... ..
12!
13!
e2x+3y = 1 + 2x+ 3y + 2x2 + 6xy +9
2y2 +
4
3x3 + 6x2y + 9xy2 +
9
2y3 + .....
-
.
5. TAY LOR 45
5.2.2. Taylor F (x, y) = ex+y (2, 1). 3 .
. (x 2)m (y 1)n.
ex+y = e3e(x2)+(y1).
e(x2)+(y1)
e(x2)+(y1) = 1 + ((x 2) + (y 1)) + 12!
((x 2) + (y 1))2 + 13!
((x 2) + (y 1))3
= 1 + (x 2) + (y 1) + 12!
(x 2)2 + (x 2) (y 1) + 12!
(y 1)2 +1
3!(x 2)3 + 1
3!3 (x 2)2 (y 1) + 1
3!3 (x 2) (y 1)2 + 1
3!(y 1)3 + ....
, ex+y e(x2)+(y1)
e3
ex+y = e3 + e3 (x 2) + e3 (y 1) + e3
2(x 2)2 + e3 (x 2) (y 1) + e
3
2(y 1)2 +
e3
6(x 2)3 + e
3
2(x 2)2 (y 1) + e
3
2(x 2) (y 1)2 + e
3
6(y 1)3 + ....
5.2.3. Taylor F (x, y) = cos (x2 + y2) (0, 0). 8 .
.
cos (z) = 1 z2
2!+z4
4!+ ....
z = x2 + y2
cos(x2 + y2
)= 1 1
2!
(x2 + y2
)2+
1
4!
(x2 + y2
)4+ ... .
, (x2 + y2)2 (x2 + y2)4 ( (a+ b)4;)
cos(x2 + y2
)= 1 1
2x4 x2y2 1
2y4+
1
24x8 +
1
6y2x6 +
1
4y4x4 +
1
6y6x2 +
1
24y8 + ......
5.2.4. Taylor F (x, y) = xy cos (x2 + y2) (0, 0). 6 .
. F (x, y) = G (x, y)H (x, y) G (x, y) = cos (x2 + y2) H (x, y) = xy, F (x, y) G (x, y), H (x, y) -. , 4
G (x, y) = cos(x2 + y2
)= 1 1
2x4 x2y2 1
2y4 + ... .
-
.
5. TAY LOR 46
, Taylor H (x, y) = xy
H (x, y) = xy = xy.
F (x, y) = G (x, y)H (x, y) =
(1 1
2x4 x2y2 1
2y4 + ...
) xy
= xy 12x5y x3y3 1
2xy5 + ...
5.2.5. Taylor F (x, y) = ex
1y (0, 0). 4 .
. , F (x, y) = G (x, y)H (x, y) G (x, y) = ex H (x, y) = 1
1y .
G (x, y) = ex = 1 + x+1
2!x2 +
1
3!x3 + ... .
H (x, y) =1
1 y= 1 + y + y2 + y3 + ...
F (x, y) = G (x, y)H (x, y) =
(1 + x+
1
2!x2 +
1
3!x3 + ...
)(1 + y + y2 + y3 + ...
) . , . , 1, 2 ... ( ).
F (x, y) =
(1 + x+
1
2!x2 +
1
3!x3 + ...
)(1 + y + y2 + y3 + ...
)= 1 + x+ y +
1
2x2 + xy + y2 +
1
6x3 +
1
2yx2 + y2x+ y3
+1
24x4 +
1
6x3y +
1
2y2x2 + y3x+ y4 + ...
5.2.6. Taylor F (x, y) = ex+y sin (x+ y) (0, 0). 3 .
.
G (x, y) = ex+y = 1 + x+ y +1
2x2 + xy +
1
2y2 + ...
H (x, y) = sin (x+ y) = x+ y 16x3 1
2x2y 1
2xy2 1
6y3 + ... .
-
.
5. TAY LOR 47
F (x, y) = G (x, y)H (x, y)
=
(1 + x+ y +
1
2x2 + xy +
1
2y2 + ...
)(x+ y 1
6x3 1
2x2y 1
2xy2 1
6y3 + ...
)= x+ y + x2 + 2xy + y2 +
1
3x3 + yx2 + xy2 +
1
3y3 + ...
. .. 6 , , F (x, y) 6 ( 3 G (x, y) H (x, y)).
5.2.7. Taylor F (x, y) = ex2+y2/ (1 x y) (0, 0). 3 .
.
G (x, y) = ex2+y2 = 1 + x2 + y2 +
1
2x4 + x2y2 +
1
2y4 + ...
H (x, y) =1
1 x y= 1 + x+ y + x2 + 2xy + y2 + x3 + 3x2y + 3xy2 + y3 + ... .
F (x, y) = G (x, y)H (x, y)
=
(1 + x2 + y2 +
1
2x4 + x2y2 +
1
2y4 + ...
)(
1 + x+ y + x2 + 2xy + y2 + x3 + 3x2y + 3xy2 + y3 + ...)
= 1 + x+ y + 2x2 + 2xy + 2y2 + 2x3 + 4x2y + 4xy2 + 2y3 + ...
5.2.8. Taylor F (x, y) = 5x+ 4y 6xy y2 + x3 (0, 0). .
. ,
5x+ 4y 6xy y2 + x3
. . Taylor (0, 0) xmyn F (x, y) . , :
F (x, y) = F (0, 0)+Fx (0, 0)
1!x+
Fy (0, 0)
1!y+
Fxx (0, 0)
2!x2 +
2Fxy (0, 0)
2!xy+
Fyy (0, 0)
2!y2 +...
-
.
5. TAY LOR 48
F (x, y) = 5x+ 4y 6xy y2 + x3, F (0, 0) = 0,Fx (x, y) = 5 6y + 3x2, Fx (0, 0) = 5,Fy (x, y) = 4 6x 2y, Fy (0, 0) = 4,Fxx (x, y) = 6x, Fxx (0, 0) = 0,Fxy (x, y) = 6, Fxy (0, 0) = 6,Fyy (x, y) = 2, Fyy (0, 0) = 2
... 0 Fxxx (0, 0) = 6 ( ;).
F (x, y) = 0 + 5x+ 4y +0
2!x2 +
2 (6)2!
xy +22!y2 +
6
3!x3 = 5x+ 4y 6xy y2 + x3.
5.2.9. Taylor F (x, y) = 5x+ 4y 6xy y2 + x3 (1, 2). .
. (x 1)m (y 2)n.
F (x, y) = 5x+ 4y 6xy y2 + x3, F (1, 2) = 2Fx (x, y) = 5 6y + 3x2, Fx (1, 2) = 4Fy (x, y) = 4 6x 2y, Fy (1, 2) = 6Fxx (x, y) = 6x, Fxx (1, 2) = 6Fxy (x, y) = 6, Fxy (1, 2) = 6Fyy (x, y) = 2, Fyy (1, 2) = 2...Fxxx (x, y) = 6, Fxxx (1, 2) = 6.
F (x, y) = 2 4 (x 1) 6 (y 2) + 3 (x 1)2 6 (x 1) (y 2) (y 2)2 + (x 1)3
( )
2 4 (x 1) 6 (y 2) + 3 (x 1)2 6 (x 1) (y 2) (y 2)2 + (x 1)3 =5x+ 4y 6xy y2 + x3 = F (x, y) .
5.2.10. Taylor F (x, y, z) = ex+y+z (0, 0, 0). 2 .
.
eu = 1 + u+u2
2!+ ...
, u = x+ y + z
ex+y+z = 1 + x+ y + z + (x+ y + z)2 + ...
= 1 + x+ y + z + x2 + y2 + z2 + 2xy + 2yz + 2zx+ ..
-
.
5. TAY LOR 49
5.2.11. Taylor F (x, y) = ex+y
1xz (0, 0, 0). 2 .
.
ex+y = 1 + x+ y +1
2!
(x2 + 2xy + y2
)+ ...
1
1 x z= 1 + x+ z + x2 + 2xz + z2 + ...
,
ex+y
1 x z=
(1 + x+ y +
1
2!
(x2 + 2xy + y2
)+ ..
)(1 + x+ z + x2 + 2xz + z2 + ...
)= 1 + 2x+ y + z +
5
2x2 + 2xy + 3xz +
1
2y2 + yz + z2 + ...
5.2.12. F (x, y) = x2 + y2.. .
Fx = 2x = 0
Fy = 2y = 0.
(x1, y1) = (0, 0).
Fxx (x, y) = 1, Fxx (x1, y1) = 1 > 0
D (x, y) =
Fxx FxyFyx Fyy = 1 00 1
= 1 > 0. (x1, y1) . F (x, y) = x2 + y2.
5.2.13. F (x, y) = 4xyx23y2 + 3x+ 4.
. .
Fx = 4y 2x+ 3 = 0Fy = 4x 6y = 0.
, (x1, y1) = (9/2,3).
Fxx (x, y) = 2, Fxx (x1, y1) = 2 < 0
D (x, y) =
Fxx FxyFyx Fyy = 2 44 6
= 4 < 0. (x1, y1) F (x, y).
-
.
5. TAY LOR 50
5.2.14. F (x, y) = x3 + y3 3xy.
. .
Fx = 3x2 3y = 0
Fy = 3y2 3x = 0.
x2 = y x4 = y2 = x x (x3 1
)= 0
{(x1, y1) = (0, 0)(x2, y2) = (1, 1)
.
Fxx (x, y) = 6x, D (x, y) =
Fxx FxyFyx Fyy = 6x 33 6y
= 36xy 9.
Fxx (x1, y1) = 6 0 = 0, D (x1, y1) = 9 < 0.Fxx (x2, y2) = 6 1 > 0, D (x2, y2) = 27 > 0.
(0, 0) (1, 1) F (x, y).
5.2.15. F (x, y) = x3 + y3 3xy.
. .
Fx = 3yx2 + 24x = 0
Fy = x3 8 = 0.
x = 2 12y + 48 = 0 y = 4. . (x1, y1) = (2,4).
Fxx (x, y) = 6x, D (x, y) =
Fxx FxyFyx Fyy = 6xy + 24 3x23x2 0
= 9x4.
Fxx (x1, y1) = 6 2 = 12 > 0, D (x1, y1) = 9 24 < 0.
(2,4) F (x, y).
5.2.16. F (x, y) = x3y +12x2 8y.
. .
Fx = 3yx2 + 24x = 0
Fy = x3 8 = 0.
-
.
5. TAY LOR 51
x = 2 12y + 48 = 0 y = 4. . (x1, y1) = (2,4).
Fxx (x, y) = 6x, D (x, y) =
Fxx FxyFyx Fyy = 6xy + 24 3x23x2 0
= 9x4.
Fxx (x1, y1) = 6 2 = 12 > 0, D (x1, y1) = 9 24 < 0.
(2,4) F (x, y).
5.2.17. F (x, y) = x4 + y4 4xy + 2.
. .
Fx = 4x3 4y = 0
Fy = 4y3 4x = 0.
x3 = y x9 = y3 = x x (x8 1
)= 0 x
(x4 1
)(x4 + 1
)= 0
x (x2 1
)(x2 + 1
)(x4 + 1
)= 0
(x1, y1) = (0, 0)(x2, y2) = (1, 1)
(x3, y3) = (1,1).
Fxx (x, y) = 12x2, D (x, y) =
Fxx FxyFyx Fyy = 12x2 44 12y2
= 144x2y2 16.
Fxx (x1, y1) = 12 0 = 0, D (x1, y1) = 16 < 0.Fxx (x2, y2) = 12 1 > 0, D (x2, y2) = 128 > 0.Fxx (x3, y3) = 12 1 > 0, D (x3, y3) = 128 > 0.
(0, 0) (1, 1) , (1,1) F (x, y).
5.2.18. F (x, y) = (1 + xy) (x+ y).
. .
Fx = y(x+ y) + 1 + xy = 0
Fy = x(x+ y) + 1 + xy = 0.
-
.
5. TAY LOR 52
x = 0 y = 0. y (x+ y) = x (x+ y) , x = y, x = y. x = y 3x2 + 1 = 0. x = 1.
0 + 1 x2 = 0{
(x1, y1) = (1,1)(x2, y2) = (1, 1)
.
Fxx (x, y) = 2y, D (x, y) =
Fxx FxyFyx Fyy = 2y 2x+ 2y2x+ 2y 2x
= 4x2 4xy 4y2.
Fxx (x1, y1) = 2 > 0, D (x1, y1) = 4 < 0.Fxx (x2, y2) = 2 < 0, D (x2, y2) = 4 > 0.
