.

3IV

'.

3 3

nI0 .

. .

=

[,. ()] Vl+y?(X)d~- [' (x)]Vl+y~2(x) dx+4

d

d.

+ '(, . ()]' 1 +y~a () dx,

d

.

.

-;7.>, ;, .. =-

f

11 (),

I

== I1 (),

=

li ().

1.

.

20.

~

()

.

l:ieKa

.ll

8, ~.

MN.

'

Oz,

nn

.II (.

145).'

g)'

=

= ''

,

z

' '

~

N

w '

" -

, =-

0r-;-__~____~__

z=

(,

JI).I)

ds

q ,

w,

n

-

.

145

.II

ds, dJ = zds

= (, ) ds ,1)z

.ll

= z ds - (, ) ds.

,

n

.ll

- + - = 1,59

l

l

> ,

z> ...

'

z=yds,

(."

146). (6)

'

==- zds=

f

.

146

.( ' 1)

r + ... 1.5 9

' IIe

~ z ... (, ), (, ) 06n , .

1) ~ OCHOHneds . MH-z-P (,,).

&

x={5cost, y=3sinj,

(6),

(5),tr

== 3sintV5sinJt+9costtdt.U '

h

cos t -

,-1

- sin t dt = du,

h 1

- - '14 3 + du=6 !V4u%.:t5dU=9+ 1;10g5. ,I

.

-

h :

10. h x~ d-:;, r + % lJ

"'" 3 (, ) ( - , ).

R)yra -

cos

t,

=

sin t,

f f '

X1dS -

COS2

t Va%cos l t+0 8 sin8 t dt'

: 8 08 2 tdt,.. ~201.

f

=='15

-f

4

dx ~lo [ _.! _1 :: ' ]' 7 +3 '15 V5xl-8.t+16 ~ 5->V5V5-8+16=lg 2 2

- ~ - 2 (, - 2) (4, ).

30.

t

rXYdS = Vd%+d2 - dx + 4' . dy

f

4

Z

-f

. 2 . dx -

.; . ;': =2

24,

4

1,

r

7

,

= ,

-

,

- 4,

.... 2.

Kpyr

= cos

t,

=

sin t.

50.

'1 2 ds == 2a'la

f

.

(1- cos t) dt

=

'n(.

cr ....

n

a(t-sin t),

= (l-cos t)16 y~

93).

xyzds= -

!I

tlV8tI6

l +2 t+fI dt =

143 '

t,

8 t l = --'

z -- 2".

t'

1=0 t= 1.

70.

l + 2z2ds

=

"-2 Jddt=a 8n1 2" , ,3'

.

()

n =

cos t,

=

sin t,

z -- t.

80. .... (8),

&.

n

(, ) ds= ( cos8, sin &) 'l+ . d&.

9',111

n

n

.

+l+ Z2 =

RI

I

n '

n . h

'

~ (x,y,z), . h

.rds,,

ch 1f (R, , )

= --,

R

~ ' n

, > n

=

Rcos 1fchcp

,

='

R siri q. ch l'

z=

Rlh1',

8 .! .t>

r_

~

100.

ds =- RV2 Jc~: -2R~2[arct, "- :]. . .' .

n.II. l +,.

w

z

-=

R + -R

..

R

1

n

~

- [Zd' -1( ~)YtU"Hy'- R J(R + R'C;s't)dt ,

3xR'.

- Rcost,y - Rsint.8 9

. 1141. n n ,.- 2 ,

z ==

-"'- n

,

, l

- 2 . n n + l

120.

=. RI 1

2 Rz -

13',

41f

zdr -

~1 Ydx,+ dy'+ .1 - RI n

-

R' [ 2 tdt= R'.

l n. l

./l

n

+ 2 + Zl -== RI~

== R 1, = R 8 t. . + l - Rx n

== 4

z ds

= 4R

V RZ -

2

RI 08 d6 == 4R3

f

2

~

sin d8 - .. RI

=1) l+yI- ,

R 08 .

RI

-

' . .

1.

ptI808IIllCUten

,

98 cos2 t.

14'.

(-') ds -(r-r) Vdr + dr

= ,.

rAe

l+.

- 4.(1 1 ) '=:

15.

fXdS -= Vdr + dy'

'12

f

1

xdx =

'12'2,

(0,0)

.

.

-

(, )

(1,1).,

219,

! , . h

-I+ 2-- ydx~ ydX--rd, m. bIIIl '{.. .n ( ).

1. Jl8OJl8:":' m-u

8- ~l) (.

150),

.8

. m ,.ll . ,.ll m ,.ll : ,.ll

.

.

150 .

151

m(

(C.1I. 151),

... - ydx .... - YdX- dx,

,

\

.

n '

1 .

(C.1I. 152),)

. ~-,~'h"7.:1

~--'

(. 8m)

.

