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(104013) 2

:2014 28

. " , !!

. " //

1

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6 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( ) 2.311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( ) 2.413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a

b c

2.514 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.316 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.417 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.319 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.419 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.320 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.120 . . . . . . . . . . . . . . . . . . . . . . . . . . ) ( 6.221 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

22 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.122 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.424 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.525 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.625 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.927 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1027 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11

28 Rn 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . () 9

2

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31 33 Rk 34 36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

38 41 41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1343 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.143 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f : R2 R 1444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1550 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ) ) 1652 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1755 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1857 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2063 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2167 . . . . . . . . . . . . . . . . . . . . . . . . . . 2268 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.169 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2369 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.169 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.270 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4

72 72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' 1 " 2472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2574 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2674 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.280 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.382 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2882 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' 1 " 28.184 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(x, y)

(u, v) 28.2

85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.385 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.1

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85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.287 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.388 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28.3.489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3.590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.191 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R3 29.292 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.3

96 96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3096 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.196 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.1.197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3297 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.398 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.499 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( ) 3399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.199 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3.1100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3.2101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.1101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.2101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.3103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.4103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.5104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R2 35108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.1108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.2

110 110 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37112 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.1113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.2114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . () 40118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.1119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . / 40.2123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1124 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2124 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2

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127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .R3 42131 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1131 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.1132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.1135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F = r

r3 43.3

139 139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.0.1139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.0.2140 . . . . . . . . . . . . . . . . . . . . . . . . . . . " " 44.0.3140 . . . . . . . . . . . . . . . . . . . . . . 44.0.4142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.1142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.1.1143 . . . . . . . . . . . . . . . . . . . . . 44.1.2144 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' 44.2.1148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' 44.2.2

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1

1

. () , , :. () , , :

a = a= a = a =

AB

. .|a| :

a =

AB :1

. " a= b . ,0 .0 ) (

.1 ) (. ( (

. ( ( 1.1

: . x, y, z ,i j, k ) ).x, y, z

:i j, j k, k ii

=j

=k

= 1

: R3 ( )

a= a1i + a2j+ a3k= (a1, a2, a3)

6

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2

:2

) )|a| =

a21+ a

22+ a

23

: .a= (a1, a2, a3) ,b= (b1, b2, b3) ) ( R a= (a1, a2, a3) .1

a + b= (a1+ b1, a2+ b2, a3+ b3) .2

.i

{1, 2, 3

} ai= bi

a,b .3

2 2.1

.|m| |a| ,(a m < 0 .m > 0 )a m a

:3

.a .1 a 1|a| a a :

7

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2.2 2

:4

: .a= (a1, a2, a3) ,b= (b1, b2, b3) = 0 : ai= bi b a

2.2). )

:5

: a + b = |a|2 + b2 2 |a| b cos :

a b a +b

:a + b= b + a .1

.2a +

b + c

=

a + b

+ c ()m (na) = n (ma) = (mn)a ()

.3(m + n) a= ma + na ()

m

a + b

=ma + mb ()

8

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( ) 2.3 2

: .a= (a1, a2, a3) ,b= (b1, b2, b3) a + b= (a1+ b1, a2+ b2, a3+ b3) .1

.B (0, 1, 5) A (1, 2, 1) ::

OA + AB= OB

AB= OB OA = (0, 1, 5) (1, 2, 1) = (1, 1, 6)

,b cos a a b \) ) . :

B A :6

:7

( ) 2.3a b= |a|

b cos :

9

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( ) 2.3 2

. .1.a b a .2

. .3.a b= 0 .4

) ( a b |a| b|cos | 1 :

a b= b a .1

a b + c = a b + a c .2a

mb

= (ma) b= m

a b .3

a a= a2 = |a|2 .4: .5

i j =j k= k i= 0i i=j j = k k= 1

) ( a + b |a| + b:

a + b2 = a + b a + b = |a|2 + 2a b + b2 |a|2 + 2 |a|

b

+b2

=|a| +

b

2

: .a= (a1, a2, a3) ,b= (b1, b2, b3) a b= a1b1+ a2b2+ a3b3

10

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( ) 2.4 2

( ) 2.4

c= a b

.|c| = |a|b sin :c

. ,a,b :c

:8

:. .1

.a,b a b .2: .3

i i= i i= i i= 0i j = k, j k= i, k i=j

j i= k, k j = i, i k= j

:9

11

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( ) 2.4 2

a b= b a .1

a

b + c

= a b + a c .2

m

a b = (ma) b= a mb .3a

b c

=

a b

c : .4

:10

a b c = (a c)b b ca : .a= (a1, a2, a3) ,b= (b1, b2, b3)

a b=

i j ka1 a2 a3b1 b2 b3

= (a2b3 a3b2) i(a1b3 a3b1) j+ (a1b2 a2b1) k

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a

b c 2.5 2

A (1, 2, 1) , B (0, 1, 5) , C(1, 2, 1) : ABC : AC AB

AB= (1, 1, 6) , AC= (2, 0, 2)

AB AC=

i j k1 1 62 0 2

= (2, 10, 2)SABC=

1

2

AB AC =12

4 + 100 + 4 =

27

a b c

2.5

a

b c

=

a1 a2 a3b1 b2 b3c1 c2 c3

.1a

b c

= c

a b

= b (c a) .2

.a,b, c a b c .3.a

b c

= 0 ,a,b c .4

:11

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3

3

3.1. N= (A ,B,C) M0(x0, y0, z0)

.M(x,y,z) . MM0 = (x x0, y y0, z z0)

.MM0 N = 0 : .MM0 , , N:

A (x x0) + B (y y0) + C(z z0) = 0

(). N M0 D= Ax0 By0 Cz0 :

Ax + By + Cz + D= 0

:12

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3.2 3

Q (1, 2, 3) , R (1, 1, 1) , S(1, 2, 0) :

QR= (0,

1,

2)

QS= (2, 0, 3)

N=QR QS=

i j k0 1 2

2 0 3

= (3, 4, 2)3 (x 1) 4 (y 2) + 2 (z 3) = 0

3x 4y 2z+ 5 = 0

: .xy ,k

N= (A,B, 0)

C= 0 .1

.xy ,k N= (0, 0, C) A = B = 0 .2

A = B = 0 :z )( C = 0 :z )( :13

3.2. Ax + By + Cz + D= 0 , ,M0(x0 y0, z0)

. M1(x1, y1, z1) , N M1M0 d

d=M1M0 cos =

M1M0 N N

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3.3 3

:d M1M0 = (x0 x1, y0 y1, z0 z1)

d=|A (x0 x1) + B (x0 x1) + C(x0 x1)|

A2 + B2 + C2

: ,D= Ax0 By0 Cz0 d=

|Ax0+ Bx0+ Cx0+ D|A2 + B2 + C2

:14

3.3.N1 = (A1, B1, C1) N1= (A2, B2, C2) " A1x + B1y + C1z + D1 = 0 A2x + B2y + C2z + D2 = 0

.A1A2

=B1B2

=C1C2

3.4

cos =

N1 N2 N1 N2 = |A1A2+ B1B2+ C1C2|A21+ B

21 + C

21

A22+ B

22 + C

22

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4

:15

4 4.1

.a M0 . M0(x0, y0, z0) , a= (,m,n) = 0 M0M=a a M0M M(x,y,z)

.

x x0 = y y0 = mz z0 = n

: M0M= (x x0, y y0, z z0)

.(x0, y0, z0) (,m,n) .

x= x0+

y= y0+ m

z= z0+ n

:. () = (x0+ ,y0+ m,z0+ n) :

. . x x0

= y y0m

= z z0n

:.x x0

=

z z0n

, y= y0 : ,m= 0 .,m,n = 0 ). ) :

M0 :16

17

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4.2 4

.a= (3, 1, (2 (1, 2, 3) :

.

x= 1 + 3

y= 2

z= 3 + 2.x 1

3 =

y 21 =

z+ 3

2 :