(1,1) (1, 1) F (x, y).5.2.19. F (x, y) = xy (1 x y).
. .
Fx = y(x+ 1 y) xy = 0Fy = x(x+ 1 y) xy = 0.
(x1, y1) = (0, 0). x = 0, y 6= 0
y (1 y) = 0 (x2, y2) = (0, 1) (x3, y3) = (1, 0). , (x, y) 6= (0, 0)
x+ 1 y = xx+ 1 y = y
(x4, y4) = (1/3, 1/3). . .
Fxx (x, y) = 2y,
D (x, y) =
Fxx FxyFyx Fyy = 2y 1 2x 2y1 2x 2y 2x
= 4x2 4xy + 4x 4y2 + 4y 1.
Fxx (x1, y1) = 0, D (x1, y1) = 1 < 0.Fxx (x2, y2) = 2 < 0, D (x2, y2) = 1 < 0.Fxx (x3, y3) = 0, D (x3, y3) = 1 < 0.Fxx (x4, y4) = 2/3 < 0, D (x4, y4) = 13 > 0.
(0, 0) , (0, 1) , (1, 0) F (x, y) (1/3, 1/3) .
-
.
5. TAY LOR 53
5.2.20. F (x, y) = x2+y2z2.. .
Fx = 2x = 0
Fy = 2y = 0.
Fz = 2z = 0
, (x1, y1, z1) = (0, 0, 0). Hesssian
D (x, y, z) =
Fxx Fxy FxzFyx Fyy FyzFzx Fzy Fzz
= 2 0 00 2 0
0 0 2
.
D1 (x, y, z) = Fxx (x, y, z) = 2 > 0
D2 (x, y, z) =
Fxx (x, y, z) Fxy (x, y, z)Fyx (x, y, z) Fyy (x, y, z) = 2 00 2
= 4 > 0D3 (x, y, z) =
Fxx (x, y, z) Fxy (x, y, z) Fxz (x, y, z)Fyx (x, y, z) Fyy (x, y, z) Fyz (x, y, z)Fzx (x, y, z) Fzy (x, y, z) Fzz (x, y, z)
= 8 < 0. (0, 0, 0) F (x, y, z).
5.2.21. F (x, y) = x2y2z2 + 2xy + xz 3y.
. .
Fx = 2x+ 2y + z = 0Fy = 2y + 2x 3 = 0.Fz = 2z + x = 0
Cramer (x1, y1, z1) = (6, 9/2, 3). Hesssian Fxx Fxy FxzFyx Fyy Fyz
Fzx Fzy Fzz
= 2 2 12 2 0
1 0 2
.
D1 (x, y, z) = 2 = 2 < 0
D2 (x, y, z) =
2 22 2 = 0
D3 (x, y, z) =
2 2 12 2 01 0 2
= 2 > 0. (x, y, z) (x1, y1, z1) = (0, 0, 0, ). (x1, y1, z1) F (x, y, z).
-
.
5. TAY LOR 54
5.2.22. F (x, y) = x4 + y4 +z4 4xyz.
. .
Fx = 4x3 4yz = 0
Fy = 4y3 4xz = 0.
Fz = 4z3 4xy = 0
(x1, y1, z1) = (0, 0, 0) (x2, y2, z2) = (1, 1, 1). x 6= 0 yz 6= 0 , ..,
x3
y3=y
x x4 = y4 y = x
z = x.
x3 = yz, y = x, z = x
( )
(x3, y3, z3) = (1,1,1)(x4, y4, z4) = (1, 1,1)(x5, y5, z5) = (1,1, 1) .
Hesssian Fxx Fxy FxzFyx Fyy FyzFzx Fzy Fzz
= 12x2 4z 4y4z 12y2 4x4y 4x 12z2
.
D1 (x, y, z) = 12x2 = 12x2
D2 (x, y, z) =
12x2 4z4z 12y2 = 144x2y2 16z2
D3 (x, y, z) =
12x2 4z 4y4z 12y2 4x4y 4x 12z2
= 192x4 + 1728x2y2z2 128xyz 192y4 192z4.1. (x1, y1, z1) = (0, 0, 0, )
D1 (0, 0, 0) = D2 (0, 0, 0) = D3 (0, 0, 0) = 0
(0, 0, 0).
2. (x2, y2, z2) = (1, 1, 1)
D1 (x2, y2, z2) = 12 > 0
D2 (x2, y2, z2) = 128 > 0
D3 (x2, y2, z2) = 1024 > 0.
(x2, y2, z2) F (x, y, z).
-
.
5. TAY LOR 55
3. (x3, y3, z3) = (1,1,1)
D1 (x2, y2, z2) = 12 > 0
D2 (x2, y2, z2) = 128 > 0
D3 (x2, y2, z2) = 1280 > 0.
(x3, y3, z3) F (x, y, z).
4. (x4, y4, z4) = (1, 1,1) (x5, y5, z5) = (1,1, 1) -, .
5.2.23. F (x, y) = x3 + y3 z2 xyz.
. .
Fx = 3x2 yz = 0
Fy = 3y2 xz = 0.
Fz = 2z xy = 0
(x1, y1, z1) = (0, 0, 0) (x2, y2, z2) = (6,6,18). Hesssian Fxx Fxy FxzFyx Fyy Fyz
Fzx Fzy Fzz
= 6x z yz 6y xy x 2
.
D1 (x, y, z) = 6x = 6x
D2 (x, y, z) =
6x zz 6y = 36xy z2
D3 (x, y, z) =
6x z yz 6y xy x 2
= 6x3 2xyz 72xy 6y3 + 2z2. (x1, y1, z1) = (0, 0, 0, )
D1 (0, 0, 0) = D2 (0, 0, 0) = D3 (0, 0, 0) = 0
(0, 0, 0). (x2, y2, z2) = (6,6,18)
D1 (x2, y2, z2) = 6 (6) = 36 < 0
D2 (x2, y2, z2) =
6 (6) 1818 6 (6) = 972 > 0
D3 (x2, y2, z2) =
6 (6) 18 6
18 6 (6) 66 6 2
= 1944 > 0. (x2, y2, z2) F (x, y, z).
-
.
5. TAY LOR 56
5.2.24. F (x, y) = x2 +2y2 x2 + y2 = 1.
. , /Lagrange. F (x, y) = x2+2y2
L (x, y) = F (x, y) + G (x, y) = x2 + 2y2 + (x2 + y2 1
).
:
Lx = 2x+ 2x = 0
Ly = 4y + 2y = 0
L = x2 + y2 1 = 0.
2x = 2x4y = 2y
x = 0 = 1. x = 0, y = 1. = 1, y = 0 x = 1. : (0, 1), (0,1), (1, 0), (1, 0).
F (0, 1) = F (0,1) = 2 F (1, 0) = F (1, 0) = 1.
(0, 1) (0,1) (1, 0) (1, 0).
5.2.25. F (x, y) = 3x+ 4y G (x, y) = x2 + y2 1 = 0.
. , /Lagrange. F (x, y) = 3x+4y
L (x, y) = F (x, y) + G (x, y) = 3x+ 4y + (x2 + y2 1
).
:
Lx = 3 + 2x = 0
Ly = 4 + 2y = 0
L = x2 + y2 1 = 0.
x, y, 6= 0.
x = 32, y = 2
9
42+
4
2= 1 = 5
2.
-
.
5. TAY LOR 57
:
(x1, y1) =
( 3
21, 2
1
)=
(3
5,4
5
)(x2, y2) =
( 3
22, 2
2
)=
(3
5,4
5
).
F (x1, y1) = 3x1 + 4y1 = 33
5+ 4
4
5= 5
F (x2, y2) = 3x2 + 4y2 = 335
+ 445
= 5.
. F (x, y) (x1, y1) =(
35, 4
5
) (x2, y2) =
(3
5,4
5
).
5.2.26. F (x, y, z) = x+ y+ z - xyz = 1.
. , /Lagrange. F (x, y, z) = x + y + z
L (x, y, z) = F (x, y, z) + G (x, y, z) = x+ y + z + (xyz 1) .
:
Lx = 1 + yz = 0
Ly = 1 + xz = 0
Lz = 1 + xy = 0
L = xyz 1 = 0.
xyz = 0 ( -) x 6= 0, y 6= 0, z 6= 0.
= 1yz
= 1xz
= 1xy x = y = z.
xyz = 1
x = y = z = 1. F (x, y, z) =x + y + z ( G (x, y, z) = xyz 1 = 0). , (x, y, z) =
(x, y, 1
xy
).
5.2.27. (3, 1,1) x2 + y2 + z2 = 4.
. (x, y, z) . G (x, y, z) = x2 + y2 + z2 4 = 0.
-
.
5. TAY LOR 58
() d (x, y, z) =
(x 3)2 + (y 1)2 + (z + 1)2 () F (x, y, z) = (x 3)2 + (y 1)2 +(z + 1)2.
L (x, y, z, ) = (x 3)2 + (y 1)2 + (z + 1)2 + (x2 + y2 + z2 4
).
Lx = 2 (x 3) + 2x = 0Ly = 2 (y 1) + 2y = 0Lz = 2 (z + 1) + 2z = 0
L = x2 + y2 + z2 4 = 0.
x, y, z
x =3
1 , y =
1
1 , z = 1
1
( 6= 1 ... ;). (3
1
)2+
(1
1
)2+
(1
1
)2= 4
(1 )2 = 114 = 1
11
2.
1 = 1 +
11
2 (x1, y1, z1) =
(611,
211, 2
11
)2 = 1
11
2 (x2, y2, z2) =
( 6
11, 2
11,
211
)
d (x1, y1, z1) =
(611 3)2
+
(211 1)2
+
( 2
11+ 1
)2' 1. 316 6
d (x2, y2, z2) =
( 6
11 3)2
+
( 2
11 1)2
+
(+
211
+ 1
)2' 5. 316 6
(x1, y1, z1) (x2, y2, z2).
-
.
5. TAY LOR 59
5.2.28. F (x, y, z) = x + 2y + 3z G1 (x, y, z) = x y + z 1 = 0 G2 (x, y, z) = x2 + y2 1 = 0.
.
L (x, y, z, 1, 2) = x+ 2y + 3z + 1 (x y + z 1) + 2 (x2 + y2 1
).
1 + 1 + 22x = 0
2 1 + 22y = 03 + 1 = 0
x y + z 1 = 0x2 + y2 1 = 0
1 = 3 x = 12 y = 5
22.
1
22+
25
422= 1
22 = 29/4 2 =
29/2. 1 + 0.371 39 + 0.928 48 = 2. 299 9
x1 = 2/
29 ' 0.371 39y1 = 5/
29 ' 0.928 48
} z1 = 1 x1 + y1 ' 2. 3
x2 = 2/
29 ' 0.371 39
y2 = 5/
29 ' 0.928 48
} z1 = 1 x1 + y1 ' 0. 3.
(x1, y1, z1)
F (x1, y1, z1) ' F (0.37139, 0.92848, 2.3) = 8. 385 6
(x2, y2, z2)
F (x2, y2, z2) ' F (0.37139, 0.92848, 2.3) = 2. 385 6.
F (x, y, z) (x1, y1, z1).
5.3
5.3.1. Taylor F (x, y) = ex+y (0, 0). 3 .
. 1 + x+ y + 12x2 + xy + 1
2y2 + 1
6x3 + 1
2x2y + 1
2xy2 + 1
6y3 + ...
5.3.2. Taylor F (x, y) = ex+y (1, 2). 3 .
. e3 +e3 x+e3 y+ e32x2 +e3 xy+ e3
2y2 + e
3
6x3 + e
3
2x2y+ e
3
2xy2 + e
3
6y3 + ...
-
.
5. TAY LOR 60
5.3.3. Taylor F (x, y) = sin (x2 + y2) (0, 0). 6 .
. 1 16x6 1
2x4y2 1
2x2y4 1
6y6 + ....
5.3.4. Taylor F (x, y) = x2y sin (x2 + y2) (0, 0). 9 .
. x2y 16x8y 1
2x6y3 1
2x4y5 1
6x2y7 + ...
5.3.5. Taylor F (x, y) = 21x+ 42y 6xy 12y2 + 4y3 109 (5, 1). .
. 15 (x 5) 6 (x 5) (y 1) + 4 (y 1)3 .
5.3.6. Taylor F (x, y) = 11+x2+y2
(0, 0). 8 .