152

1 "'" - d~-" ydx- ydxaRlb1)

f

.1'dx, ,

/Jd :

(je~,.,

,.

2)

' &. ~ were IIe. (:1 . CD " . 8. . bI. (16.

.

r .....

P1Jtl.

--

f

ydx- ydx- [YdX Ihd d

dx.bntJ

(1.1

j~p ".I

1

Ct

(16),

Il.flIl. n

-

1'.

~m

- t, - . sin t, ll

..

2'1',

(,) , t

..

0"0 2.'11:. 20. mI

ll DestesII (..ll

101)

'+8-3 axy~) ...

- l+t"y- 1+1' '

t

.

(16), ll

- ~ tly-ytlx - ~ ![X(x+Jt'fJ-t') tlt-

_.!.jxt28.JIC6alN. -

'

dt _ 9 oI tI dt ___ ["

2.

(1 + t')I

2 (1 + 1') 02

8 " ]: 3.11'

.

..h 8lTt:

r cos '.

-. llh ([6)h 08 l+yI..;,,, &_ - r t. .08m; :u'tOll (CJI.98) (xt+yI)' - '('.,....yI) ':os28 h,) .... (1'6) 8h y-xtgt, .- t

2'. . . ,. ;" .

1): tu-

_ tJC.4

~.

&)

,.... t ..p.paOJ1 0";'0 -!. _

' ....

0"'.8

qx Al7iiO

,.1IpIIOII

,1.,

80.lCII

21

2 l dt 2 - cos't ~ 3'.

"i

1/ l!!..4

1- tg t dt (l+tg't)1 cos1t

= .

,

x .... 2acost+acos2t. y=,2asint+asin2t. , (l).

..

- ~ ~Xdl- ydx221. Oreen,oBaAll

60'[(1 +/)-

d/ - 6 ....

.Jl .

,. (,) Q(x,y)AII

-

-

~Q

, ' 1 (. 89). 81(17)8

J(:~hn'8

-::

)dX dy =-

(.)d + Q(x.y}dy

. Jr,OCPYKor

YJl

. , Iflf lf~rp.iAY. . 8 ,.1).,

81 1.

::

dx dy,

811.

,.uoct

'... '

f".f 1.

dx. -

.

"dx '" ." ...:'f~' '....... . ,d' - "P(x.,,)dx - "" P(X,y,)d~ti.,;\;~, ..

.

~- (,,)dx1)

f"

,.

-1

!'..

'1,.'

/1

P(J!;yo}dx.

.u(17)'lCiI jOat 00""-08& .... Rl,malln-oa .... ~~"i: ':~"' ,),.,-(~~~ ,.~~:- ~~.,.,

22Jl

r4

f (, .)tI

dx,_

f P(~'Y2)dx,.

tI

Jl

Jl Jl ' 1 '. '.',

Jl

-

-

(18)

J~: dxdy= - (,)d,1.r. Jl Jl Jl

, (JI.

89).,

(19)

f I:r;1.

dx dy

= Q(x,y) dy,

Jl

rJl

. h

(18) (19), h Jl (17).

Jl, u ~

. TaJti n8 (17) n (17) .

3'

Q-

,

... -, n, (17) (16)

2 dxdy = dy-ydx == 2 ,',.

,, -, 6h

m J 1 3

Q =- , == - Q

JdXdY--d=, fdX dy = d=.. ''

11

(17) . ' ~ -

" (.

153),

+ Q dY) ds = (,)d+Q(,)d='(d ds ds

-

p(PCOS.+QCDS~)d$- ~'(PCOSi>+QSin.)d$,

I.

On8 rern

. 23

(17)

J(~~ - :;)dX dy =

. ___ + - - - = 2,;Xl+yll+l

-ydx

xdy

26 l + l

rnaBa

= ,.

. ,n- nl

(17),

=

"'" (,)=--,-

xl+y.

Q(x,y)----. + .

8 n

+ . =,~ (. 154), 08. = '1' 1 2

8

6

I

6

+ =0. .6

. 154

n8 2

,n- n (17), .

+ = J(~; - :~)dXdY ~ 0,1) , .

Q --=-=-

l_

(l+l)1

222. Jl Jl . - V ' (, ") .'

V

i

01.

155

'

Ouv,

11 (, ) n .

(.

(21)

x=f(u,v),')

155). y .... ,(u,v)

n

-- _, I

n .' .

1.

~ Jl

27

' 1 ) '" , 2 ' , (.:

155).

10. h (21) ' .1)

20.

,(,

11)

1 /dV ="11.I'J

11.,

.

r0/

11.,

F(x,y)IP(x,y)/dV. D(x, ")

.

,

(25')

. 11. (28) (, ) dx dy =JdXJF(X, )/ D (, ) / dv = (, ) D (, ) dxdv~ D (, ") D (, ") 11.

'

'...

Oxv.