4.2.x + 1

2 =

y 21 =

z 13 M(1, 1, 3)

:: .M0M , a d

d= M0M a|a| =

i j k2 3 22 1 3

4 + 1 + 9=|(7, 2, 4)|

14=

=

49 + 4 + 16

14=

69

14

4.3.x + 2y z = 5 () = (1 + , 2, 3)

: , (x,y,z)

x + 2y z= 5x= 1 +

y= 2

z= 3

(x,y,z) = (3, 4, 6)

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4.4 5

:17

4.4.3x + 2y z+ 2 = 0 x + y 3z = 0

. , N1 = (3, 2, 1) ; N2 = (1, 1, 3)

a= N1 N2 =

i j k1 1 33 2 1

= (5, 10, 5)

: , (x,y,z) , . 3x + 2y z+ 2 = 0x + y 3z = 0

, ,( , ( .M(4/5,11/5, 0) : .y x .z = 0

: M(4/5,11/5, 0) a= (5, 10, 5) x 4/5

5 =

y+ 11/5

10

=z

5

. :

5. x x0

=

y y0m

=z z0

n , Ax + By + Cz + D= 0

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5.1 6

a= (,m,n) , N= (A ,B,C) : 5.1

A + Bm + Cn = 0 a N= (,m,n) (A ,B,C) = 0 a N

5.2: .

sin = a N|a| |N|

= A + Bm + Cn

2 + m2 + n2

A2 + B2 + C2

5.3A + Bm + Cn

2 + m2 + n2

A2 + B2 + C2 = 1 /2 a N

6. x x1

1=

y y1m1

=z z1

n1,x x0

=

y y0m

=z z0

n

a1 = (1, m1, n1),a= (,m,n) : 6.1

:a, a1 cos =

a a1|a| | a1|

. \ \ ) ( 6.2

.L0 M0 L1 M1 a, a1, M0M1 M0M1 (a a1) = 0

x1 x0 y1 y0 z1 z0 m n

1 m1 n1

= 0

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7

:18

7b= c a b= a c

a

b c

a

b c

= 0

.1

(a b "): , a b .2

a b c+ c a b+ b (c a) = 0. .( )

: .3a b2 + a b2 = |a|2 b2

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8

.F(x,y,z) = 0

: 8 8.1

z =x2

a2+

y2

b2

). ( z = 0 . z= h 0 .zy x= 0 .xz y= 0

:19

8.2z2 =

x2

a2 +

y2

b2

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8.3 8

:20

8.3x2

a2+

y2

b2 +

z2

c2 = 1

:21

8.4x2

a2+

y2

b2 z

2

c2 = 1

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8.5 8

:22

8.5x2

a2 +

y2

b2 z

2

c2 = 1

:23

24

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8.6 8

:24

8.6(x a)2 + (y b)2 + (z c)2 =r2

.r (a,b,c)

:25

8.7: , (x,y,z) ,

F(x, y) = 0

F(y, z) = 0

F(z, x) = 0

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8.8 8

8.8x2

a2+

y2

b2 = 1

:26

8.9x2 = 2py OR y2 = 2px

:27

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8.10 8

8.10z =

x2

a2 y

2

b2

:28

8.11'. x x0 x : .1

.b a : .2. ): ( .3

27

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Rn Rn = {x= (x1, x2, , xn) : xi R}

). ) Rn : Rn

: .1x y= x1y1+ x2y2+ + xnyn

). R3 (:): (Rn .2

|x| = x x= x21+ x22+ + x2n): (Rn .3

d (x, y) = |x y| =

(x1 y1)2 + (x2 y2)2 + + (xn yn)2

: .4cos =

x y

|x| |y|

. n " Rn: "

:= .1|x + y| |x| + |y|

:CS .2

|x

y| |

x| |

y|

B (xo, r) = {x Rn | d (x, xo)< r}

.xo = (xo1, xo2, , xon) r

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.R b1, b2, , bn a1, a2, , an {x Rn | i ai xi bi}

.R

n n : ,B (xo, r) xo

{x | d (x, xo)< }

.{x | |x x0| < } R.

(x, y) |

(x x0)2 + (y y0)2 <

R2

. :. , :

. A Rn

.A xo A xo .1

.A ,A xo A xo .2. A , A xo .3

. ,A A .4

.A0 A .1.A A .2

.A0 A = A A .3:

R3 A= B (o, r) = x R3 | d (o, r)< rA = {x | d (x, o) = r}A= {x | d (x, o) r}

A0 =A

:A=

(x, y) R2 | y= x2

A0 =

A = A

A= A

5/23/2018 - 2

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) ) 9

:A=

(x, y) R2 | 1< x2 + y2 4

A0 =

(x, y) R2 | 1< x2 + y2

5/23/2018 - 2

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.f : Rn

Rm

. f : R2 R . ) ) ,f : R2 R

z= f(x, y)

f(x, y) = x2 + y2

. , :.z= x2 ,y= 0

.c z= f(x, y) xy .f(x, y) = xy :

xy= 0 x= 0OR y= 0xy= 1 y= 1

x

xy= 2 y= 2x

xy= 1 y =1x

xy= 2 y =2x

z = xy :29

, x = y .z = x2 y2 () . =x y

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.c S f(x,y,z) ) ( R3 S f(x,y,z) = x2 + y2 z :

c= 0 z = x2 + y2 c= 1 z 1 = x

2

+ y

2

c= 1 z+ 1 =x2 + y2

32

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Rk .

|xn

L

|< n > N N >0 xn

n

L R . xn

.d (xn, L)< n > N N >0 xn n

L : :

{xn}n=1 = {(xn1 , xn2 , , xnk)}n=1L = (L1, L2, , Lk)

.xni n Li 1 i k xn n L : ,R2 , :

xn L1 AND yn L2 (xn, yn) (L1, L2)

. > 0 :|xn L1|

(xn L1)2 + (yn L2)2 <

. > 0 :d ((x

n, yn

) , L) = (xn L1)2 + (yn L2)2 |xn L1| + |yn L2| <

2+

2<

: .Rk R . ( ) (Rk) ():

:.A ,A " (A ) A .1

A " ) ) A .2.A

33

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.

|f(x)

L

|<

= 00 >0 lim

xa

f(x) = L (R) .(a, b) f : R2 R

= 0 0 > 0 lim(x,y)(a,b)

f(x, y) = L .|f(x, y) L| <

:.0< d (x, a)< 0 0 > 0 lim(x,y)(a,b)

f(x, y) = L .|f(x, y) L| < = (x, y) = (a, b)

: .3: (a, b) = (xn, yn)

n (a, b) " lim

(x,y)(a,b)f(x, y) = L

.f(xn, yn)n L

. :

.f(x, y) =

x sin 1y + y sin

1x x = 0, y= 0

0 x= 0OR y= 0 1

: |y 0| < |x 0| < ,= /2 . >0 . lim(x,y)(0,0)

f(x, y) = 0

|f(x, y)

0

| x sin1

y

+ y sin1

x

x sin1

y + y sin1

x

|x| + |y| = |x 0| + |y 0| < /2+ /2=

*|sin| 1 **

() L lim(x,y)(a,b)

f(x, y) = L .(a, b)

34

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. , (a, b) (x, y) f(x, y) :. lim(x,y)(0,0)

f(x, y) .f(x, y) =

xyx2+y2 (x, y) = (0, 0)0 (0, 0)

2 : ) ) y = mx

f(x, y) = f(x,mx) = x mx

x2 + (mx)2 =

m

1 + m2

,y = x (0, 0) : , , . 8

1+82

=89 y= x8 , 11+12 = 12

!