. 1 x6 3x4y2 + x4 3x2y4 + 2x2y2 x2 y6 + y4 y2 + ... .
5.3.7. Taylor F (x, y) = sinx sin y (/4, /4). 2 .
. 12+1
2(x /4)+1
2(y /4)1
4(x /4)21
4(y /4)2+1
2(x /4) (y /4)+
...
5.3.8. Taylor F (x, y) = ex sin y (0, 0). 3 .
. y + xy + 12x2y 1
6y3 + ...
5.3.9. Taylor F (x, y) = ex ln (1 + y) (0, 0). 3 .
. y 12y2 + xy + 1
2x2y 1
2xy2 + 1
3y3 + ...
5.3.10. Taylor F (x, y) = cosx1+x2+y2
(1, 2). 8 .
. 1 124x10 1
8x8y2 + 13
24x8 1
8x6y4+ 19
12x6y2 37
24x6 1
24x4y6 + 37
24x4y4 97
24x4y2+
3724x4 + 1
2x2y6 7
2x2y4+ 5
2x2y2 3
2x2 y6 + y4 y2 + ... .
5.3.11. Taylor F (x, y) = 11xy+xy
(0, 0). 3 .. 1 + x+ y + x2 + xy + y2 + ... .
5.3.12. Taylor F (x, y) = ln (1 x) ln (1 y) (0, 0). 5 .
. xy + x2y2
+ xy2
2+ x
2y2
4+ x
2y3
6+ x
3y2
6+ ... .
5.3.13. Taylor ( (1, 1) ) z (x, y) z3 + yz xy2 x3 = 0. 2 .
. 1 + (x 1) + 14
(y 1) 18
(x 1) (y 1) + 964
(y 1)2 + ... .
5.3.14. Taylor F (x, y, z) = sin (x+ y + z) (0, 0, 0). 3 .
. x+ y+ z 16x3 1
2x2y 1
2x2z 1
2xy2xyz 1
2xz2 1
6y3 1
2y2z 1
2yz2 1
6z3 + ...
.
-
.
5. TAY LOR 61
5.3.15. Taylor F (x, y, z) = 11+x+y+z
(0, 0, 0). 2 .
. 1 x y z + x2 + 2xy + 2xz + y2 + 2yz + z2 + ... .
5.3.16. F (x, y) = 2xyx22y2 + 3x+ 4
. (3, 3/2).
5.3.17. F (x, y) = 2x3 + xy2 + 5x2 + y2
. (0, 0), (5/3, 0), (1, 2), (1,2).
5.3.18. F (x, y) = 8x3 + y312xy + 8
. (8) (1, 2) (0) (0, 0) .
5.3.19. F (x, y) = xy (a x y). (0, 0) , (0, a), (a, 0),
(a3, a
3
).
5.3.20. F (x, y) = 2x2 + y22xy 4x+ 3
. (1) (2, 2).
5.3.21. F (x, y) = (2ax x2) (2by y2). (0, 0), (0, 2b), (2a, 0), (a, b), (2a, 2b).
5.3.22. F (x, y) = x44xy+2y2 5.
. (0, 0), (1,1), (1, 1) .
5.3.23. F (x, y) = x2 + xy + y2 + a3
x+ a
2
y
. (a/ 3
3, a/ 3
3).
5.3.24. F (x, y) = 13x2 +16xy + 7y2 + 10x+ 2y 5
. (1, 1).
5.3.25. F (x, y) = cos (x+ y)2x2 2y2 + 8x 8y + 4xy
. .
5.3.26. F (x, y) = x4 + y4 2x2 4xy 2y2..
(2,
2)
(
2,
2).
5.3.27. F (x, y) = x3 + y2 6xy 39x+ 18y + 20.. (5, 6) .
5.3.28. F (x, y) = x3y2 (12 x y).. (6, 4).
-
.
5. TAY LOR 62
5.3.29. F (x, y) = x3 + y3 3xy.. (0, 0), (1, 1).
5.3.30. F (x, y) = 2xy x2 2y2 + 3x+ 4.
. (8, 5),(3, 3
2
).
5.3.31. F (x, y, z) = 2x2 + y2 + 2z xy xz.. (2, 1, 7).
5.3.32. F (x, y, z) = 3 lnx+2 ln y+5 ln z+ln (22 x y z) .. (6, 4, 10).
5.3.33. F (x, y) = xm + ym ( m > 1) x+ y = 2.
. (1, 1).
5.3.34. F (x, y) = xy x2 +y2 = 2a2.
. (a, a), (a,a) (a,a), (a,a) .
5.3.35. F (x, y) = 1x
+ 1y
1x2
+ 1y2
= 1a2
..
(a
2, a
2)
(a
2,a
2).
5.3.36. F (x, y, z) = x+y+z 1x
+ 1y
+ 1z
= 1.. (3, 3, 3).
5.3.37. F (x, y, z) = xyz x+ y + z = 5.
. (
53, 5
3, 5
3
).
5.3.38. F (x, y, z) = xyz xy + yz + zx = 8.
. (
53, 5
3, 5
3
).
5.3.39. F (x, y) = x3 3xy2 + 18y 3x2 y3 6x = 0.
. (
3,
3)
(
3,
3).
5.3.40. (x, y, z) 2x6y+3z = 22 (3,3, 1)
. (1, 3,2).
5.3.41. (x, y, z) 3x + 2y + 3z = 5 (2, 3, 5)
. (1, 1, 2).
-
.
5. TAY LOR 63
5.3.42.
l1 :x 1
2=y 3
1=z + 7
2 l2 :
x+ 3
3=y + 4
1=z 5
1.
. 10
2.
5.3.43. z 2x2 + 2y2 + z2 + 8xz z + 8 = 0.
. (2, 0), (16/7, 0).
5.3.44. z 5x2 + 5y2 + 5z2 2xy 2xz 2yz 72 = 0.
. (1, 1) (1,1).
-
. 6
6.1
6.1.1. C D ( ). S D. ( .7.1).
7.1
6.1.2. S - : D S
limx,y0
S = lim
x,y0
xy = lim
x,y0
yx.
, S
S =
D
dS. (6.1)
64
-
.
6. 65
dS dS = dxdy.
6.1.3. -
S =
D
dS =
y2y1
x(y2)x(y1)
dxdy =
x2x1
y(x2)y(x1)
dydx. (6.2)
6.1.4. , f (x, y) limx,y0
f (x, y) S)
D
f (x, y) dS =
y2y1
x(y2)x(y1)
f (x, y) dxdy =
x2x1
y(x2)y(x1)
f (x, y) dydx. (6.3)
6.1.5. , (6.3) D {(x, y, f (x, y)) : (x, y) D}.
6.1.6. ( ):
a
D
f (x, y) dxdy =
D
af (x, y) dxdy D
[f (x, y) + g (x, y)] dxdy =
D
f (x, y) dxdy +
D
g (x, y) dxdy
, (x, y) D f (x, y) g (x, y), D
f (x, y) dxdy
D
g (x, y) dxdy.
6.1.7. , : f (x, y) D S, (x0, y0) D
D
f (x, y) dxdy = f (x0, y0) S.
6.1.8. (6.1)(6.3) - " , y (x1), y (x2) . C. .
6.1.9. . (x, y) - (u, v)
x = x (u, v) , y = y (u, v) .
f (x, y) = f (x (u, v) , y (u, v))
-
.
6. 66
dS
dS = dxdy =
drdu drdv = D (x, y)D (u, v)dudv.
D
f (x, y) dxdy =
D
f (x (u, v) , y (u, v))D (x, y)
D (u, v)dudv.
6.1.10. .., , u = , v = D(x,y)D(u,v)
= , dxdy = dd
D
f (x, y) dxdy =
D
f (x (, ) , y (, )) dd.
6.2
6.2.1. 1
0
20y2dydx.
. 10
20
y2dydx =
10
(y3
3
)y=2y=0
dx =
10
8
3dx =
(8x
3
)x=1x=0
=8
3.
6.2.2. 1
0
x0y2dydx.
. 10
x0
y2dydx =
10
(y3
3
)y=xy=0
dx =
10
x3
3dx =
(x4
12
)x=1x=0
=1
12.
6.2.3. 1
0
x2xy2dydx.
. 10
x2x
y2dydx =
10
(y3
3
)y=x2y=x
dx =
10
(x6
3x3
3
)dx
=
(x7
21 2
15xx3)x=1x=0
=1
21 2
15= 3
35.
6.2.4. y1 (x) =x,
y2 (x) = x2.
. x = x2, . x = 0 x = 1.
[0, 1] x2 x.
10
xx2
dydx =
10
(y)y=x
y=x2 dx =
10
(x x2
)dx =
(2x3/2
3 x
3
3
)x=1x=0
=2
3 1
3=
1
3.
-
.
6. 67
6.2.5. : x = 0, y = 0, 12x+ y = 1.
. (0, 0), (0, 1), (2, 1) ( x = 0, y = 0, 1
2x+ y = 1).
E =
20
1x/20
dydx =
20
(y)y=1x/2y=0 dx =
20
(1 x
2
)dx
=
(x x
2
4
)x=2x=0
= 2 1 = 1.
6.2.6. : xy = 1, y = x2, x = 1, x =2.
. 21
1/xx2
dydx =
21
(y)y=1/x
y=x2 dx =
21
(1
x x2
)dx =
(lnx x
3
3
)x=2x=1
= ln 2 83(
0 13
)= ln 2 +
7
3.
6 4 ln 2.
6.2.7. D y = 4xx2 y = 0, y = 3x+ 6.
. D , D1, y = 4x x2 x = 2, y = 3x + 6 D2, .y = 4x x2, x = 2, y = 0 D2.
D
dydx =
D1
dydx+
D2
dydx
(
2 42 433
)(
2 22 233
)= 16
3 D1
dydx =
21
4xx23x+6
dydx =
21
(4x x2 (3x+ 6)
)dx
=
(2x2 x
3
3+
3
2x2 6x
)x=2x=1
=13
6 D2
dydx =
42
4xx20
dydx =
42
(4x x2
)dx =
(2x2 x
3
3
)x=4x=2
=16
3
D
dydx =
D1
dydx+
D2
dydx =13
6+
16
3=
45
6=
15
2.
-
.
6. 68
6.2.8. (0, 0) R.. x2 + y2 = R2, x y [
R2 x2,
R2 x2
]. R2x2
R2x2
dydx.
-. () = R 2
0
R0
dd =
20
(2
2
)=R=0
d =
20
R2
2d =
(R2
2
)2=0
= R2
.
6.2.9. = cos 2, [/2, /2]. [/4, /4].
4
/4/4
cos 20
dd = 4
/4/4
(2
2
)=cos 2=0
d = 4
/4/4
cos2 2
2d
= 4
/4/4
1 + cos 4
4d = ( + cos 4)
=/4=/4 =
2.
6.2.10. 2 = cos 2, [/4, /4]().
. [0, /4].
4
/40
cos 20
dd = 4
/40
(2
2
)=cos 2=0
d = 4
/40
cos 2
2d = (sin 2)
=/4=0 = 1.
6.2.11. = cos 3, [0, ].. [/6, 6] ( ;).
3
/6/6
cos 30
dd = 3
/6/6
(2
2
)=cos 3=0
d =3
2
/6/6
cos2 3d
=3
2
/6/6
1 + cos 6
2d =
3
4()
=/6=/6 =
4.
6.2.12.
Dxydxdy D : x2 + y2 R2.
-
.
6. 69
. 20
R0
xydd =
20
R0
cos sin dd
=1
2
20
R0
3 sin 2dd
=1
2
20
(4
4
)=R=0
sin 2d
=1
2
20
R4
4sin 2d
=R4
8
20
sin 2d = 0
( [0, 2] ).
6.2.13.
D(x2 + y2) dydx, D x2 +y2
2Rx.. , D
x2 + y2 = 2Rx x2 2Rx+R2 + y2 = R2 (xR)2 + y2 = R2
. (0, R) R. ... (0, R), .
x = R + cos , y = sin .
-
dxdy =D (x, y)
D ()dd =
cos sin sin cos dd = dd
(. ). 2
0
R0
((R + cos )2 + sin2
)dd =
20
R0
(R2 + 2 cos + 2
)dd
=
20
(R22
2+ 2
3
3cos +
R4
4
)=R=0
d
= 2 R4
2+ 2 0 + 2 R
4
4=
3R4
2.
-
.
