文档内容
例8.12 设 f (x)具有一阶连续的导数，则
(2,2)
xf (x2 + y2 )dx + yf (x2 + y2 )dy =( A ).
(0,0)
1 1
8 4
(A) f (x)dx (B) f (x)dx (C) f (8) − f (0) (D)0
2 0 2 0
Pfixtg
ax fixty / de
7B = +
35
=:
I c fix a
=
E
fitdt
=
fixax例8.13 设函数 f (x)在 （− ，+ ） 内具有一阶连续导数，L是上半平面
( y 0)内的有向光滑曲线，其起点为(a,b)，终点为(c,d)，记
1 x
I = [1 + y2 f (xy)]dx + [ y2 f (xy) − 1]dy，(1)证明曲线积分I 与路径
L y y2
I 无关；
= = 29 fIY)
EPNY) [HYfNxe] Quyl 1)
4 )
+
· = %+m - 17 + y y fixy) Y = f(x) - ( + xy fixy)
#
ETHY f(x)
+224f(xy)
y fixy)
x]
= + +
-
.
- f(xy) f(xy)
+ 2 Xy fry)
-
=
EA15
· 110(2)当ab = cd 时，求 I 的值.
CGd)
·O
x(z : X= 2
4 y b
/ "In : =
2
=
"
caib) (2
b)
y b X = 2 ,
=
+
= Hax f(y)
May
-
b Ifyay
fixax I
ca
+ + a
= -
= 12 fixb)axb 1 figdy +
+ +(2)当ab = cd 时，求 I 的值.
= In flaxb 1 fidy +
+ +
#
#
/ flat fat-
E
+
=
X-f
=
-
=例8.14 设为抛物面z = 9 − (x2 + y2)在 x o y
↑
面上方的部分，则
2x2 + 2 y2 + xy + z
X
dS =____________.
4x2 + 4 y2 + 1
x
: a
:_
X
,
x
44
+ E
wy
z
/ 2x +
T + I XOYFBD x y -9
: +
=
4 447
+
(x y z 2xzy
2 z = q - + = - - z
d) zx zaxay
= H+ + = 1+ 4
+
442 axay
2 x 92 9- x" 42.
TB 11 + 2 + -
:
= x e axay
+
xEym2 +< +g + 12 x 92 9- x" 42.
TB 11 + 2 + -
:
= x e axay
+
#442 1
=729 + +
x
Il
+ 9) axay
= (x +
x y 29
+
100% 102
97 pap
+
= .
=x2 y2 z2
例8.15 设 : x2 + y2 + z2 = R2 ，则 ( + + )dS = .
2 3 4
: EE E * d /yas = za
=
1 E 5 #) x as ----
= + +
5(i 4)((x y
zids
5
+ + + +
=
5 (I 5 q)(d)
+ +
=
= (i + R PART
)x4T
=↑
例8.16 计算I = x2 yzdx + (x2 + y2 )dy + (x + y + 1)dz，其中L为球面
L
x2 + y2 + z2 = 5与旋转曲面z = 1+ x2 + y2 的交线，从z轴看为逆时针方
I
向.
=
=
/ My S
35 :
-
=
-----
-
%X + yay
y 2 dx
PS1 I
. .
=
L
X
#L
3
=
-
O
//2x-ex T No
I (axay
: = = - I /
x=
3)例8.16 计算I = x2 yzdx + (x2 + y2 )dy + (x + y + 1)dz，其中L为球面
L
x2 + y2 + z2 = 5与旋转曲面z = 1+ x2 + y2 的交线，从z轴看为逆时针方
↑
向.
I , AAA .
- : =2
=
exey
axdz
agaz
/l
I
=
&
= *
07
z
x xtRx
yz +y +/ L
/
aydz xy-1)axdz (2x-xE)
= + + dy
11
/l (x-2x)aay -
(2x-2xYaxay
= =
=
x=y z例8.17 计算I = xzdxdy + xydydz + yzdzdx，其中是平面 x + y + z = 1
↑
在第一卦限部分的上侧．
GJE AB ..
# 2 .: X = 0 . [2 = Y =0 Is = z = 0 177
, , ,
P
11
1 xzaxay xyaydz yzdzax -
+ +
1)
10 y
1
[
+2,+22+2 ,
D
y
(((x
+ y + z/m o - 0 - 0 Did) x+
-
=
X
M
*
way
3 l =
xar yaz x(x-y)
/I ayi
=
=
*
3(ax)" 5
= x(+ x y(dy
- - =例8.18 计算曲面积分I = (8 y + 1)xdydz + 2(1 − y2 )dzdx − 4 yzdxdy，其
z = y − 1
中是有向曲线 ,(1 y 3)绕 y轴旋转一周所成的曲面，它
x = 0
的法向量与 y轴正向夹角恒大于 .
2
347↑
例8.19
计算[ f (x, y, z) + x]dydz + [2 f (x, y, z) + y]dzdx + [ f (x, y, z) + z]dxdy
其中 f (x, y, z)为连续函数 是平面 x − y + z = 1在第四卦限部分的上侧.
↑
# : z = x - y + z = = z = 1- x + Y
-Ex
dxay axar
Mydz
·= =
=
11
= (X-7 + z) axaly axay = 1 axay
=