)( )(

f(x, y) = xyx

2+y2 (x, y) = (0, 0)0 (0, 0)

:30

. lim(x,y)(0,0)

f(x, y) .f(x, y) =

x2yx4+y2 x = 0, y= 00 x= 0OR y = 0

3 : y= mx

f(x, y) = f(x,mx) = x2 mxx4 + (mx)

2 = mx

x2 + m2x0

0

.(0 ): y= x2 , !!

f(x, y) = f

x, x2

= x4

x4 + x4 =

1

2 1

2

. ( ) : . lim

(x,y)(a,b)g (x, y) = K lim

(x,y)(a,b)f(x, y) = L ()

35

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11

lim(f+ g) = L + K

lim f g= LK

(K= 0 g (x, y) = 0 ) limfg

= L

K

: ,(a, b) h (x, y) f(x, y) g (x, y) (') lim

(x,y)(a,b)h (x, y) = lim

(x,y)(a,b)g (x, y) = L

. lim(x,y)(a,b)

f(x, y) = L

10limyb

limxa

f(x, y)

! . ,

. , , . , ,(0 (0 , 2

.(0, (0 , 1 . , lim

(x,y)(a,b)f(x, y)

11:

x= r cos y= r sin

ORr= x

2 + y2

= arctan yx

:31

36

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11

: .f(x, y) : R2 R f(r cos , r sin ) = F(r) G () .1

G () .2F(r) r0 0 .3

.f(x, y) (x,y)(0,0)

0 :

. > 0

|f(x, y) 0| = |F(r)| |G ()| < M

M=

.G ()< M M , G. r |F(r)| < /M , F

.f(x, y) =

x2y2

x2+y2 (x, y) = (0, 0)0 (0, 0)

:

f(r cos , r sin ) = r2 cos2 r2 sin2

r2 =r2 cos2 sin2

: F(r) = r2, G () = cos2 sin2 f(r cos , r sin ) = F(r) G () G ()

F(r) r0

0

. lim(x,y)(0,0)

f(x, y) = 0 :

. .1.r 0 ' 2 , .2

37

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. lim

(x,y)(a,b)f(x, y) = f(a, b) (a, b) f : R2

R

: >0 >0 .1

|f(x, y) f(a, b)| < =

(x a)2 + (y b)2 <

.|y b| < |x a| < , ,1 .2.f(xn, yn)

nf(a, b) (xn, yn)

n (a, b) .3

. D f

.f(x, y) = sin xsin y

xy x

=y

cos x x= y ). ) f , x =y

.(xn, yn)n (x, x) . .(x, x)

: xn= yn f(xn, yn) = f(xn, xn) = cos xn

n cos x= f(x, x)

: xn=yn , f(xn, yn) =

sin xn sin ynxn yn =

sin xnyn2 cosxn+yn

2xnyn

2

n cos

x= f(x, x)

. f (x, x) f :

. f (). D f :D R ( ( f (). D f : D R ( (

.). (t) ) [a, b] x (t) , y (t) ) )

.t [a, b] (x (t) , y (t)) () D R2 . f :D R

). f ) . (t) = f(x (t) , y (t)) .[a, b]

R R2 f R

t (x (t) , y (t)) f(x (t) , y (t))

38

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.(x1, y1) , (x2, y2) D . f :D R , () D R2 ) ( .f(x0, y0) = z0 (x0, y0) D f(x1, y1) f(x2, y2) z0

:). D ) D (x1, y1) , (x2, y2)

.a

t

b (t) = (x (t) , y (t))

(x1, y1) = (a) = (x (a) , y (a))

(x2, y2) = (b) = (x (b) , y (b))

:

f(x1, y1) = f(x (a) , y (a)) = (a)

f(x2, y2) = f(x (b) , y (b)) = (b)

1') " , ) .( ( [a, b] . (t0) = z0 a < t0 < b

z0 = (t0) = f(x (t0) , y (t0))

.f(x0, y0) = z0 (x0, y0) D .x0 = x (t0) , y0 = y (t0) :

39

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.f(x, y) =

x2y2x2+y2 (x, y) = (0, 0)0 (0, 0)

. /2

f(P1) = f(2/3, 0) = 1

f(P2) = f(0, 2/3) = 1

.1 1 f ,( ) f.(0, (0 f (0, 0)

:32

40

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12

12 12.1

x= x (t)

y= y (t) a t bz= z (t)

.1

r (t) = (x (t) , y (t) , z (t)) .2

r (t) = x (t) i + y (t) j+ z (t) k .3

r (t) = (cos t, sin t) , 0 t 2 .1

. r (t) = (cos t, sin t, t) , 0 t .2

.( ) r (t) =

t, t2

, 0 t 1 .3

.y= x2 r (t) =

sin t, sin2 t

, 0 t /2 .4

. , r (t) =

cos t, cos2 t

, 0 t /2 .5

. , r (t) = (t sin t, 1 cos t) , 0 t 2 .6

. r (t) =

cos3 t, sin3 t

, 0 t 2 .7

. .x2/3 + y2/3 = 1

! .1.R2 (t, f(t)) .2

. .3. :

. , , , : :

41

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13

13. ,'1 "

. ( ) ,

r= r (t) r (t0)

. r

t =

r (t) r (t0)t t0

.

=x (t) x (t0)

t t0 i +y (t) y (t0)

t t0j+

z (t) z (t0)t t0 k

t0x(t) + y(t) + z(t) = r(t)

0). ) . (r(t0 ,

:: .1

r(t0). |(|r(t0

. , = r(t0)|r(t0)| .2. t0 , .r(t0) = 0 t0 t0 : .3

.r(t) r(t0) .4.t r(t) = 0

:.r (t) = (t, t) , 1 t 1 .1

. .r(t) = (1, 1).r (t) =

t3, t3

, 1 t 1 .2

: , ! t= 0 ,r(t) = 3t

2, 3t2." " .r (t) = (t, |t|) , 1 t 1 .3

.r(t) =

1, t|t|

=

(1, 1) t > 0

(1, 1) t > 0). , ("", t= 0

5/23/2018 - 2

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13.1 f : R2 R 14

13.1.y= f(x0) (x x0) + f(x0) ,'1 "

: r (t0) r(t) = 0 : (t) = r(t0) (t

t0) + r (t0)

: ,r(t) = (1, f(t)) r (t) = (t, f(t)) , (t) =

x= 1 (t t0) + t0 = ty= f(t0) (t t0) + f(t0)

: .t= t0 , x x (t0)

x(t0) =

y y (t0)y(t0)

=z z (t0)

z(t0)

: (x (t0) + t x(t0) , y (t0) + t y(t0) , z (t0) + t z(t0))

f : R2 R 14:f : R R

f(x0) = limh0

f(x0+ h) f(x0)h

.(x0, y0) , f(x, y) ,x0 x

.x= x0 : 1', "

f

y(x0, y0) = lim

h0f(x0, y0+ h) f(x0, y0)

h

: (x0, y0) ,y0 y f

x(x0, y0) = lim

h0f(x0+ h, y0) f(x0, y0)

h

.(x0, y0) (y x) f . , ,(x0 y0)

:.f : R3 R .1

: .2/. ,

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14.1 f : R2 R 14

:.f(x, y) =

xyx2+y2 (x, y) = (0, 0)0 (x, y) = (0, 0)

fx (0, 0) = limh0 f(h, 0) f(0, 0)h = 0

: f

y(0, 0) = 0

:(x0, y0) = (0, 0) x f

x(x0, y0) =

y

x2 + y2 2x (xy)

(x2 + y2)2

(x0,y0)

= y30 x20y0(x20+ y

20)

2

.fy

(x0, y0) .(0, (0 :

. : 14.1

:A =

f

x(x0, y0)

B = f

y(x0, y0)

z0 = f(x0, y0)

f (x0, y0, z0) :y= y0 x= x0

:1

x= x0

z= B (y y0) + z0

x= x0zz0B =y y0

a1 = (0, 1, B)

:2

y= y0

z = A (x x0) + z0

y= y0zz0A =x x0

a2 = (1, 0, A)

: , N=

i j k0 1 B1 0 A

= (A,B, 1)

5/23/2018 - 2

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15

: (x0, y0, f(x0, y0))

A (x x0) + B (y y0) 1 (z f(x0, y0)) = 0

:z= f(x0, y0) +

f

x(x0, y0) (x x0) + f

y(x0, y0) (y y0)

.