6. 70
6.2.14. D
sinxxdxdy
D = {(x, y) : 0 x 1, 0 y x}
. D
sinxxdxdy -
sinxxdx, . -
, . D
sinx
xdxdy =
D
sinx
xdydx.
D
sinx
xdydx =
10
( x0
sinx
xdy
)dx =
10
(sinx
xy
)y=xy=0
dx
=
10
(sinx
xx sinx
x0
)dx =
10
sinxdx = 1 cos 1..
6.2.15. D
x cos (xy) dydx
D = {(x, y) : 0 x , 0 y 1} .
. D
x cos (xy) dydx =
0
( 10
x cos (xy) dy
)dx =
0
( 10
cos (xy) d (xy)
)dx
=
0
(sin (xy))y=1y=0 dx =
0
sinxdx = ( cosx)x=x=0 = 2.
6.2.16. D
xydxdy D
y = x 1 y2 = 2x+ 6,.
. y = x 1 y2 = 2x+ 6
y2 = 2 (1 + y) + 6 y1 = 2 y2 = 4.
D =
{(x, y) : 2 y 4, y
2
2 3 x y + 1
}.
42
( y+1y2
23xydx
)dy =
42
(x2y
2
)x=y+1x= y
2
23dy =
42
(y
5
4+ 4y3 + 2y2 8y
)dy
=1
2
(y
6
24+ y4 +
2
3y3 4y2
)y=4y=2
= 36.
-
.
6. 71
6.2.17. D
(3x+ 4y2) dxdy D
D ={
(x, y) : x2 + y2 1 x2 + y2 4 y 0}.
. ( ) D
(3x+ 4y2
)dxdy =
0
21
(3x+ 4y2
)dd
=
0
21
(3 cos + 42 sin2
)dd
=
0
(3 cos + 4 sin2
)=2=1
d
=
0
(3 cos + 4
1 cos 22
)=2=1
d
=
0
(7 cos +
15
2(1 cos 2)
)d
=
(7 sin +
15
2
)==0
=15
2.
6.2.18. D x2 +y2 = 1 x2 +y2 =5.
D
(x2 + y
)dxdy
. D
(x2 + y
)dxdy =
51
20
(2 cos2 + sin
)dd
=
51
2( 2
0
( cos2 + sin
)d
)d
=
51
2( 2
0
(
1 + cos 2
2+ sin
)d
)d
=
51
2(
2+ sin 2
4 cos
)=2=0
d
=
51
2 () d =
(4
4
)=5=1
= 6.
6.2.19. : x = 0, y = 0, z = 0, x = 2,y = 1 z = x2 + y2 + 1
-
.
6. 72
. 20
10
(1 + x2 + y2
)dydx =
20
(y + x2y +
y3
3
)y=1y=0
dx
=
20
(4
3+ x2
)dx =
(4x
3+x3
3
)x=2x=0
=16
3.
6.2.20. x = 0, y = 0,z = 0, x+ y + z = 1.
. ( !). (x, y, z) = (0, 0, 0), (x, y, z) = (1, 0, 0), (x, y, z) = (0, 1, 0) ( x = 0, y = 0, x+y = 1). z z = 1 x y. 1
0
1x0
(1 x y) dydx = 1
0
(y xy y
2
2
)y=1xy=0
dx
=
10
(1 x x (y) 1 x (1 x)
2
2
)dx =
1
6.
6.2.21. x2 + y2 = R2. z = 2, z = 5.
. ( D x2 + y2 = R2)
D
(z2 (x, y) z1 (x, y)) dydx = D
(5 2) dydx = 3R2
( D
dydx = R2, ).
6.2.22. x2 +2y2 + z2 = 16 x = 0, x = 2, y = 0, y = 2, z = 0.
. . 0 x 2, 0 y 2. 2
0
20
(16 x2 2y2
)dxdy =
20
(16x x
3
3 2y2x
)x=2x=0
dy =
20
(88
3 4y2
)dy = 48.
6.2.23. z = sinx cos y z = 0.
. . 0 x ,
2 y
2.
0
/2/2
sinx cos ydxdy =
( 0
sinxdx
)
( /2/2
cos ydy
)= (cosx)x=x=0 (sin y)
x=/2y=/2 = 4.
-
.
6. 73
6.2.24. z = x2 + y2
D y = 2x y = x2.. . D ( xy)
D ={
(x, y) : 0 x 2, x2 y 2x}.
20
2xx2
(x2 + y2
)dydx =
20
(x2y +
y3
3
)y=2xy=x2
dx
=
20
(2x3 +
8x3
3 x4 x
6
3
)y=2xy=x2
dx
=
(2x4
4+
8x4
12 x
5
5 x
7
21
)x=2x=0
dx =216
35.
6.2.25. z = 0 z = 1 x2 y2.
. z = 0 ( z = 0) x2 +y2 =1. 2
0
10
(1 2
)dd =
20
(2
2
4
4
)=1=0
d =
20
1
4d =
2
6.2.26. z = 0, z = x2 + y2 x2 + y2 = 1.
. ( ) D
(x2 + y2
)dxdy
D = {(x, y) : x2 + y2 = 1} ( D , - S = {(x, y, z) : x2 + y2 = 1} !!!). - 2
0
10
2dd =
20
(4
4
)=1=0
d =
2.
6.2.27. z = 0, z = x2 + y2 x2 + y2 = 2x.
. ( !)
C : x2 + y2 = 2x C : (x 1)2 + y2 = 1.
x = 1 + cos , y = sin
-
.
6. 74
dxdy = dd.
20
10
(x2 + y2
)dd =
20
10
((1 + cos )2 + 2 sin2
)dd
=
20
10
(2 + 2 cos + 1
)dd
=
20
10
(3 + 22 cos +
)dd
=
20
(4
4+
2
33 cos +
2
2
)=1=0
d
=
20
(3
4+
2
3cos
)d
=
(3
4+
2
3sin
)=2=0
=3
2.
6.2.28. . y = 0, y = 2x x = 1 d (x, y) = 6x+ 6y + 6.
. , (x1, y1) = (0, 0), (x2, y2) = (1, 0), (x3, y3) = (1, 2) ( ; !). 1
0
2x0
(6x+ 6y + 6) dydx =
10
((6xy + 3y2 + 6y
)y=2xy=0
dy)dx
=
10
(24x2 + 12x
)dx =
(8x3 + 6x2
)x=1x=0
= 14.
6.2.29. (0, 0), R .
. x2 + y2 R2. d (x, y) =d0x2+y2
.
R2x2R2x2
d0x2 + y2
dydx.
20
R0
d0dd =
20
(d0)=R=0 d =
20
d0Rd = (d0R)2=0 = 2Rd0.
-
.
6. 75
6.2.30. (0, 0), R (x, y) d0y.
. [0, R], [0, ].
0
R0
d0ydd =
0
R0
d0 sin dd =
0
(d03
3
)=R=0
sin d
=
0
d0R3
3sin d =
(d0R3
3cos
)2=0
=2
3d0R
3.
6.3
6.3.1. a
0
x0
dydx.. 2
3a3/2.
6.3.2. 4
2
2xx
yxdydx.
. 9.
6.3.3. 2
1
ln y0
exdxdy.. 1/2.
6.3.4. : x = 0, y = 0, x+ y = 1.. 1/2.
6.3.5. : y2 = b2
ax, y = b
ax.
. ab6.
6.3.6. : y =x, y = 2
x, x = 4.
. 16/3.
6.3.7. : x = y, y = 5x, x = 1.. 2.
6.3.8. : xy = 4, y = x, x = 4.. 6 4 ln 2.
6.3.9. : y2 = 4 + x, x+ 3y = 0.. 100/6.
6.3.10. : ay = x2 2ax, y = x.. 9a2/2.
6.3.11. : y = ln x, x y = 1, y =1.
. e22e
.
6.3.12. : xy = a2/2, xy = 2a2, y =x/2, y = 2x.
. 32a2 ln 2.
-
.
6. 76
6.3.13. : (x2 + y2)2 = 2ax3.. 5
8a2.
6.3.14.
Dx3y2dxdy D : x2 + y2 R2.
. 0.
6.3.15.
Dxydxdy, D = {(x, y) : 0 x 1, 0 y 2}
. 1.
6.3.16.
Dex+ydxdy, D = {(x, y) : 0 x 1, 0 y 1}
. (e 1)2.
6.3.17.
Dx2
1+y2dxdy, D = {(x, y) : 0 x 1, 0 y 1}
. /12.
6.3.18.
D(x2 + y2) dxdy D : y = x2, x = y2.
. 33/140.
6.3.19.
Dx2
y2dxdy D : x = 2, y = x, xy = 1.
. 9/4.
6.3.20.
Dx sin (x+ y) dxdy, D = {(x, y) : 0 x , 0 y /2}
. 2.
6.3.21.
Dx2exydxdy, D = {(x, y) : 0 x 1, 0 y 2}
. 2.
6.3.22.
Dx3y2dxdy, D = {(x, y) : x2 + y2 R2}.
. 0.
6.3.23.
Dx2
y2dxdy, D x = 2, y = x
xy = 1.. 9/4.
6.3.24.
Dcos (x+ y) dxdy, D x = 0,
y = x = y.. 2.
6.3.25.
D
1 x2 y2dxdy, x 0,
y 0 x2 + y2 1.. /6.
6.3.26. R
0
R2x20
ln (1 + x2 + y2) dydx.. /4 . ..
6.3.27.
D
1x2y21+x2+y2
dydx, D x 0,y 0 x2 + y2 1.
. 22
8 . .
-
.
6. 77
6.3.28.
D(h 2x 3y) dydx, D x2 + y2 R2.
. R2h . .
6.3.29.
D
R2 x2 y2dydx, D x2 + y2 Rx.
. R3
3 . .
6.3.30.
Dxydydx, D x
2
a2+ y
2
b2 1, x 0, y 0.
. a2b2
8 . .
6.3.31. R
0
R2x20
ln (1 + x2 + y2) dydx.. /4 . .
6.3.32. R
0
R2x20
ln (1 + x2 + y2) dydx.. /4 . .
6.3.33. R
0
R2x20
ln (1 + x2 + y2) dydx.. /4 . .
6.3.34. R
0
R2x20
ln (1 + x2 + y2) dydx.. /4 . .
6.3.35. R
0
R2x20
ln (1 + x2 + y2) dydx.
-
. 7
7.1
7.1.1. A D ( ). V D. ( .8.1).
7.1
7.1.2. V - : V V
limx,y,z0
V = lim
x,y,z0
xyz.
, V
V =
D
dV. (7.1)
dV dV = dxdydz.
7.1.3. -
V =
D
dV =
z2z1
y2(z2)y1(z1)
x(y2,z2)x(y1,z1)
dxdydz. (7.2)
78
-
.
7. 79
7.1.4. , f (x, y, z) (. f (x, y, z) V )
V =
D
dV =
z2z1
y2(z2)y1(z1)
x(y2,z2)x(y1,z1)
f (x, y, z) dxdydz. (7.3)
7.1.5. :
a
D
f (x, y, z) dxdydz =
D
af (x, y) dxdydz D
[f (x, y, z) + g (x, y, z)] dxdydz =
D
f (x, y, z) dxdydz +
D
g (x, y, z) dxdydz
, (x, y, z) D f (x, y, z) g (x, y, z), D
f (x, y, z) dxdydz
D
g (x, y) dxdydz.
7.1.6. , : f (x, y, z) D V , (x0, y0, z0) D
D
f (x, y, z) dxdy = f (x0, y0, z0) V.
7.1.7. (7.1)(7.3) - " , - A. .
7.1.8. . (x, y, z) - (u, v, w)
x = x (u, v, w) , y = y (u, v, w) , z = z (u, v, w)
D D.
f (x, y, z) = f (x (u, v, w) , y (u, v, w) , z (u, v, w))
dV
dS = dxdydz =D (x, y, z)
D (u, v, w)dudvdw.
D
f (x, y, z) dxdydz =
Df (x (u, v, w) , y (u, v, we))
D (x, y, z)
D (u, v, w)dudvdw.
-
.
7. 80
7.1.9. .., , u = , v = , w = z
x = cos , y = sin , z = z
D (x, y, z)
D (, , z)= .
dxdydz = dddz D
f (x, y, z) dxdydz =
Df (x (, , z) , y (, , z) , z (, , z)) dddz.
7.1.10. , , u = r, v = , w = z
x = r cos sin, y = r sin sin, z = r cos
D (x, y, z)
D (, , )= r2 sin.
dxdydz = r2 sin drdd D
f (x, y, z) dxdydz =
Df (x (r, , ) , y (r, , ) , z (r, , )) r2 sindrdd.
