文档内容
第八章
数一专题例8.1 设有三张不同平面的方程a x + a y + a z = b ,i = 1,2,3,它们所组
i1 i2 i3 i
成的线性方程组的系数矩阵与增广矩阵的秩都为 2，则这三张平面可
: GREE
rC1 3
能的位置关系为 ( B ).r C) = = 2
RIYE FigRE F
F
VIA U(A) = 3 r(A) = 2 < r(A) = 3 r(A) = (A)
=
"例8.2 设函数 f (x, y)在点(0,0)附近有定义，且 f I(0,0) = 3, f I(0,0) = 1，
x y
则（ C ）.
X
（A）dz | = 3dx + dy #7 AidBR
(0,0)
. .
X （B）曲面z = f (x, y)在点(0,0, f (0,0))的法向量为{3,1,1}
z = f (x, y)
（C）曲线 在点(0,0, f (0,0))的切向量为{1,0,3}
y = 0
z = f (x, y)
X
（D）曲线 在点(0,0, f (0,0))的切向量为{3,0,1}
y = 0
/2 fNxY)
(B) E) fIXY ) > =0 FIXYiz) = - z
- ,
↑ (0 (fx
0 fro) = 10 , 0) fylad - 1) = 3 . 1 . - 1)
. · ,(DIGEfi) ↑ Y
(P
.
f(x
z d)
= ,
10 0 fixd) In 5 = (1 . 0 , fx10 . %) = (1 . 0 . 3)
. ·例8.3 设有一小山，取它的底面所在的平面为 xoy坐标面，其底部所占
的区域为D = (x, y) x2 + y2 − xy 75 ，小山的高度函数为
h(x, y) = 75 − x2 − y2 + xy.（1）设M(x , y )为区域
0 0
D 上的一点，问
h(x, y)在该点沿平面上什么方向的方向导数最大？若记此方向导数的
最大值为 g(x , y )，试写出 g(x , y )表达式.
0 0 0 0
F
(1) h(X) T M(Xo 40)
. M
0
Y
- >
h'x hy(m .
grad = (
=
( 2y + y -24+ x)
, ,
m - --
:
&
M
240tX0) L
32 (-2x0 + 40 *. ↓
, = y
x xy 75
- =
E 9N0 Yol (NotHol 1240x01
= +
·（2）现欲利用此小山开展攀岩活动，为此需要在山脚寻找一上山坡度
最大的点作为攀登的起点.也就是说，要在 D 的边界线 x2 + y2 − xy = 75
上找出使（1）中的 g(x, y)达到最大值的点.试确定攀登起点的位置.
EPcTiYE X + Y - xY = 75 ElF7 FE91X Y) = ( 2x + y)" + 72y + Xi
, .
M
& FI Y x) ( - 2x + y( + ( 2 + x + x(x + y - x4 - 75)0 - > Y
=
. . .
- --
:
Fx = - x( -2x+ y) + 2( y + x) + 2xx - xy = 00 & M
L
I ↓
=
y
#y x - xy = 75
= 2) - 2x + y) - 4( 24 + x) + 2xy - xX = 00
y ③
Fx x + xy - 15 = 0
= -Fx = - x( -2 x+ y) + 2( y + x) + 2xx - xy = 00
I
#y
y) 4( 24 x) 2xy
= 2) - 2x + - + + - xX = 02
Fx x + y xy - 15 = 0 ③
= -
0 + 0 = - 2( - 2x + y) - 2( - 2y + x) + xX + xy = 0
=> 2X + 2 + xX + xy = 0
9
=> (x+ y) 2 + (x+y) x = r
=> (x+ Y) (x + 2) =0
casel x= -
Y At St x0 =
9554&
X =- J
.
5
=EX -2 At Q
cusez =
.
X-Y
=
Six
- SEESE ST
⑮50
9 55 55)
9) 55 55) 0 - - =
= .
.
9) 15
, -
5)
=
Eso
,
91 -5
.