15 ( )

f(x0) = limh0

f(x0+ h) f(x0)h

,f(x0+ h) f(x0) = Ah+ (h) h A R x0 f(x) . (h)

h00

::

f(x0+ h) f(x0)h

=A + (h) h0

A

.f(x0) = A , f :

: .A= f(x0) f(x0+ h) f(x0)

h

h0A

f(x0+ h) f(x0)h

A h0

0

: , (h) = f(x0+h)f(x0)h A (h)

h= f(x0+ h)

f(x0)

Ah

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15

.(x0, y0) f(x, y) : ,A B R (x0, y0) f

f(x0+ h, y0+ k) f(x0, y0) = Ah + Bk + (h, k)

h2 + k2

(h, k) (h,k)(0,0)

0 :

f=f(x0+ h, y0+ k) f(x0, y0)x= h

y= k

f=Ah + Bk + (h, k) h + (h, k) k

.(h, k) (h,k)(0,0)

0 , (h, k) (h,k)(0,0)

0

.(x0, y0) " (x0, y0) f(x, y) :

: k= 0 f(x0+ h, y0) f(x0, y0) = Ah + (h, k) h

:h f(x0+ h, y0) f(x0, y0)

h =A + (h, k)

h0A

: f

x(x0, y0) = A

f

y(x0, y0) = B

46

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15

.(0, (0 .f(x, y) =

x2y

(x2+y2)1/2

(x, y) = (0, 0)0 (0, 0)

: :

f

x(0, 0) = lim

h0f(h, 0) f(0, 0)

h = 0

f

y(0, 0) = lim

h0f(0, h) f(0, 0)

h = 0

: ,A B :

f(h, k) f(0, 0) = (h, k)

h2 + k2

: 0 (h, k) :

(h, k) =f(h, k) f(0, 0)

h

2 + k2 =

h2k

h2 + k2

:

(r cos , r sin ) = r2 cos2 r sin

r2 =r cos2 sin

. 0 : ,r 0 , cos2 sin .(x0, y0) (x0, y0) f

:

lim(h,k)(0,0)

(f(x0+ h, y0+ k) f(x0, y0)) =

= lim(h,k)(0,0)

Ah + Bk + (h, k)

h2 + k2

= 0

: f(x0+ h, y0+ k)

(h,k)(0,0)f(x0, y0)

.(x0, y0) f

47

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15

(0, (0 ,(0, (0 f(x, y) =

xyx2+y2 (x, y) = (0, 0)0 (0, 0)

,.((0, (0 " )

.(x0, y0) f (x0, y0) " f(x, y)

:33

: ! : .f(x, y) =

x2 + y2

sin 1

x2+y2(x, y) = (0, 0)

0 (0, 0): " .1

.f(x, y) = |x| : .2.f(x, y) = |x|:" .3

.f(x, y) =

xyx2+y2 (x, y) = (0, 0)0 (0, 0)

: " .4

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15

:

f=f(x0+ h, y0+ k) f(x0, y0) == (f(x0+ h,y0+ k)

f(x0,y0+ k)) + (f(x0, y0+ k)

f(x0, y0)) =

. ( ) : '

=f

x(x0+ th,y0+ k) h + f

y(x0, y0+ sk) k =

: ,( )

(h, k) =f

x

(x0+ th,y0+ k)

f

x

(x0, y0)

(h,k)(0,0)0

(h, k) =f

y(x0, y0+ sk) f

y(x0, y0)

(h,k)(0,0)0

:

=f

x(x0, y0) h + (h, k) h + f

y(x0, y0) k+ (h, k) k

: x0 f ,f : Rn R f=

ni=1

Aixi+ni=1

ixi

.i, i (x1, ,xn)0

0 ,Ai= fxi

x0

: P0 S P0 .1

.P0 P 0 P P0 .2 (z ) z= f(x, y)

.(x0, y0) f P0 = (x0, y0, f(x0, y0))

5/23/2018 - 2

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) ) 16

) ) 16:

R

R2

R

t (x (t) , y (t)) f(x (t) , y (t))

.(f C1) D " f(x, y) .(x (t) , y (t)) D t I I x (t) , y (t)

: .F(t) = f(x (t) , y (t)) dF

dt =

f

x

dx

dt +

f

y

dy

dt

: :

f(x, y) = x2y y2x (t) = t2

y (t) = 2t

: dF

dt = (2x (t) y (t))(2t) +

(x (t))

2 2y (t)

2 == 8t4 + 2t4 8t= 10t4 8t

50

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) ) 16

:dF

dt = lim

t0F(t + t) F(t)

t =

= limt0 f(x (t + t) , y (t + t)) f(x (t) , y (t))t =

:x = x (t + t) x (t)y = y (t + t) y (t)

:= lim

t0f(x + x, y+ y) f(x, y)

t

= limt0

ft

: , f f=

f

xx +

f

y+ x + y

: t f

t =

f

x

x

t +

f

y

y

t +

x

t +

y

t

: x

t t0

dx

dt ;

y

t t0

dy

dt ; ,

t00

:lim

t0f

t =

f

x

dx

dt +

f

y

dy

dt

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17

17

:34

.C1 ,( ) .1

: .2. ()

. (). ()

: .3lim

x x0y y0

f(x, y) f(x0, y0) A (x x0) B (y y0)(x x0)2 + (y y0)2

= 0

.(f C1) D " f(x, y)

.(x (t) , y (t)) D t I I x (t) , y (t) : .F(t) = f(x (t) , y (t))

dF

dt =

f

x

dx

dt +

f

y

dy

dt

5/23/2018 - 2

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17

: F(t) = fx x(t) + fy y(t)

z = f(x0, y0) +A (x x0) +B (y y0) ,(x0, y0) f(x, y) .P0(x0, y0, f(x0, y0)) z = f(x, y)

r (t) = (x (t) , y (t) , z (t)) ; t

: , .z = f(x, y) z (t) = f(x (t) , y (t))

. , ? .P0 :

:r(t) = (x(t) , y(t) , z(t))

=

x, y, fxx+ fyy

: , N= fx, fy, 1

N r = fxx+ fyy

fxx+ fyy

= 0 , ,P0 . , ,

. y (u, v) x (u, v) ,f(x, y) ,F(u, v) = f(x (u, v) , y (u, v))

.(C1 (:

F

u =

f

x

x

u+

f

y

y

u

F

v =

f

x

x

v+

f

y

y

v

f(x, y) = ex

2y ; x (u, v) = uv : y (u, v) = 1v

F

u = 2xyex

2y

v

2

u+ x2ex

2y 0

= 2

uv

v eu

v

2

u=eu

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17

F

v = 2xyex

2y

u

2

v+ x2ex

2y

1v2

= 2uv

v eu u

2

v uveu 1

v2

= u

veu u

veu = 0

54

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18

18.(|u| = 1) u= (u1, u2)

:f

u(x0, y0) = limh0f(x0+ h

u1, y0+ h

u2)

f(x0, y0)

h

.(x0, y0) u f ( ) .fy u= (0, 1) .fx u= (1, 0)

: ,(x0 y0) u f ,(x0, y0) f f

u(x0, y0) =

f

x(x0, y0) u1+ f

y(x0, y0) u2

: f(x0+ hu1, y0+ hu2) f(x0, y0) = Ahu1+ Bhu2

+ (hu1, hu2)

(hu1)

2+ (hu2)

2

:f(x0+ h u1, y0+ h u2) f(x0, y0)

h =Au1+ Bu2

+ (hu1, hu2)

0

|h|h

h0

fx

u1+ fy

u2

55

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18

.(0, 0) u f(x, y) = 3xy2 :

f

u

(0, 0) = limh0

3

hu1(hu2)

2

h

= 3u1u22.fy (0, 0) = 0 ;

fx (0, 0) = 0 , !