7.2
7.2.1. 3
0
20
50dzdydx.
. 30
( 20
( 50
dz
)dy
)dx =
30
( 20
(z)z=5z=0 dy
)dx =
30
( 20
5dy
)dx
=
30
(5y)y=2y=0 dx =
30
10dx = (10x)x=3x=0 = 30.
7.2.2. 3
0
20
50xyzdzdydx.
. 30
( 20
( 50
xyzdz
)dy
)dx =
30
( 20
xy
(z2
2
)z=5z=0
dy
)dx =
30
( 20
xy25
2dy
)dx
=
30
x
(25
2
y2
2
)y=2y=0
dx =
30
100
4xdx =
(25x2
2
)x=3x=0
=225
2.
7.2.3. P
xyzdzdydx P
(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1).
-
.
7. 81
. ( ). .. Ax+ By + Cz +D = 0 (0, 0, 0),(1, 0, 0), (0, 1, 0),
A 0 +B 0 + C 0 +D = 0A 1 +B 0 + C 0 +D = 0A 0 +B 1 + C 0 +D = 0
. A = D,B = D, D = 0, Cz = 0 z = 0. x = 0, y = 0 x+ y+ z = 1. Q ( z = 0) z = 0 ( z = 0)
Q = {(x, y) : 0 x 1, 0 y 1 x} .
P = {(x, y, z) : 0 x 1, 0 y 1 x, 0 z 1 x y} .
10
( 1x0
( 1xy0
xyzdz
)dy
)dx =
10
( 1x0
xy
(z2
2
)z=1xyz=0
dy
)dx
=
10
( 1x0
xy(1 x y)2
2dy
)dx
=
10
(1
4xy2 1
2x2y2 1
3xy3 +
1
4x3y2 +
1
3x2y3 +
1
8xy4)y=1xy=0
dx
=
10
(1
24x 1
6x2 +
1
4x3 1
6x4 +
1
24x5)dx
=
(1
48x2 1
18x3 +
1
16x4 1
30x5 +
1
144x6)1x=0
=1
720..
7.2.4.
D(x2 3y + 2z) dzdydx D
D = {(x, y, z) : x 0, y 0, z 0, x+ y + z 1} .
. 10
1x0
1xy0
(x2 3y + 2z
)dzdydx =
10
1x0
(x2z 3yz + z2
)z=1xyz=0
dydx
=
10
1x0
((x2 3y
)(1 x y) + (1 x y)2
)dydx
=
10
(y3 +
1
2
(x2 3 + 3x
)y2 + x2 (1 x) y 1
3(1 x y)3
)y=1xy=0
dx
=
10
(1
6+
1
2x 5
6x3 +
1
2x4)dx = 1
40
-
.
7. 82
7.2.5.
D(y + z) dxdydz, D
y =x, y = x2, z = 0 x+ z = 1.
. y =x, y = x2 . D
. 10
xx2
1x0
(y + z) dzdydx =
10
xx2
(yz +
z2
2
)z=1xz=0
dydx
=
10
xx2
(y (1 x) + (1 x)
2
2
)dydx
=
10
(y2
2(1 x) + y (1 x)
2
2
)y=xy=x2
dx
=
10
(x
2(1 x) +
x
(1 x)2
2
(x4
2(1 x) + x2 (1 x)
2
2
))dx
=
10
(1
3x3 +
1
4x2 +
1
7x
72 2
5x
52 +
1
3x
32 +
1
12x6 1
5x5 +
1
4x4)dx
=53
420.
7.2.6. D z = x2 + 3y2
z = 8 x2 y2.. D z
x2 + 3y2 z 8 x2 y2.
x y. , .
8 x2 y2 = x2 + 3y2 2x2 + 4y2 = 8 x2 + 2y2 = 4.
,
4 x2
2 y
4 x2
2, 2 x 2.
-
.
7. 83
22
4x22
4x22
8x2y2x2+3y2
dzdydx =
22
4x22
4x22
(8 x2 y2 x2 3y2
)dydx
=
22
4x22
4x22
(8 2x2 4y2
)dydx
=
22
(8y 2x2y 4y
3
3
)y= 4x22
y=
4x22
dx
=
22
2 (8 2x2)4 x22 8
3
(4 x2
2
)3 dx=
4
2
3
22
(4 x2
2
)3/2dx = 8
2.
7.2.7.
D ={
(x, y, z) : 2 z 5, x2 y x}.
. y = x2 y = x . P z = 2
P ={
(x, y, z) : z = 2, x2 y x, 0 x 1}
( 0 1 y = x2, y = x). 10
xx2
52
dzdydx =
10
xx2
3dydx =
10
(3x 3x2
)dx =
1
2.
7.2.8. z = 4 x2 y2 z = 0.
. . 20
20
4x2y20
dzdd =
20
20
(4 x2 y2
)dd
=
20
20
(4 2
)dd
=
20
20
(4 3
)dd
=
20
(22
4
4
)=2=0
d
=
20
(8 4) d = 8.
-
.
7. 84
7.2.9. z = x2 + y2,x2 + y2 = 4 z = 0.
. . x2 + y2 = 4 . . 2
0
20
4x2y20
dzdd =
20
20
(z)z=2
z=0 dd
=
20
20
3dd
=
20
(4
4
)=2=0
dd
=
20
4d = 8.
7.2.10. D r = 1 = /3 ( ).
. ( ) 20
/30
10
r2 sindrdd =
20
/30
1
3sindd
=
20
(1
3cos
)=/3=0
d
=
20
(1
3 1
6
)d =
3.
7.2.11. ( - ) = 2 sin x2 + y2 + z2 = 4.
. . . z
x2 + y2 + z2 4 2 + z2 4 |z|
4 2
-
.
7. 85
0
2 sin 0
42
42dzdd =
0
2 sin 0
2
4 2dd
= 2
/20
2 sin 0
(4 2
)1/2d(2)d
= 2
/20
(2
3
(4 2
)3/2)=2 sin =0
d
=4
3
/20
(8 8 cos3
)d
=32
3
/20
d 323
/20
(1 sin2
)cos d
=32
3
/20
d 323
/20
(1 sin2
)d (sin )
=16
3 32
3
(sin sin
3
3
)=/2=0
=16
3 32
3
(1 1
3
)=
16
3 64
3.
7.2.12.
S1 :x2
5+y2
5 z
2
4= 1
S2 : x2 + y2 = 25
. S1 S2 ( ).
x2
5+ y
2
5 z2
4= 1
x2 + y2 = 25
} 5 z
2
4= 1 z = 4.
S1 S2 :
C1 ={
(x, y, z) : x2 + y2 = 25, z = 4}
C2 ={
(x, y, z) : x2 + y2 = 25, z = 4}
V , V1 . V1 = R2h = 52 8 =200.
(z) =x2 + y2 =
5 (
1 +z2
4
)
-
.
7. 86
V2 =
442dz =
44
5
(1 +
z2
4
)dz =
280
3.
V = V1 V2 =200
1 280
3 =
320
3.
7.2.13. S1 : x
2 + y2 = a2 S2 : x2 + z2 = a2.. ,
.
V = 8V = 8
a0
a2x20
a2x20
dzdydx
= 8
a0
a2x20
a2 x2dydx = 8
a0
a2 x2
a2 x2dx
= 8
a0
(a2 x2
)dx = 8
(a2x x
3
3
)x=ax=0
= 8a3 8a3
3=
16a3
3.
7.2.14.
D(x2 + y2) dzdydx D
.. ( ) 2
0
/30
10
(x2 + y2
)r2 sindrdd =
20
/30
10
(r2 sin2 cos2 + r2 sin2 sin2
)r2 sindrdd
=
20
/30
10
r4 sin3 drdd
=
20
/30
1
5sin3 dd
=1
5
20
/30
(1 cos2
)sindd
=1
5
20
/30
(1 cos2
)d cosd
=1
5
20
/30
(cos cos
3
3
)=/3=0
d
=1
5
20
(1
2+ 1 +
1
24 1
3
)d =
12.
7.2.15.
D = {(x, y, z) : 0 x 1, 0 y 1, 0 z 1}
.
-
.
7. 87
. d (x, y, z) = x2 + y2 + z2 10
10
10
(x2 + y2 + z2
)dxdydz =
10
10
(x3
3+ y2x+ z2x
)x=1x=0
dydz
=
10
10
(1
3+ y2 + z2
)dydz
=
10
(y
3+y3
3+ z2y
)y=1y=0
dz
=
10
(1
3+
1
3+ z2
)dz
=
(2z
3+z3
3
)z=1z=0
dz = 1.
7.2.16. R .
. . d (r) =kr 2
0
/20
R0
kr r2dr sindd = 2
0
/20
kR4
4sindd
= kR4
4
20
( cos)=/2=0 d
= kR4
4
20
(0 + 1) d =kR4
2.
7.2.17.
Dz (y2 + z2) dxdydz, D
(0, 0, 0) 2.
-
.
7. 88
. 22
4x2
4x2
4x2y20
z(y2 + z2
)dzdydz =
22
4x2
4x2
4x2y20
z(y2 + z2
)dzdydx
=
22
4x2
4x2
(y2z2
2+z4
4
)z=4x2y2z=0
dydx
=
22
4x2
4x2
(y2 (4 x2 y2)
2+
(4 x2 y2)2
4
)dydx
=1
4
22
4x2
4x2
((4 x2
)2 y4) dydx=
1
4
22
((4 x2
)2y y
5
5
)y=4x2y=
4x2
dx
=4
5
20
(4 x2
)5/2dx ( x = 2 sinu)
=4
5
u=/2u=0
64 cos6 udu = 8.
7.2.18. = /6 ( ) z = a, d (r, , ) = r cos.
. . r.
0 z a 0 r cos a 0 r acos
.
20
/60
a/ cos0
d (r, , ) r2 sindrdd =
20
/60
a/ cos0
r3 cos sindrdd
=
20
/60
(r4
4
)r=a/ cosr=0
cos sindd
=
20
/60
a4
4 cos3 sindd
=a4
4
20
/60
cos3 d (cos) d
=a4
4
20
(cos2
2
)=/6=0
d =a4
24
20
d =a4
12.
7.2.19. r = 5 ( ) z = 4, d (r, , ) = r1.
. . r. ( )
4 z 4 r cos 4cos
r r 5.
-
.
7. 89
0 arccos 45. 2
0
arccos(4/5)0
54/ cos
d (r, , ) r2 sindrdd =
20
arccos(4/5)0
54/ cos
r sindrdd
=
20
arccos(4/5)0
(r2
2
)r=5r=4/ cos
sindd
=
20
arccos(4/5)0
(25
2 8
cos2
)sindd
= 2
0
arccos(4/5)0
(25
2 8
cos2
)d (cos) d
= 2
0
(25
2cos+
8
cos
)=arccos(4/5)=0
d
=
20
(25
2+ 8 25
2 4
5 8
4/5
)d
=
20
1
2d =
7.2.20.
D(2x+ 3y)2 dxdydz D =
{(x, y, z) : x
2
4+ y
2
9+ z
2
16 1}.
. .
x = 2u, y = 3v, z = 4w.
dxdydz = D(x,y,z)D(u,v,w)
= 24
D ={
(x, y, z) : u2 + v2 + w2 1}
, u, v, w:.
u = r cos sin
v = r sin sin
w = r cos
D
(2x+ 3y)2 dxdydz =
D
(4u+ 9v)2 24dudvdw
=
20
0
10
(4r cos sin+ 9r sin sin)2 24r2 sindrdd
= 24
20
0
10
(4 cos + 9 sin )2 r4 sin3 drdd
= 24
( 10
r4dr
)( 0
sin3 d
)( 20
(4 cos + 9 sin )2 d
)=
4656
15.
-
.
7. 90
7.3
7.3.1. 1
0
20
30dzdydx.
(. 6)
7.3.2. a
0
b0
c0
(x+ y + z) dzdydx.(. 1
2bca (a+ c+ b))
7.3.3. a
0
x0
y0xyzdzdydx.
(. a6/48)
7.3.4. a
0
x0
xy0x3y3zdzdydx.
(. a12/144)
7.3.5.
Ddxdydz
(x+y+z+1)3, D
x = 0, y = 0, z = 0, x+ y + z = 1.(. 1
2
(ln (2) 5
8
))
7.3.6.
Dxydxdydz, D
z = xy, x+ y = 1 z = 0.(. 1/180)
7.3.7.