5)
= 50
+E -5) * (- 5) E
(5 5
: .
.↑
例8.4 设 f (x)在 x = 0点处可导，且 f (0) = 0, f (0) = 1,
1
: x2 + y2 + z2 t2 ,则lim f ( x2 + y2 + z2 )dxdydz =_________.
+
t4
t→0
/
D F
35- :
,
=
↓If y = fast
a
...
L
(5 1 51 E M
. .
M
=
t
do"
an
33 11f(* firs risure
zm
: +y = a
+ .
!
2/smeay far /
firrar
=
=far
Mr
T
: =
to
[4 -B
f
Im
fi
=例8.5 设 = {( x, y, z) | z x2 + y2 3z,0 z 4},计算 zdxdydz.
/odz/1 Z
Tis z axay
= /
=
Y23z
② X
x
12 (7-3E Tzdz
- -
'Y
O
L
( . 22 X
dz
=
12872
=-
例8.6 设曲线L为球面 x2 + y2 + z2 = 1与平面 x + y + z = 0的交线，则
(xy + yz + zx)ds = ( )
A
L
(A) − (B) (C) − (D) 2.
2
z) x = y z
(x + y + = + + + 2xy + 242 + 2x2
zi (8H+ 23 "
(x+y + - ;
: Xy yz + zX
+ =
cir
2 d)
,
zP (x Y z] us
~ TB 6 (xTY + - + + P Eas
= =
-
, .
2
E
2 7
= . = -
-例8.7 L为由直线 y = x及抛物线 y = x2 所围成的区域的整个边界，则
xds =( ).
L
1 1
(A) (5 5 + 6 2 + 1) (B) (5 5 + 6 2 − 1)
12 12
1 1
(C) (5 5 − 6 2 − 1) (D) (−5 5 + 6 2 − 1)
12 12
↑
( / (1)
TB xas
xas +
- =
x
4 y
, , -: =
n =x=(Ly
↳: Y X 0 : X = 1 ds = Hyax = Edx
=
.
E
(0
0)
.
↳ = = x o = X = /di = 14* &X
(bx
/!X
TB +
= Exx + nxax = (555 + Gr - 1)
.例8.8 设质点在变力F = (3x + 4 y)i + (7x − y)j的作用下，沿椭圆
ax2 + y2 = 4的逆时针方向运动一周所做的功等于 6 ，则a = ________
E
axtY
32L 4 3
: = 1
=
, [
# FR
7
G
W (3x+ 4y)ax +(7X y)ay 67
- =
=
,
1 & Plaxay 3 .. =
# W 2
=
= 6
,
+
.: G = 4例8.9 计算 y2 x2 + y2dx，其中 L为圆周 x2 + y2 = a2 上由点
L
a a a a
B( , )按逆时针方向到点 A(− , )的一段弧.
2 2 2 2
B
a)
T ) 7 42ax At t
xynax
= =
, ,
&
/xiax
T Gat
35- : = a = -
Iz
I
S
x y a
33 == ( : + = E
37
T Cashtdacost-
:
=例8.10 计算I = y3dx + 3 y − x2 dy，其中 L是正向圆周 x2 + y2 = 4.
L↑
例8.11 设在上半平面D = {(x, y) | y 0}内，函数 f (x, y)具有连续偏导
数，且对任意t 0都有 f (tx,ty) = t−2 f (x, y)，证明：对D内任意分段光
滑的有向简单闭曲线 L，都有 yf (x, y)dx − xf (x, y)dy = 0.
L
& PNMl = Y fixm) Q(X - Y) = - X fix)
xf (M) fy) yfy
Ex f(x)
- = +
= -
-flex tH) t fixM) FITsi
=
,
,
fictx fi(X E fNxM)
: , +y) · X + . + y) Y = z . / ==
,
fi(x fr fixy)
↓
.
ul
.
x+ (x 41
.
Y
=
-2
: fix . + fix . 41 . 4 = - fix , Y) - fiN71 X
.:
T = - = 0
=