. :

limx 0y 0

f(x, y) f(0, 0) fx (x 0) fy (y 0)(x 0)2 + (y 0)2

= lim

x 0y 0

3

xy2

x2 + y2= lim

r

0

r 3

cos sin

2

r =

3

cos sin2

.(0, (0 .0 ,

.f grad (f) f

f

x,f

y

,f

. = ( ): "

.f

u= f u= f

.fu = |f| |u| cos = |f| ,f u fu .1

.f u fu .2.f , .f u fu = 0 .3

.xy "" fx, fy ,f : R2 R .1

.2

f : R2

R2

(x, y) fx(x0, y0) , fy(x0, y0). /

). ) f ,f : R3 R .3.0 .(x0, y0) f(x, y)

56

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19

19

y f

x = 2f

yx=fxy = fxy

x

fx

=

2f

x2 =fxx= fxx

.f(x,y,z) = exy + z cos x 2 " :

:1 " f

x=yexy z sin x

f

y =xexy

fz

= cos x

:2 " 2f

x2 =y2exy z cos x

2f

yx=exy + yxexy

2f

zx= sin x

2f

xy =exy

+ xyexy

2f

y2 =x2exy

2f

zy = 0

2f

xz = sin x

2f

yz =

2f

z2 = 0

! fyx 2xy

.2f

xy =

2f

yx f(x, y) C2

. 2 ," f : Rn R .1

57

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19

. , , .2

: S(x, y) f(x0+ x, y0+ y) f(x0+ x, y0)f(x0, y0+ y) + f(x0, y0)

: (y0 ( y

g (x) f(x, y0+ y) f(x, y0) S(x, y) = g (x0+ x) g (x0)

x (x0, x0+ x) ', g(x) =

g (x0+ x) g (x0)x

:S(x, y) = g(x) x

=

f

x(x, y0+ y) f

x(x, y0)

x

h (y) =

f

x(x, y)

y (y0, y0+ y) ' h(y) =

fx (x, y0+ y) fx (x, y0)

y

S(x, y) = h(y) yx= 2f

yx(x, y) xy

2f

yx(x, y) =

S(x, y)

xy

: 2fyx 2f

yx(x0, y0) = lim

x 0y 0

S(x, y)

xy

: 2f

xy(x0, y0) = lim

x 0y 0

S(x, y)

xy

58

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19

. , () 2 " .1

(17831707). .2. ,

.f(x, y) =

xy x

2y2x2+y2 (x, y) = (0, 0)

0 (0, 0)

. , (0, (0 , 2 "

59

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20

20.[a, b] [c, d] f(x, y) ) / )

.[a, b] F .F(x) =d

c

f(x, y) dy :: F , fx

F(x) =

dc

f

x(x, y) dy

x

d

c

f(x, y) dy=

d

c

f

x(x, y) dy

.F(x) .F(x) =2

1

sin(xey) dy

.f

x=ey cos(xey) f(x, y) = sin (xey)

: ,R2 fx f

F(x) =

2

1

ey cos(xey) dy =t= xey

dt= xeydy

xe2

xe

1

xcos tdt

= 1

xsin t

xe2

xe

= 1

x sin

xe2

sin(xe)

F(y) =

(y)

(y)

f(x, y) dx

: , . , . F(y) =

ya

f(t)

dt = F(y) = f(y)

:

F(y) =

(y)

(y)

f(t) dt

F(y) = f((y)) (y) f( (y)) (y)

60

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20

.[c, d] (y) (y) ,[a, b] [c, d] f(x, y) C1 : , F(y) =

(y)

(y)

f(x, y) dx

F(y) =

(y)

(y)

f

y(x, y) dx + f((y) , y) (y) f( (y) , y) (y)

:

(s,t,y) =

t

s

f(x, y) dx

: F(y) = ( (y) , (y) , y)

: ,

s = f(s, y)

t =f(t, y)

y =

t

s

f

y(x, y) dx

:F(y) =

dF

dy =

s

s

y+

t

t

y+

y

y

y

= f( (y) , y) (y) + f((y) , y) (y)

+

(y)

(y)

f

y(x, y) dx

61

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20

1

0

xdx

(1 + 2x)2

: :

F(a) =1

0

xdx

(1 + ax)2

f(x, a) :f

a =

x

(1 + ax)2

:

F(a) =

1

0

f

a(x, a) dx=

d

da

1

0

f(x, a) dx

. ::a fa = x(1+ax)2 f(x, a)

x

(1 + ax)2 da=

1

1 + ax=f(a, x)

. 2 fa f .[0, 1]x

[1, 3]a

:

F(a) =

10

xdx(1 + ax)

2 = dda

10

dx1 + ax

= dda1

aln(1 + ax)|x=1x=0 =

= dda

ln (1 + a)

a

= a

11+a ln (1 + a)

a2

: F(2) = 1

6+

ln 3

4

62

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21

21

.(x y ,y x ( .x2 + y2 = 1 : , y =

1 x2

y =

1 x2

.(1, (0 (1, 0) , (x0, y0) x y y= f(x) ") ,) . ,F(x, y) = 0

: (x0, y0) F(x, y) F(x0, y0) = 0 .1

(x0, y0) F C1 .2Fy (x0, y0) = 0 .3

: y= f(x) y0 x0 f(x0) = y0 .1

x0 x F(x, f(x)) = 0 .2: x0 f .3

f(x) = Fx (x, f(x))Fy (x, f(x))

.y5 + y3 + y+ x= 0 ?y= f(x) .y x

). " , ) 5 .x0 (y) = y5 + y3 + y+ x0

(y) = 5y4 + 3y2 + 1> 0 y

. (y).y0 , (y)

.F(x0, y0) = 0 y0 x0 ,F(x, y) = y5 + y3 + y+ x . , R ,y= f(x) !x0 " y0

63

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21

: Fy ,F C1 ,2 .() Fy (x0, y0)> 0 (") .Fy (x0, y0)= 0 ,3

,F(x0, y0) ,1 . F(x0, y) , x0 .(x0, y0) Fy >0 .F(x0, y2)< 0 y2 F(x0, y1)> 0 y1

: x0 x F

F(x, y1) > 0

F(x, y2) < 0

y ,(y ) F(x, y) ,Fy > 0 () , x .F(x, y) = 0

:(2) y= f(x) .1

() F(x, f(x)) = 0

Fx

dx

dx+

F

y

y

x= 0

dydx

= FxFy

f Fx = 0 Fy = 0 . () ,y = f(x) .2.. F(x, f(x)) = 0 ,y= f(x)

: F

x

dx

dx+

F

y

y

x= 0

?Fx= 0 Fy = 0 . f F .