Dy cos (z + x) dxdydz, D
y =x, y = 0, z = 0 x+ z = /2.
(. 2
16 1
2)
7.3.8. 1
0
1x2
1x2 a
0dzdydx.
(. a/2)
7.3.9. 2
0
2xx20
a0zx2 + y2dzdydx.
(. 8a2/9)
7.3.10. RR
R2x2R2x2
R2x2y2R2x2y2
(x2 + y2) dzdydx.
(. 4R5/15)
7.3.11. 1
0
1x20
1x2y20
x2 + y2 + z2dzdydx.
(. /8)
7.3.12.
D(x2 + y2) dzdydx D
z 0, r2 x2 + y2 + z2 R2.(.
4(R5r5)15
)
7.3.13.
D1
x2+y2+(z2)2dzdydx D
x2 + y2 + z2 1.(. 2
3)
-
.
7. 91
7.3.14.
D1
x2+y2+(z2)2dzdydx D
x2 + y2 1.(.
(3
10 + ln
21101
2 8
))
7.3.15.
0
0
0
1(1+x+y+z)4
dzdydx.(. 1
6)
7.3.16. : x = 0, y = 0, z = 0, 3x +2y + z = 6.
. 6.
7.3.17. : z = 0, 2y = x, x = 1, y =0, z = x2 + 3y + 1.
. 1/2.
7.3.18. : z = 0, y = x, y = 2x, y =3, z = x+ y2.
. 27/2.
7.3.19. : x2 = 2y, y = 2, z = 0, z =x2 + 3y2.
. 8.
7.3.20. : x = 0, x = 1, y = 0, y =2, z = 0, z = 4 x2.
. 22/3.
7.3.21. : z = 1 y2, x2 = 4y, z = 0.. 32/21.
7.3.22. : 3y = 9 x2, z = 0, y =0, x = 3 z.
. 36.
7.3.23. : x = 0, x = 16 z2 4y2.. 64.
7.3.24. : x2 + y2 = k2, z = 0, z =k2 x2 (k > 0).
. 3k4/4.
7.3.25. : z = 0, y = 3x, y = x, z =16 x2.
. abc/6.
7.3.26. : z = 4 x2, z = 4 y2.. 32.
-
.
7. 92
7.3.27. : x2 + z2 = a2, y2 + z2 = a2.. 16a3/3.
7.3.28. : x2 + y2 + z2 = a2.. 4
3a3.
7.3.29. : x2 +y2 = a2, x2 +y2 +z2 =4a2.
. 2a3(8 3
3)/3.
7.3.30. : r = 2, y x+ 2z = 8.. 16.
7.3.31. : r = 3 sin , z = x.. 9/4.
7.3.32. : r = 1 cos , z = y.. 4/3.
7.3.33. : x2 +y2 = 1, z2 = 4x2 +4y2.. 4/3.
7.3.34. : x = 0, y = 0, z = 0, x = 4,y = 4 z = x2 + y2 + 1
. 5603.
7.3.35. : x = 0, y = 0, z = 0,xa
+ yb
+ zc
= 1.. abc
6.
7.3.36. : x = 0, y = 0, z = 0,xa
+ yb
+ zc
= 1.. 84/3.
7.3.37. : z = x + y + a, y2 = ax,x = a, z = 0, y = 0.
. 79a3/60.
7.3.38. : y = 0, z = 0, 3x + y = 6,3x+ 2y = 12, x+ y + z = 6.
. 12.
7.3.39. : x = 0, y = 0, z = 0,z = x2 + y2, x+ y = 1.
. 1/6.
7.3.40. : x2 + y2 = a2, x2 + z2 = a2.. 16a3/3.
-
.
7. 93
7.3.41. : z2 = xy, x = a, x = 0, y =a, y = 0.
. 8a3/9.
7.3.42. : x = 0, y = 0, z = 0,z = 9 y2,3x+ 4y = 12.
. 45.. /4 . .
7.3.43. z = mx, x2 +y2 = a2, z = 0.. 4ma3/3 . .
7.3.44. az = a2 x2 y2, z = 0.. a3/2 . .
7.3.45. x2 +y2 +z2 = 4a2, x2 +y2 =a2.
. 4a3
3 . .
7.3.46. x2
a2+ z
2
b2= 1, x
2
a2+ y
2
b2= 1.
. 16ab2/3 . .
7.3.47. z2 = (x+ a)2, x2 + y2 = a2.. 2a3 . .
7.3.48. z = 4x2+y2
, z = 0, x2+y2 = 1,x2 + y2 = 4.
. 8 ln 2 . .
7.3.49. az = x2 + y2, z = 0,x2 + y2 ax = 0.
. 3a3/16 . .
7.3.50. z = a, x2 + y2 = a2,x2 + y2 + z2 = 5a2 ( a z a
5).
. a3(10
5 19)/3 . .
7.3.51. r = a, z = a2 x2, z 0.. 3a4/4 . .
7.3.52. z = 12 x2 y2, z = 8.. 8 . .
-
. 8
8.1
8.1.1. , . - R R3. r (t): t ,
r (t) = x (t) i + y (t) j + z (t) k.
(x (t) , y (t) , z (t)). {(x (t) , y (t) , z (t))}tR , .
8.1.2. , F : R R3, . t:
F (t) = P (t) i +Q (t) j +R (t) k.
8.1.3. F (t) t t0
limtt0
F (t) = F0 ( > 0 > 0 : 0 < |t t0| < |F (t) F0| < ) .
8.1.4. F (t) t0
limtt0
F (t) = F (t0) .
8.1.5. F (t) = P (t) i +Q (t) j+R (t) k
F (t) =d
dtF (t) = lim
t0
F (t+ t) F (t)t
=dP
dti +
dQ
dtj+dR
dtk.
94
-
.
8. 95
d
dt(F + G) =
d
dtF +
d
dtG
d
dt(F G) = F d
dtG + G d
dtF
d
dt(FG) = F (t) d
dtG + G d
dtF
d
dt(F) = F
d
dt+
d
dtF
d
dt(F GH) = F
(G d
dtH
)+ F
(d
dtGH
)+d
dtF (GH) .
8.1.6. dF
dt=dP
dti +
dQ
dtj +
dR
dtk
:
dF = dP i + dQj + dRk
:
d (F + G) = dF + dG
d (F G) = F dG + G dFd (FG) = F (t) dG + G dF
d (F) = Fd+ dF
d (F GH) = F (G dH) + F (dGH) + dF (GH) .
8.1.7. M . , (x (t) , y (t) , z (t)). M C, :
C : (x (t) , y (t) , z (t)) , t [t1, t2] .
, C
r (t) = x (t) i + y (t) j + z (t) k , t [t1, t2] .
1.
8.1.8. r (t) , drdt
.
1 C 3- - :
C : (x, y, z) = 0, (x, y, z) = 0.
-
.
8. 96
8.1.9. r (t) , t [t1, t2] r (t1) = r (t2), t d
dtr (t) (.
), ( p 6= q r (p) 6= r (q)).8.1.10. r (t) = ix (t) + jy (t) + kz (t) r0= r (t0)= ix (t0) + jy (t0) + kz (t0)= ix0 + jy0 + kz0. R (t) = iX (t) + jY (t) + kZ (t), R (t)
(R (t) r0)dr (t0)
dt= 0
X (t) x0x (t0)
=Y (t) y0y (t0)
=Z (t) z0z (t0)
.
8.1.11. r (t) r0= r (t0)= ix (t0) +jy (t0)+kz (t0)= ix0 +jy0 +kz0. R = iX + jY + kZ,
(R r0) dr (t0)
dt= 0
x (t0) (X x0) + y (t0) (Y y0) + z (t0) (Z z0) = 0.
8.1.12. . (, ) F (t)
F (t) dt =
[P (t) dt
]i +
[Q (t) dt
]j +
[R (t) dt
]k, t2
t1
F (t) dt =
[ t2t1
P (t) dt
]i +
[ t2t1
Q (t) dt
]j +
[ t2t1
R (t) dt
]k.
. ..cF (t) dt = c
F (t) dt,
c F (t) dt = c
F (t) dt,c F (t) dt = c
F (t) dt.
8.2
8.2.1. , , F (t) = i cos t+j sin t+ k G (t) = i (t2 + 1) + jt+ ket.
. F (t) = G (t) .
cos t = t2 + 1
sin t = t
1 = et
t = 0. F (0) = G (0) = i + k, . (1, 0, 1).
-
.
8. 97
8.2.2. , , F (t) = i cos t+j + kt G (t) = it2 + j2t+ k e
t
t+1.
. F (t) = G (t) .
cos t = t2
1 = 2t
t =et
t+ 1
t = 1/2, cos 1
2= 1
4, . .
8.2.3. limt2 F (t) = i cos t+ j tt+1 + kt..
limt2
F (t) = i limt2
(cos t) + j limt2
(t
t+ 1
)+ k lim
t2(t) = i cos 2 +
2
3j + 2k..
8.2.4. ddtF (t),
ddtF (t)
, ddt|F (t)| F (t) = i cos t+ j sin t+ kt
.
d
dtF (t) =
d
dt(i cos t+ j sin t+ kt) = i sin t+ j cos t+ k
ddtF (t) = |i sin t+ j cos t+ k| = sin2 t+ cos2 t+ 1 = 2.
|F (t)| = |i cos t+ j sin t+ tk| =
cos2 t+ sin2 t+ t2 =t2 + 1
d
dt|F (t)| =
d(
t2 + 1)
dt=
tt2 + 1
.
8.2.5. F (t) = i cos t + j sin t + kt, G (t) = it2 + jt3 + k, (t) = t d
dt(F + G), d
dt(F G), d
dt(F) .
.
d
dt(F + G) =
d
dt
(i cos t+ j sin t+ kt+ it2 + jt3 + k
)=
d
dt
(i (t+ cos t) + j
(t2 + sin t
)+ k (1 + t)
)= i (1 sin t) + j (2t+ cos t) + k.
d
dt(F G) = d
dt
(t2 cos t+ t3 sin t+ t
)= t3 cos t+ 2t2 sin t+ 2t cos t+ 1.
-
.
8. 98
d
dt(F) =
d
dt
(it cos t+ jt sin t+ kt2
)= i (cos t t sin t) + j (t cos t+ sin t) + k2t.
8.2.6. F (t) = i sin t + jt2 + kt, G (t) = it + j ln t + ket, c = i2 + j3 k
(F (t) + G (t)) dt,
(F (t) G (t)) dt,
c F (t) dt.
. (F (t) + G (t)) dt =
(i (t+ sin t) + j
(t2 + ln t
)+ k
(t+ et
))dt
=
(i
(t2
2 cos t+ c1
)+ j
(t3
3+ t (ln t 1) + c2
)+ k
(t2
2+ et + c3
))= i
(t2
2 cos t
)+ j
(t3
3+ t (ln t 1)
)+ k
(t2
2+ et
)+ c
c =c1i + c2j + c3k. (F (t) G (t)) dt =
(t sin t+ t2 ln t+ tet
)dt = sin tet+1
3t3 ln tt cos t+tet1
9t3+c.
c F (t) dt =
(2 sin t+ 3t2 t
)dt = t3 1
2t2 2 cos t+ c.
8.2.7. r (t) = i (1 + t) jt2 + k (1 + t3) r (1).
. r0 = r (1)
r0 = r (1) = i2 j + k2
n0 =dr
dt|t=1 =
(i j2t+ k3t2
)|t=1 = i 2j + 3k.
R (t) r (t) r0
X (t) 21
=Y (t) + 1
2=Z (t) 2
3= t
, ,
R (t) = i (2 + t) j (1 + 2t) + k (2 + 3t) .
r (t) r0
1 (X 2) 2 (Y + 1) + 3 (Z 2) = 0
X 2Y + 3Z = 10.
-
.
8. 99
8.2.8. r (t) = i2 cos t j sin t+ kt r (/2).
. r0 = r (/2)
r0 = r (/2) = j + k
2
n0 =dr
dt|t=/2 = (i2 sin t+ j cos t+ k) |t=1 = 2i + k.
R (t) r (t) r0
X (t)
2=Y (t) 1
0=Z (t) /2
1= t
, ,
R (t) = i2t j + k (/2 + t) .
8.2.9. r (t) = i cos t +j sin t+ kt r (t) |t=/2= j + k2 .
. r (t = /2) n = drdt|t=/2= i+
k.
1 (X 0) + 0 (Y 1) + 1 (Z
2
)= 0
X + Z = /2.
8.2.10. t, r (t) = i2t3
+jt2+kt3
v = i + k.. R (t) = iX (t) + jY (t) + kZ (t) r (t)
(x0, y0, z0)
X (t) x02/3
=Y (t) y0
2t=Z (t) z0
3t2.
n (t) = i23+j2t+ k3t2
n (t) v = i + k
cos =n (t) v
n (t) ||v||=
23
+ 3t2(23
)2+ (2t)2 + (3t2)2
12 + 12
=23
+ 3t2(23
)2+ (2t)2 + (3t2)2
12 + 12
=23
+ 3t29t4 + 4t2 + 4
9
2
=23
+ 3t2(3t2 + 2
3
)22
=
2
2.
= /4, t .
-
.
8. 100
8.2.11. r (t) = i cos t + jet + k sin t - E : X + Z = 1.
. r (t)
a (t) =dr
dt= i sin t+ jet + k cos t.
E n = i + k. r (t) E
a (t) n = sin t+ cos t = 0 sin t = cos t t = 4
+ n n {0,1,2, ...} .
(
cos(4
+ n), e(
4
+n), sin(4
+ n))
=(2
2, e(
4
+n),
22
) r (t) E.
8.2.12. ddt
(F G) = F ddtG + G d
dtF.
.
d
dt(F G) = d
dt((F1i+F2j+F3k) (G1i+G2j+G3k)) =
d
dt((F1G1+F2G2+F3G3))
=
(F1dG1dt
+F2dG2dt
+F3dG3dt
)+
(dF1dt
G1+dF2dt
G2+dF3dt
G3
)= F d
dtG + G d
dtF.
8.2.13. -
S1 : x2 + y2 + z2 = 2, S2 : z = 1.
. , S1 S2. S1 z = 1 x2 + y2 = 1.
x (t) = cos t, y (t) = sin t, z (t) = 1
, r (t) = i cos t+ j sin t+ k.
8.2.14. r (t) x2 + y2 = 1 y + z = 2.
. r (t) xy x2 + y2 = 1
x = cos t, y (t) = sin t, z = 0.
(x (t) , y (t) , z (t)) r (t) x y . , y + z = 2,
z (t) = 2 y (t) = 2 sin t.
-
.
8. 101
(x (t) , y (t) , z (t)) = (cos t, sin t, 2 sin t)
r (t) = i cos t+ j sin t+ k (2 sin t) .
8.2.15. r (t) t r (t) ddtr (t) =0.
r (t) .. f (t) = r (t)2
df
dt=
d
dt
(r (t)2
)=
d
dt(r (t) r (t)) = r (t) d
dtr (t) +
d
dtr (t) r (t) = 2r (t) d
dtr (t) = 0
( ). , c
f (t) = r (t)2 = cf (t) = r (t) =
c.
, . 0
c.
8.3
8.3.1. F (t) = i sin t+ j cos t+kt2 G (t) = it+ jet + k2t.
. (0, 1, 0) .
8.3.2. F (t) = it+ jt+ kt G (t) =it+ jet + kt2.
. .
8.3.3. F (t) = it + jt2 + kt3 G (t) = it2 + jt3 + kt4. .
. (1, 1, 1). = arccos 20406
.
8.3.4. limt1 F (t) = it+ jt+ kt.. i + j + k.
8.3.5. ddtF (t),
ddtF (t)
, ddt|F (t)| F (t) = i sin t+ j cos t+ kt2
. i cos t j sin t+ k2t,
4t2 + 1, 8t3
4t4+1
.
8.3.6. ddtF (t),
ddtF (t)
, ddt|F (t)| F (t) = iet + j ln t+ kt
. iet + j1t
+ k,e2t +
(1t
)2+ 1, 1
te2tt+ln t+t2e2t+ln2 t+t2
.
8.3.7. ddtF (t),
ddtF (t)
, ddt|F (t)| F (t) = i sin t+ jt
. i cos t+ j,
cos2 t+ 1, sin t cos t+tt2+1cos2 t .
-
.
8. 102
8.3.8. F (t) = i sin t+ j cos t+ kt2, G (t) = it+ jt2 + k2, (t) = t+ 2 d
dt(F + G), d
dt(F G), d
dt(FG), d
dt(F)
.
i (cos t+ 1) j (sin t+ 2t) + k2t,3t cos t+ sin t t2 sin t+ 4t, i(2 sin t 4t3
)+ j(3t2 2 cos t
)+ k
(t2 cos t+ 3t sin t cos t
),
i (sin t+ (t+ 2) cos t) + j (cos t (t+ 2) sin t) + k(3t2 + 2t
).
8.3.9. F (t) = i ln t+ jt+ kt, G (t) = it3 + jt2 + ket, (t) = t+ 1 d
dt(F + G), d
dt(F G), d
dt(FG), d
dt(F).
.
i
(1
t+ 3t2
)+ j (1 + 2t) + k
(1 + et
),
4t2 + 3t2 ln t+ et + tet,
i(et + tet 3t2
)+ j
(4t3 1
tet (ln t) et
)+ k
(t+ 2 (ln t) t 4t3
),
i
(ln t+
t+ 1
t
)+ j (1 + 2t) + k (1 + 2t) .
8.3.10. F (t) = i ln t + jt + k2t, G (t) = it3 + jt2 + ket, c = i2 + j3 k
(F (t) + G (t)) dt,
(F (t) G (t)) dt,
(F (t)G (t)) dt,
c F (t) dt,
c F (t) dt..
i
(t ln t t+ 1
4t4)
+ j
(1
2t2 +
1
3t3)
+ k(t2 + et
),
1
4t4 ln t+
3
16t4 + 2tet 2et,
i
(tet et 1
2t4)
+ j
(et ln t+ 2
5t5)
+ k
(1
3t3 ln t 1
9t3 1
5t5),
2t ln t 2t+ 12t2,
i7
2t2 + j
(t ln t+ t 2t2
)+ k
(t2 3t ln t+ 3t
).
8.3.11. r (t) = it+ jt2 + kt3 s (t) = i sin t+j cos 2t+ kt r (0).
. = arccos(
6/6).
8.3.12. r (t) = it + j (1 t) + k (3 + t2) s () = i (3 ) + j ( 2) + k ( 2).
. r (t = 1) = s ( = 2) = (1, 0, 4) = arccos(
3/3).
-
.
8. 103
8.3.13. r (t) = i (1 + t) jt2 + k (1 + t3) r (1).
. i (2 + t) j (1 + 2t) + k (2 + 3t) X 2Y + 3Z = 10.
8.3.14. r (t) = i t
4
4+ j t
3
3+ k t
2
2.
. r (t0): i(t20t+
t404
)+j(t0t+
t303
)+k(t+
t202
) t20X+t0Y +Z =
t604
+t403
+t202.
8.3.15. r (t) = it+ j t
2
2+ k t
3
3 (6, 18, 72).
. i (t+ 6) + j (6t+ 18) + k (36t+ 72) X + 6Y + 36Z = 2706).
8.3.16. r (t) = i2 sin 3t+ jt+k cos 3t r ().
. Y 6X = .
8.3.17. r (t) = it + jt2 + kt3
r (1).. X + 2Y + 3Z = 6.
8.3.18. r (t) = it3 + j3t + kt4 6X + 6Y 8Z = 1;
. r (1)=(1,3, 1).
8.3.19. r (t) = i cos t + j sin t + ket -
3X + Y = 4.
.(
3/2, 1/2, e/6)
8.3.20.
r (t) = i
(t3
3+ t2 t
)+ j
(t3
3 t2 t
)+ k
(t3
3 5t
) X Y + Z = 4.
. t = 1 t = 5.
8.3.21. r (t) = it cos t + jt sin t + kt z2 = x2 + y2. .
8.3.22. r (t) = i sin t + j cos t + k sin2 t z = x2 x2 + y2 = 1. .
8.3.23. r (t) = it2 + j (1 3t) + k (1 + t3) (1, 4, 0) (9,8, 28).
8.3.24. x2 +y2 = 4 z = xy. .
. i2 cos t+ j2 sin t+ k2 sin 2t.
-
.
8. 104
8.3.25. z =x2 + y2
z = 1 + y. .. it+ j t
212
+ k t2+12.
8.3.26. z = 4x2+y2 y = x2. .
. it+ jt2 + k (4t2 + t4) .
8.3.27. ddt
(FG) = F (t) ddtG + G d
dtF.
8.3.28. ddt
(F) = F ddt+ d
dtF.
8.3.29. ddt
(F (h (t))) = dhdt(ddh
F).
8.3.30. r (t) = i cos t + j sin t r (t) dr(t)dt
= 0 (. ).
8.3.31. r (t) t r (t) ddtr (t) = a r (t),
a . r (t) a.
8.3.32. r (t) t d2
dt2r (t) d
dtr (t) = 0.
r (t) .
8.3.33. r (t) t r (t) = c (). r (t) d
dtr (t) = 0.
-
. 9
9.1
9.1.1. C r (t) =x (t) i + y (t) j + z (t) k(t1 t t2). C . ds
dt= |r (t)| ds = |r (t)| dt.
9.1.2. (x, y, z). , t2
t1
(x (t) , y (t) , z (t)) |r (t)| dt (9.1)
. |r (t)| = dsdt,
: C
(x, y, z) ds (9.2)
C r (t) , . r (t1) = r (t2),
C
(x, y, z) ds.
(9.1) (9.2) , . ,
C (x, y, z) ds
C, . .
9.1.3. ( t1 t2) C:
S =
t2t1
(dx
dt
)2+
(dx
dt
)2+
(dx
dt
)2dt =
t2t1
1 |r (t)| dt.
105
-
.
9. 106
9.1.4. , a , a t
s (t) =
ta
(dx
d
)2+
(dy
d
)2+
(dz
d
)2d.
9.1.5. - . .
1. .
C
(x, y, z) ds =
C (x, y, z) ds
2. C C1, C2, ..., CK( C = C1C2...CK )
C
(x, y, z) ds =
C1
(x, y, z) ds+
C2
(x, y, z) ds+ ...+
CK
(x, y, z) ds.
9.1.6. . , . C r (t) =x (t) i+y (t) j+z (t) k t1 t t2. (x, y, z).
m =
C
(x, y, z) ds.
, (x0, y0, z0)
x0 =1
m
C
x (x, y, z) ds
y0 =1
m
C
y (x, y, z) ds
z0 =1
m
C
z (x, y, z) ds.
, x, y, z
Ix =
C
(y2 + z2
) (x, y, z) ds
Iy =
C
(z2 + x2
) (x, y, z) ds
Iz =
C
(x2 + y2
) (x, y, z) ds.
-
.
9. 107
xy, yz, zx
Ixy =
C
z2 (x, y, z) ds
Iyz =
C
x2 (x, y, z) ds
Izx =
C
y2 (x, y, z) ds.
9.1.7. C r (t) =x (t) i+ y (t) j+ z (t) k (t1 t t2) F (r) = iP (x, y, z) + jQ (x, y, z) + kR (x, y, z). , t2t1
P (x (t) , y (t) , z (t))x (t) dt+
t2t1
Q (x (t) , y (t) , z (t)) y (t) dt+
t2t1
R (x (t) , y (t) , z (t)) z (t) dt
(9.3) .
C
P (x, y, z) dx+
C
Q (x, y, z) dy +
C
R (x, y, z) dz. (9.4)
C
F (r) dr. (9.5)
C C
F (r) dr. (9.3), (9.4) (9.5)
, .
9.1.8. , C
F (r) dr C, . .
9.1.9. F (r).
9.1.10. . .
1. -.
C
F (r) dr = C
F (r) dr
2. C C1, C2, ..., CK( C = C1C2...CK )
C
F (r) dr =C1
F (r) dr +C2
F (r) dr + ...+CK
F (r) dr.
-
.
9. 108
3. (x, y, z) F = A R3. , r1 / C1 r2 / C2 A (r1 (t1) = r2 (t1) r1 (t2) = r2 (t2)),
C1
F (r) dr =C2
F (r) dr = (r1 (t2)) r1 (t1) .
. C
F (r)dr - ( C A) . r2
r1
F (r) dr (x2,y2,z2)
(x1,y1,z1)
F (r) dr.
4. 3, 0:
C
F (r) dr = 0.
9.1.11. ( Green) C r (t) = ix (t) + jy (t) .1 C , D R2. F (x, y) = iP (x, y) + jQ (x, y) P Q D.