(x0, y0) ,y = f(x) , .3.f(x0)

?F(x, y) = 0 .F(x, y) = (x y)3

. ,y= x

!fy " Fx (0, 0) = 3 (x y)2

(0,0)

= 0 (0, (0

: N0 = x0, y0 ,F(x y) C1 F(N0) = 0 ;

F

y (N0) = 0

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21

, F(x, f(x)) = 0 ,f C1 ,y0 = fx0 N0 y = f(x) f

xi=

FxiFy

, i {1, , n}

: 3x2y yz2 4xz 7 = 0

.yx (1, 2) y= f(x, z) (1, 1, 2)

:.N0 = (1, 1, 2) ,F(x,y,z) = 3x2y yz2 4xz 7 :

:F C1

F(N0) = 0F

y = 3x2 z2

F

y (N0) = 3 4 = 1 = 0

! , : ,

f

x=

FxFy

f

x(1, 1, 2) =

Fx (N0)Fy (N0)

= 6xy 4z|N01 = 14

:.yx (1, 2) = 14 ,y = 7+4xz3x2z2 "

.Fz (N0) = 0 ! , z= f(x, y) " .g (x0, y0, z0) = c0 .g (x,y,z) C1 ,

g (x0, y0, z0)

= 0 .g (x,y,z) = c0 S

: .F(x,y,z) = g (x,y,z) c0 .M0 gz= 0 ,0

F(M0) = 0

F C1F

z (M0) = 0

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21

F(x,y,f(x, y)) = 0 z0 = f(x0, y0) ,f C1 z = f(x, y) , ,.g (x,y,f(x, y)) = c0 .M0 , f .f(x, y) S ,M0 ,

: .M0 z = z0

f(x0,y0)

+ fx (x0, y0) (x x0) + fy (x0, y0) (y y0)

: f

x(x0, y0) =

Fx (x0, y0)Fz (x0, y0)

f

y(x0, y0) =

Fy (x0, y0)Fz (x0, y0)

: z = z0

gx (x0, y0)gz (x0, y0)

(x x0) gy (x0, y0)gz (x0, y0)

(y y0)

gx

(x0, y0) (x x0) + gy

(x0, y0) (y y0) + gz

(x0, y0) (z z0) = 0

: .M0 c0 f(x,y,z) S g (x x0, y y0, z z0) = 0

.M0 g (x,y,z) C1 g (M0) = 0 :

.f(x, y) f :.(0, 0, R) x2 + y2 + z2 =R2

: F C1 .F(x,y,z) = x2 + y2 + z2 R2 F = (2x, 2y, 2z)

F(0, 0, R) = (0, 0, 2R) = 0

: (0, 0, 2R) (x 0, y 0, z R) = 0

2R (z R) = 0z = R

!

66

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22

" " 3x2y yz2 4xz 7 = 0

: , .F(x, f(x, y) , z) = 0 " 3x2f(x, z) f(x, z) z2 4xz 7 = 0

:x 6xf(x, z) + 3x2fx(x, z) fx(x, z) z2 4z = 0

:fx(x, z) =

4z 6xf(x, z)3x2

z2

: ,f(1, 2) = 1 : (1, 2) fx(x, z) =

8 + 6

3 4= 14

22

F1(x1, x2, , xn, z1, z2, , zm) = 0F2(x1, x2,

, xn, z1, z2,

, zm) = 0

Fm(x1, x2, , xn, z1, z2, , zm) = 0

:

= det

F1z1

F1zm

Fmz1

Fmzm

1 i m zi = x0, z0

x0, z0

= 0

. " ,f(x1, , xn)

67

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22.1 22

xu + yvu2 = 2

xu3 + y2v4 = 2

.x, y v u (x,y,u,v) = (1, 1, 1, 1) :

F1(x,y,u,v) = xu + yvu2 2

F2(x,y,u,v) = xu3 + y2v4 2

=

F1u F1vF2u

F2v

= x + 2yuv yu23xu2 4y2v3

|(1,1,1,1)=3 13 4

= 9 = 0. v= f2(x, y) u= f1(x, y)

: x .ux (1, 1)

f1+ xf1x + y

f2x (f1)

2+ yf2f1

f1x = 0

(f1)3

+ 3x (f1)2 f1x + 4y

2 (f2)3 f2x = 0

: ,(u, v) = (1, 1) (x, y) = (1, 1) 1 + f1x +

f2x + 2

f1x = 0

1 + 3 f1x + 4f2x = 0

3f1x + f2x = 1

3f1x + 4f2x = 1

: , f1x

(1, 1) = 13

; f2

x (1, 1) = 0

22.1:

y1= f1(x1, , xn)

yn= fn(x1, , xn)i, fi C1

yi. xi .x= ln y y = ex : ,

5/23/2018 - 2

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23

:

F1(y1, , yn, x1, , xn) = f1(x1, , xn) y1 = 0

Fn(y1, , yn, x1, , xn) = fn(x1, , xn) yn= 0: x0 = 0

=

f1x1

f1xn

fnx1

fnxn

x0

=J(f)

x0

= (f1, , fn)(x1, , xn)

.

23 23.1

f(x, y) f(x0, y0) = fx

(x0, y0) (x x0) + fy

(x0, y0) (y y0) +

+

xxx0

, yyy0

(x)2 + (y)2

. (x, y) (x,y)(0,0)

0

23.2f(x, y) = f(x0, y0) +

f

x(x0, y0) (x x0) + f

y(x0, y0) (y y0)

P1

+ R1

:R1 = (x, y)

(x)

2+ (y)

2

.(x0, y0) f .(x0, y0) f P1: R1 ,

R1(x, y)(x x0)2 + (y y0)2

x x0y y0

0

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23.3 23

23.3:(0 ) ,

g (t) = g (0) +g (0)

1! +

g (0)2!

t + + g(n) (0)

n! tn + Rn

): n + 1 f ) ' Rn Rn=

g(n+1) (c)

(n + 1)! tn+1 ; 0< c < t

. n + 1 " .(0, 0) f(x, y) : .f (a, b) R2

g (t) = f(at,bt) ; t [0, 1]

:g (t)

g (0) = f(0, 0)

g(t) = fxa + fybg(0) = afx(0, 0) + bfy(0, 0)

g(t) = d

dt(afx+ bfy)

= a (afxx+ bfxy) + b (afyx+ bfyy)

= a2fxx+ 2abfxy+ b2fyy

g(0) = a2fxx(0, 0) + 2abfxy(0, 0) + b2fyy(0, 0)g(0) = a3fxxx(0, 0) + 3a2bfxxy(0, 0) + 3ab2fxyy(0, 0) + b3fyyy (0, 0)

'.: ,

g (t) = f(0, 0) +

+ [afx(0, 0) + bfy(0, 0)] t

+1

2!

a2fxx+ 2abfxy+ b

2fyy

t2

+1

3!

a3fxxx(0, 0) + 3a

2bfxxy(0, 0) + 3ab2fxyy(0, 0) + b

3fyyy(0, 0)

t3

+ + Rn

: ,g (t) = f(at,bt) x= at, y= bt f(x, y) = f(0, 0) +

+xfx(0, 0) + yfy(0, 0)

+1

2!

x2fxx+ 2xyfxy+ y

2fyy

+1

3!

x3fxxx(0, 0) + 3x

2yfxxy(0, 0) + 3xy2fxyy(0, 0) + y

3fyyy (0, 0)

+ + Rn

70

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23.4 23

:.(x, y) (0, 0) ,(0, (0 " Rn .1

.y (y

y0) x (x

x0) (0, 0) (x0, y0) .2. .3

23.4 . x .x0 = x01, x02, , x0k Cn+1 f(x) : Rk R

: ,x x0 c Rkf(x) = f

x0

+ df

x0

+ 1

2!d2f

x0

+ + 1n!

dnf

x0

+ 1

(n + 1)!dn+1f

x0

:dnf=

x1dx1+

x2dx2+ +

xkdxk

nf

:d2 =

x1dx1+

x2dx2+ +

xkdxk

2f

=ni=1

nj=1

2

xixjdxidxj

dxi = xi x0i

71

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25

1' " 24

:35

ni=1

f(ci) xi = Riemann Sum

b

a

f(x) dx

25:z= f(x, y)

:36

72

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25

:

V =

D

f(x, y) dxdy

.f : Rn R .1

. .2:

:37

P(ci, dj) f(P) , "" :

"" :38

:

5/23/2018 - 2

74/151

26

D :

i jf(ci, di) xiyi

""

D

f(x, y) dxdy

26: P D

.D S1(D)

.D S2(D), .S2(D) S1(D)

infp

S2(D) supp

S1(D)

" , .() inf p S2(D) = supp S1(D) D .D

.0 D

S2(D)

S1(D)< P >0

() .1

( > >0 0 ).0 .2

. ). ) " ,

: .3D= {(x, y) | x, y [0, 1] Q}

![0, 1] [0, 1] D : D 26.1

: D= {(x, y) | a x b, (x) y (x)}

: ,[a, b] ,

D= {(x, y) | a y b, (y) x (y)}

74

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26.1 26

:39

. ( ) , (!): ,

: , f () D = [a, b] [c, d]

D

f(x, y) dxdy=

ba

dc

f(x, y) dy

dx: .D= [1, 2]

4 , 3 ,f(x, y) = xy

D

f(x, y) dxdy =

2

1

/3

/4

xydy

dx=

2

1

xy2

2

y=/3y=/4

dx

=

2

1

x2

2

1

9 1

16

dx

= 2

2 19 116 x22 x=2

x=1

= 2

2

1

9 1

16

3

2=

72

192

75

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27

27 27.1

.D f(x, y) . D= [a, b] [c, d]

P .1xi = xi+1 xi .2yj =yj+1 yj .3

Rij =

(x, y) | xi x xi+1

yj y yj+1

.4

(P) = maxi,j

xiyi .5

Mij = maxRij

f(x, y) .6

mij = minRij

f(x, y) .7

Rij :40

: S =

i,jMijxiyj

S =i,j

mijxiyj

SR =i,j

f(si, tj) xiyj

) SR , S , S)76

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27.1 27

" " :41

.infP

S supP

S ,S S ,P

: " .D f infP S= supP S

D

f(x, y) dxdy

>0 >0 I R D f : (P)< P

i,jf(Si, tj) xiyj I

<

.yj tj yj+1 xi Si xi+1 .D f(x, y) . , D R2

: .D A f(x, y) = f(x, y) (x, y) D

0 (x, y) A\D

: D

f(x, y) dxdy= A

f(x, y) dxdy. :

.A :

77

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27.2 27

27.2.D f D f .1

.0 f D f .2.D

=D1

+D2

(0 D1 D2 ( D1, D2 D= D1 D2 : .3

: ,D ,f g : .4

D

(f+ g) =

D

f+

D

g

.

D

f

D

g f g : .5

. fg

f, g .6

.

D

f

D

|f| ||f f .7

m S(D)

D

f M S(D) f .8). D) D S(D) M= max

Df, m= min

Df :

: (x0, y0) D D f : .9.

D

f=f(x0, y0) S(D)

.

: , f () D = [a, b] [c, d]

D

f(x, y) dxdy=

b

a

dc

f(x, y) dy

dx. ba , dc

78

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27.2 27

: :

m

j=1

n

i=1f(ci, dj) xi

dj

ba

f(xi, dj) dx

yj

.

b

a

f(xi, dj) dx= F(dj) :

: d

c

F(y) dy m

j=1F(dj) dy ""

ba

dc

f(x, y) dy

dx

: ,

D

f .1

a x b d

c

f(x, y) dy .2

c y d

ba

f(x, y) dx

: f ) ( b

a

dc

f(x, y) dy

dx= dc

ba

f(x, y) dx

dy

:y2(x)

y1(x)

f(x, y) dy a x b ,

D

f .D f

D

f(x, y) dxdy=

b

a

y2(x)y1(x)

f(x, y) dy

dx

79

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27.3 27

:.R= [a, b] [c, d] : ,R

:

g (x, y) = f(x, y) (x, y) D0 (x, y) R\D. g = 0

R\D

g= 0 ::

D

f=

R

g=

ba

dc

f(x, y) dy

dx=

=

ba

y1(x)c

f(x, y) dy 0

+

y2(x)y1(x)

f(x, y) dy+

dy2(x)

f(x, y) dy

0

dx=

=

b

a

y2(x)y1(x)

f(x, y) dy

dx:y1(x) y y2(x) g= f

D

f(x, y) dxdy=

b

a

y2(x)

y1(x)

f(x, y) dy

dx

27.3D=

(x, y) | 0 x 4x

2 y x

,

D

x3 + y3

dxdy .1

=

4

0

y=x

y=x2

x3 + y3

dy

dx=

4

0

x3y+

y4

4

y=x

y=x2

dx=

=

4

0

x4 +

x4

4 x

4

2 x

4

64

dx=

4

0

47

64x4dx=

= 47

64

x5

5

40

=752

5

80

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27.3 27

1 :42

D=

(x, y) | 0 x 1x y 1

,f(x, y) = ey

2

.2

=

1

0

y=1y=x

ey2

dy

dx, !

=

1

0

x=yx=0

ey2

dx

dx= 10

xey

2x=yx=0

dy

=

1

0

yey2dy=

1

2ey2

1

0

=

1

2 1e 1.y = 1, x= y2, x + 2y+ 1 = 0 : D .3

.

D

1

D=

D

1 =

1

1

x=y2

x=2y11dx

dy= : .4

I =

1

0

11y

0

f(x, y) dx

dy+ 10

4y2

1+

1y

f(x, y) dx

dy

+

2

1

4y2

0

f(x, y) dx

dy

81

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28

.D ()

D :43 .

f(x, y) dy

dx I ()

I=

2

0

y=4x2

y=x2+2x

f(x, y) dy

dx.D ()

S(D) =

20

y=4x2

y=x2+2x1dy dx

=

2

0

4 x2 x2 + 2x dx

=

2

0

4 x2dx + x

3

3

20

+ x220

= +8

3 4

281' " 28.1

ba

f(x) dx=

f(x (t))dx

dtdt

5/23/2018 - 2

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1' " 28.1 28

. x (t) ,a= x () , b= x () :

D

f(x, y) dxdy =

E

f(x (u, v) , y (u, v))

(x, y)

(u, v)

dudv

: " .uv D E

. xy uv . x (u, v) , y (u, v)

: , .1

(r, ) = r cos x

, r sin y

: E D :44

. ,: R2 R2 1'. " " .2

.(0, (0 r= 0 ": (r, ) ,) ( " T A22 T : R2 R2 .3

.|A| = 0 "

83

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(x, y)

(u, v) 28.2 28

(x, y)

(u, v) 28.2

:45

B1 = (x (u, v) , y (u, v))

B2 = (x (u + u, v) , y (u + u, v))

B4 = (x (u, v+ v) , y (u, v+ v))

B1B2 = (x (u + u, v) x (u, v) , y (u + u, v) y (u, v))B1B4 = (x (u, v+ v) x (u, v) , y (u, v+ v) y (u, v))

B1B2

xuu,

yuu

B1B4

xv v,

yv v

B1B2 B1B4 =det

i j kxuu

yuu 0

xv v

yv v 0

=

det xu yuxv

yv

uv

84

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28.3 28

,|J| uv xy () ,uv ,uv ,. =J (x, y)

(u, v)= det

xu

xv

yu

yv

E

f(x (u, v) , y (u, v)) |J| dudv : ,

f(x (u, v) , y (u, v)) |J| uv : .

D

f(x, y) dxdy

28.3 u, v E " (C1) y (u, v) x (u, v)

: .E J= (x, y)(u, v)

, .x, y D

D

f(x, y) dxdy =

E

f(x (u, v) , y (u, v))

(x, y)

(u, v)

dudv

.

28.3.1). f ) u, v

). ) 0 J= 0 . J= 0 . , ,

). , ) , T ". y yx= 0 1', "

. " J= 0 ,

28.3.2: D

D

arctany

xdxdy .1

85

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28.3 28

D :46

: .r

1 r 24 3

D

D

arctany

xdxdy=

2

1

3

4

r|J|

d

dr= 72192

: : J=

xr

x

yr

y = cos r sin sin r cos =r

.y= 4x y= 9x y2 = 6x ,y2 = 3x D

D

xydxdy .2

D :47

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28.3 28

: y=

v

x, y2 =ux

.4 v 93 u 6 u, v : ,v u y x

y = v

x x= v

y

y2 = ux=uv

y y3 =uv

y = u1/3v

1/3

x = u1/3v2/3

: J = xu xvy

uyv

= 13u4/3v2/3 23u1/3v1/313

u2/3v1/3 13

u1/3v2/3 = 1

9u1 2

9u1 = 1

3u

: , u, v J= 0

D

xydxdy =

9

4

63

v

1

3udu

dv=

1

3

9

4

v

ln u

6

3

dv

= ln 2

3

9

4

vdv

= ln 2

3 2

3v3/2

94

= 2 l n 2

9 (27 8)

= 38

9 ln 2

28.3.3. " J= 0 .1

). ( " , J= 0 : T .2

T

uv

=

A a bc d

uv

=

au + bvcu + dv

=

x (u, v)y (u, v)

87

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28.3 28

T J= 0 , !J = A ? J .|A| = 0 T .