C
F (r) dr =D
(Q
x Py
)dxdy (9.6)
C
(P (x, y) dx+Q (x, y) dy) =
D
(Q
x Py
)dxdy.
9.1.12. , C D F (x, y) = iP (x, y) + jQ (x, y).
Q
x=P
y(9.7)
(x, y) D, (9.7) C
(P (x, y) dx+Q (x, y) dy) =
D
(Q
x Py
)dxdy = 0.
1 xy.
-
.
9. 109
9.1.13. C1 C2 D . F (x, y) = iP (x, y) + jQ (x, y).
Q
x=P
y(9.8)
(x, y) D, C1
(P (x, y) dx+Q (x, y) dy) =
C2
(P (x, y) dx+Q (x, y) dy) , (9.9)
. (9.9) .
9.1.14. .2 (9.7) P (x, y) dx+Q (x, y) dy . . (9.7), C
(P (x, y) dx+Q (x, y) dy) =
C
d = (x (t) , y (t))|t=t2t=t1 = (x (t2) , y (t2)) (x (t1) , y (t1)) = 0
, C (x (t2) , y (t2)) = (x (t1) , y (t1)).
9.1.15. Riemann Green S D C r (t):
S =
C
xdy = C
ydx =1
2
C
(xdy ydx) .
9.2
9.2.1. r (t) = iR cos t+ jR sin t t [0, 2].. () .
S =
t2t1
(dx
dt
)2+
(dx
dt
)2+
(dx
dt
)2dt
=
20
(R sin t)2 + (R cos t)2dt =
20
Rdt = 2R.
.
9.2.2. r (t) = i cos t+ j sin t+ kt t [0, 2]..
S =
t2t1
(dx
dt
)2+
(dx
dt
)2+
(dx
dt
)2dt
=
20
( sin t)2 + (cos t)2 + (1)2dt =
20
2dt = 2
2.
-
.
9. 110
9.2.3. r (t) = i12t+ j8t3/2 + k3t2 t [0, 1]..
S =
t2t1
(dx
dt
)2+
(dx
dt
)2+
(dx
dt
)2dt
=
10
122 + (12t1/2)
2+ (6t)2dt
=
10
144 + 144t+ 36t2dt =
=
10
6 (2 + t) dt =(12t+ 3t2
)t=1t=0
= 15.
9.2.4. C
(2 + x2y) ds C x2+y2 = 1.. C x (t) = cos t, y (t) = sin t.
dx
dt= cos t, dy
dt= sin t, t [0, ]
ds =
(dx
dt
)2+
(dy
dt
)2dt = dt.
C
(2 + x2y
)ds =
0
(2 + cos2 t sin t
)dt =
(2t cos
3 t
3
)t=t=0
= 2 +2
3.
9.2.5. C
2xds C = C1C2 () C1 y = x2 (0, 0) (1, 1), () C2 x = 1 (1, 1) (1, 2).
. C1 x (t) = t, y (t) = t2. ( t [0, 1])
dx
dt= 1,
dy
dt= 2t, ds =
(dx
dt
)2+
(dy
dt
)2dt =
1 + 4t2dt
C1
2xds =
10
2t
1 + 4t2dt =1
4 2
3
(1 + 4t2
)t=1t=0
=5
5 16
.
C2 x (t) = 1, y (t) = t. ( t [1, 2])
dx
dt= 0,
dy
dt= 1, ds =
(dx
dt
)2+
(dy
dt
)2dt = dt
C1
2xds =
10
2 1dt = 2.
C
2xds =
C1
2xds+
C2
2xds =5
5 16
+ 2
-
.
9. 111
9.2.6. C
(x 3y2 + z) ds C (0, 0, 0) (1, 1, 1).
. C r (t) = it+ jt+ kt. .
x (t) = t, y (t) = t, z (t) = t,
dx = dt, dy = dt, dz = dt
t [0, 1].
ds =
(dx
dt
)2+
(dy
dt
)2+
(dz
dt
)2dt =
3dt
C
(x 3y2 + z
)ds =
10
(2t 3t2
)3dt =
3(t2 t3
)t=1t=0
= 0.
9.2.7.C
(x 3y2 + z) ds C = C1C2 () C1 (0, 0, 0) (1, 1, 0), () C2 (1, 1, 0) (1, 1, 1).
. C1 : t [0, 1]:
x (t) = t, y (t) = t, z (t) = 0,
dx = dt, dy = dt, dz = 0.
C2 : t [0, 1]:
x (t) = 1, y (t) = 1, z (t) = t,
dx = 0, dy = 0, dz = dt.
C1
ds =
(dx
dt
)2+
(dy
dt
)2+ 0dt =
2dt
C1
(x 3y2 + z
)ds =
10
(t 3t2 + 0
)2dt =
2
(t2
2 t3
)t=1t=0
=
2
2..
C2
ds =
0 + 0 +
(dz
dt
)2dt = dt
C2
(x 3y2 + z
)ds =
10
(1 3 + t) dt =(2t+ t
2
2
)t=1t=0
= 2.
C
(x 3y2 + z
)ds =
C1
(x 3y2 + z
)ds+
C2
(x 3y2 + z
)ds =
2
2 3
2.
-
.
9. 112
9.2.8. - C : x2 + y2 = 1, y 0 (x, y) = 1 y.
. C x = cos t, y = sin t, t [0, ].
m =
C
(x, y) ds =
0
(1 y (t))
(dx
dt
)2+
(dy
dt
)2dt
=
(1 sin t)
sin2 t+ cos2 tdt = (t+ cos t)t=t=0 = 2
y0 =1
m
C
y (x, y) ds =1
2
0
sin t (1 sin t) dt
=1
2
( cos t t
2+
1
4sin 2t
)t=t=0
=4 2 4
.
x0 = 0 .
9.2.9. r (t) = i (t sin t) + j (1 cos t)(t [0, 2]) (x, y) = 1..
.
m =
C
(x, y) ds =
20
1
(dx
dt
)2+
(dy
dt
)2dt
=
20
(1 cos t)2 + sin2 tdt =
20
1 + cos2 t+ sin2 t 2 cos tdt
=
2
20
1 cos tdt =
2
20
sint
2dt = 8.
x0 =1
m
C
x (x, y) ds =1
8
20
(t sin t)
1 cos tdt
=1
8
20
(t sin t) sin t2dt =
8
8= .
y0 =1
m
C
y (x, y) ds ==1
8
20
(1 cos t)
1 cos tdt
=1
8
20
(1 cos t) sin t2dt =
32
8 3=
4
3.
(, 4/3).
-
.
9. 113
9.2.10. C r (t) = i cos 4t+ j sin 4t+ kt, t [0, 2]
.
m =
C
(x, y, z) ds =
20
1
(dx
dt
)2+
(dy
dt
)2+
(dz
dt
)2dt
=
20
16 sin2 4t+ 16 cos2 4t+ 1dt = 2
17.
z0 =1
m
C
z (x, y) ds =1
2
17
20
t
17dt = .
x0 = 0, y0 = 0 . (0, 0, )
9.2.11. Cy2dx+ xdy C x = 4 y2
(5,3) (0, 2).. C. C : x =
4 y2 dx = 2ydy
C
y2dx+ xdy =
C
y2 (2y) dy +(4 y2
)dy =
23
(2y3 + 4 y2
)dy
=
(y
4
2+ 4y y
3
3
)y=2y=3
=245
6.
9.2.12. C
F dr F = ix2 jxy C r (t) = i cos t+ j sin t, t [0, /2].
.
dr= (i sin t+ j cos t) dtF = i cos2 t j cos t sin t
C
Fdr = /2
0
( cos2 t sin t cos2 t sin t
)dt= 2
/20
cos2 td (cos t) =2
(cos3 t
3
)/2t=0
= 23.
9.2.13. Cydx + zdy + xdz C C
(2, 0, 0) (3, 4, 5).. C
x 23 2
=y 04 0
=z 05 0
-
.
9. 114
( t [0, 1]):
x = 2 + t, y = 4t, z = 5t,
dx = dt, dy = 4dt, dz = 5dt
C
ydx+zdy+xdz =
10
4tdt+5t4dt+(2 + t)5dt = 1
0
(4t+ 20t+ 10 + 5t) dt =
10
(10 + 29t) dt =49
2.
9.2.14. C
F dr F = i (y x2) + j (z y2) + k (x z2) C r (t) = it+ jt2 + kt3, t [0, 1].
.
dr=(i + j2t+ j3t2
)dt
F = i(t2 t2
)+ j(t3 t4
)+ k
(t t6
)
C
F dr= 1
0
((t3 t4
) 2t+
(t t6
) 3t2
)dt
=
(2t5
5 2t
6
6+
3t4
4 3t
9
9
)t=1t=0
=29
60.
9.2.15. C
F dr F = ixy + jyz + kzx C r (t) =it+ jt2 + kt3, t [0, 1].
.
dr=(i + j2t+ k3t2
)dt
F = it3 + jt5 + kt4
C
F dr = 1
0
(t3 + 2t6 + 3t6
)dt=
(t4
4+
5t7
7
)t=1t=0
=27
28.
9.2.16. C
F dr F = ix + jz + ky C r (t) =i cos t+ j sin t+ kt, t [0, /2].
.
dr= (i sin t+ j cos t+ k) dtF = i cos t+ jt+ k sin t
C
F dr = /2
0
( cos t sin t+ t cos t+ sin t) dt=(
cos2 t
2+ t sin t
)t=/2t=0
=
2 1
2.
-
.
9. 115
9.2.17. C
F dr, F = i (3 + 2xy) + j (x2 3y2), C.
. P = 3 + 2xy, Q = x2 3y2, Py = 2x = Qx C
F dr C.
9.2.18. C
F dr, F = i (ex cos y + yz)+j (xz ex sin y)+k (xy + z), C.
. (x, y, z) = F, .
x = ex cos y + yz, y = xz ex sin y, z = xy + z.
x x
=
xdx =
(ex cos y + yz) dx = ex cos y + xyz + g (y, z) .
y.
xz ex sin y = y = xz ex sin y + gy gy = 0.
y
g =
0dy = h (z)
= xyz + ex cos y + h (z) .
z
xy + z = z = xy +dh
dz dh
dz= z h (z) = z
2
2+ d.
(x, y, z) = xyz + ex cos y +z2
2+ d.
, = F C
F dr C.
9.2.19. C
F dr, F = i (3 + 2xy) + j (x2 3y2) C r (t) = iet sin t+ jet cos t, t [0, ].
. C
F dr . (x, y) F =.
i (3 + 2xy) + j(x2 3y2
)= ix + jy
.x = 3 + 2xy, y = x
2 3y2.
-
.
9. 116
(x, y) =
xdx =
(3 + 2xy) dx = 3x+ x2y + c (y) ,
c (y) x, y ( x). (x, y) = 3x+ x2y + c (y) y
x2 3y2 = y = x2 +dc
dy.
dc
dy= 3y2 c (y) = y3 + d.
(x, y) = 3x+ x2 y3 + d
C
F dr= t=t=0
dr = (x () , y ()) (x (0) , y (0))
=(
3et sin t+(et sin t
)2 (et cos t)3)t=t=0
= e3 + 1.
9.2.20. C
F dr, F = i (x y) + j (x 2), C.
. P = x y, Q = x 2, Py = 1 6= 1 = Qx C
F dr C.
9.2.21. C
F dr, F = iy2 + j (2xy + e3z) + k3ye3z, C.
. (x, y, z) = F, .
x = y2, y = 2xy + e
3z, z = 3ye3z.
x x
=
xdx =
y2dx = xy2 + g (y, z) .
y.
2xy + e3z = y = 2xy + gy gy = e3z.
y
g =
gydy = ye
3z + h (z)
-
.
9. 117
= xy2 + ye3z + h (z) .
z
3ye3z = z = 3ye3z +
dh
dz dh
dz= 0 h (z) = d.
(x, y, z) = xy2 + ye3z + d.
,
= iy2 + j(2xy + e3z
)+ k3ye3z = F
C
F dr C.
9.2.22. C
F dr, F = iyz + jxz + kxy C (1, 3, 9) (1, 6,4).
. F = = xyz. C
F dr= (1, 6,4) (1, 3, 9) = 1 6 (4) (1) 3 9 = 24 + 27 = 3.
9.2.23. C
F dr, F = iy + jx + k4 C (1, 1, 1) (2, 3,1).
. F = , = xy + 4z + c. C
F dr = (2, 3,1) (1, 1, 1) = 2