(x (u (t) , v (t)) , y (u (t) , v (t))) uv (u (t) , v (t)) J= 0 3. . y(t) x(t) v(t) u(t) .xy . , :

.. 28.3.4: 2

x= x (u, v)

y= y (u, v);

u= u (r, s)

v= v (r, s)

: x= x (u (r, s) , v (r, s))

y = y (u (r, s) , v (r, s))

): ) x

r =

x

u

u

r+

x

v

v

r

y

s =

:

xr xs

yr ys = xu xv

yu yv ur us

vr vs :

(x, y)

(r, s) =

(x, y)

(u, v)

(u, v)

(r, s)

: ) )

x= x (u, v)

y = y (u, v);

u= u (x, y)

v= v (x, y)

1 =

1 00 1

=J J1

: . J1 J1 =

1

J

: 1', " dy

dx=

1dxdy

88

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28.3 28

: , y= vxy2 =ux

:u, v ,x, y "" v= yx

u= y2

x

:J1 =

ux

uy

vx

vy

=

y2x2 2yxy x =3y2x = 3u

J =

1

3u

28.3.5. .

ex2

dx : . :

Ia =

e(x2+y2)dxdy

.a :

Ia =

2

0

a0

er2

rdr

d=

2

0

1

2er

2

a0

d

=

ea2 1

= 1 ea2

5/23/2018 - 2

90/151

29

.2a 2a , ,

e(x2+y2)dxdy =

aa

aa

ex2

ey2

dx

dy=

aa

ey2

dy a

aex

2

dx

=

aa

ex2

dx

2

:

1 ea2

=Ia aa

ex2

dx

2 I2a = 1 e(2a)2

. lima

aa

ex2

dx= : a

,lima

aa

(!) , :

. , , 29

.f(x,y,z) ,R3 V . : . V

Rijk

f(ci, dj , ek) xiyjzk

.

V

f(x,y,z) dxdydz ". (' (,

V :48

5/23/2018 - 2

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29.1 29

(): . V

(x,y,z) R3

a x by1(x) y y2(x)

z1(x, y) z z2(x, y)

: ,

V

fdxdydz=

ba

y2(x)y1(x)

z2(x,y)z1(x,y)

f dz

dy dx

? :

:

.V

1 V

. (x,y,z) . m=

V

(x,y,z) dxdydz

. , 29.1

J= (x,y,z)

(u,v,w)=

xu xv xzyu yv yz

zu zv zz

V

f(x,y,z) dxdydz=

W

f(x (u,v,w) , y (u,v,w) , z (u,v,w)) |J| dudvdw

"...) ,C1 ,J= 0 )R3 29.2

: R3 ,

x= r cos

y= r sin R2

: .1

= x2 + y2= arctan yxz = z

x= cos

y= sin

z= z

0 ; 0 2

.= a x2 + y2 =a2 ,

5/23/2018 - 2

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29.3 29

: .2

=

x2 + y2 + z2

= arccos xx2+y2

z = arccos

z

x2+y2+z2

x= cos sin

y= sin sin

z= cos 0 ; 0 2; 0

.= a x2 + y2 + z2 =a ,

:49

29.3: V 1

(0, 0, 0 ) ; (a, 0, 0 ) ; (a,a, 0 ) ; (a,a,a)

92

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29.3 29

:50

: I=

V

xyz dxdydz

: V V =

(x,y,z) R3

0 x a0 y x0 z y

.(y= z )I =

x=a

x=0

y=xy=0

z=yz=0

xyz dz

dy dx

=

x=a

x=0

y=xy=0

xyz2

2

z=yz=0

dy

dx=

x=a

x=0

y=xy=0

xy3

2 dy

dx=

x=ax=0

xy48y=xy=0

dx=

x=a

x=0

x5

8dx

= a6

48

93

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29.3 29

.(x,y,z 0) .z= h ,z = 2 x2 + y2 ,z = x2 + y2 ,y = 2x ,y= x " V 2: . .V

J=

x x xzy y yz

z z zz

=

cos sin 0sin cos 0

0 0 1

=

: W ,

sin = cos = 4sin = 2 cos = arctan 2z = 0

z = h

z = 2 =

z

z = 22 =

z/2

V

1dxdydz =W

1dddz

=

z=h

z=0

=arctan 2

=/4

=

z

=

z/2

d

d dz

=

z=h

z=0

=arctan 2=/4

2

2

=z

=

z/2

d

dz=

z=h

z=0 |

=arctan 2

=

/4

z

2z

4 dz=

1

4

arctan 2

4

z=hz=0

zdz

= h2

8

arctan2

4

: .x2 + y2 + z2 =R2 : .R 3:

J = (x,y,z)(,,)

= cos sin sin sin cos cos sin sin cos sin sin cos cos 0 sin = 2 cos sin2 sin cos + cos2 sin cos

2 sin cos2 sin2 + sin2 sin2 = 2 sin

5/23/2018 - 2

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29.3 29

Vball =

2

0

0

R

0

2 sin d

d

d

= =4R33

V: ,

V

xy

z dxdydz : 4

V =

(x,y,z) R3

z x2 + y2 3z1 xy 2

3x y 4xx, y,z >0

w= y

x ; v= xy ; u=

x2 + y2

z

: V u,v,w

W =

(u,v,w) R3

1 u 31 v 24 w 4

:

J1

=

(u,v,w)

(x,y,z) = 2xz

2yz x

2+y2

z

y x 0yx2

1x 0

= x2 + y2

z2 2y

z

|J| = xz2

2y (x2 + y2)

: xy

z xz

2

2y (x2 + y2)=

1

2V

1

u

1

w

:V

xy

z dxdydz =

4

3

2

1

3

1

v

2uw dudvdw= =3

4ln 3 ln4

3

95

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30

30

:r (t) = (x (t) , y (t) , z (t))

): ( r(t) = (x(t) , y(t) , z(t))

: , r lim

t0r

t

30.1: .r (a) , r (b) r (t)

: ( ) ""length (r) =

ni=1

|r (ti) r (ti1)| =ni=1

|ri|

=ni=0

rititi

b

a

|r(t)| dt

:

L () =

ba

|r(t)| dt=b

a

(x(t))2 + (y(t))2 + (z(t))2

: y= y (x)

L () =

ba

1 + (y(x))2dx

1'. " 30.1.1

.r(t) = ( sin t, cos t) . r (t) = (cos t, sin t) 1

=

2

0

( sin t)2 + (cos t)2dt= 2

5/23/2018 - 2

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( 32

.r(t) = (a sin t, b cos t) . r (t) = (a cos t, b sin t) 2

=

2

0

a2 sin2 t + b2 cos2 t

1/2dt

). )

31:

S(t) =

t

a

x()2 + y()2d

. a :

S(t) =

x(t)2 + y(t)2 = |r(t)|

: s x= x (s)

y= y (s); 0 s L

.(t s)

.

L

0

ds= L 1 =

x(s)2 + y(s)2

( 32. , ,f(x y)

.f(x (t) , y (t)) : f. f

32.1). ) " "

.Q=

L

0

f(x (s) , y (s)) ds ,n

i=0 f(xi, yi) si ,.f(x y) 32.2

f ds=

L

0

f(x (s) , y (s)) ds

5/23/2018 - 2

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32.3 (