==

f (, ") , v "'" v ,

D (, ) / dx dv= (, ) (, >/ D (, ")I ,

I (, ) IIID (,D

D (, ") D (, ") . ,

V)/dD dv,

"

Ouv.

,

(!! 93),

D(x, )

D (, D (, ") =--'---"-, D (, ") D (, ") D (, ")

Al 9

(29)

JJF(X,Y)/ D (,)/ dx dv -JJF(X, )" D (,)/ du dv. D (, v.) D (. 11)'

, .

32

(25),

(28)

(29),

(,) dx dy =

"

(/, ) ID D (,) I dudv(, )

==

(/, fP)/.D (1, ')I.dUdV. D(, )-

. --

10.

== ,

=

20.

2 + .

= 1

-

l -

r, =2 ,

dx dy = 8 ('- ') dudv, '

'

2 +2

= 1......

11

224.

I{ D. -

.

ABOcTpYKorz

S

.

(1) (.

=

(, ),

= (, ),

=

z (, ).

, z D v uu. 1 )

S

,

= st.

v

= st.

157).

eJI

MNRT,

(u,),h

N(u+du,v),

R(u+du,v+dv),

T(u,v+dv),

(2)

dp = MN sin6.

(1),

-

n

S

(3)-1 ' (1) " w S. lJ, (, ,,) , :! u"" (, , z) w OJi (.ll. 157).

(,

v),

(, ,,), z (, ") 08 80n

.ll'I { OIlll, oAFoBapa w .

11. 8ll4

33

(3')

{

_

(~:)2 + (: + (:)1,

_ (~:)2 + (:~)a + (::)8,%

u+du,

O~

______________________ __ 11 - const.

.

.

157

dx - -du;)

';)

+ -d." ;)11

.. + , d1 - ll

i),

dz - - . + - d,l.,,

'6Ii 1 "

dsl- dv2:

.

(, v)118

'''''' con8t. ' dll - ,

Kptl80j

tl -

const.

118

, DPeIII (3), dsl .... du."

- . li8 , ... (3),

(4) , ..

(2), IUII8~r p&lluorpt MNI!T

(4')

'; dp - sine.h- ' Yl-co .... dud'.

'(,II), .

" u ..jaIIa u - cot. 11 - con, il n8 (1), 81118 'dx--dll,i)x

'

;),;)

dv--th,

"3.,

dz--dll,"

i)z'

(du==O),(dll - ) ,

dx - -diJ,

'

d1 - -du,

iu

dz

== -

:

i.

du ,

. - - - ------3

34

,

,.

41

Oz,

V= fJ Z 2 dXdY- Z1 dxd y=

J(Z2- Z )dd =.

(16)

2

.

,.;,. JU2(x,y)-/t(,)] dXdy,"'

2

1:1

1:2

.

itt

~,

Oz

,

(16),

Z1

=

11

(, ) z~

= 12 (, )

1:.1)

.

-

10.

h 2 2 z. -+-+-=1.

2

2

2 ,

9 h,

(15),dy =

1 V = z dx dy 8.

=

.

11 V1 :-, :l 2

-

y~ dx :!

=

f f'

!.-'---" _ _ _ __

dx

1/1 -

V

2 - l dy2 2

= _7t_G_b_c ')

6'

20.

h

,

(12)

~

(11).

h

~

V

= JVR 2 -x2 -y2 dxdy =JdeJpVR2 -2 dp""2:Ir

" Rcose 2"

"

= ~RI(I-Si8)d = ~R'(~ _~)3 323

i

"

=

!!,R8 _ 2R86

9'

t) 1: ' ! II . 1: ZI - 11 (:. ) Z. - 12 (, ), 1: , ,! . 1:1 1:2 ( ). ' 1: . 1:. 1) . (! 141). . 1 2

42

(,

2 . .

-

(11)

( ..

158).

8 h

2 :ri RI (2 R)3 v=----3 9'. i

ie

i10

s il

i

llipeher

.

.

- h : 10. 3 ~ + L

+!.. """ 1

2'.

3,

,

n

=

2 "'" z - 2,

30. Oxyz (0,1, ')). Oz.3;

(l,O,O)~

n

,

;

z =

+.

,

40. Oxyz' (1, , ),

(, ~, .).

Oz. 3, l--

lIaMa Z .. 2 + + 1 , 1

v- !!(X' +Y +l)dXdY '" dX!(X + Y +l)dY = ~.2.

11. Jl

43

50. OB=R=2 xt +yl-2x == , Oxz,nOBpWIiHQM

(.

158).

3,

xl +yt-2x =

z=x . V=

,

l dx dy =.

cosJ

sin d . d =

.

2!

5" .+ Zl = 2

4

60.

3,

ql

x=~, 2

' v = 4 dx V2px- d = .'JI;

v'"" ..-------.

7'.

l

+ '= ,1,

X2+Z1 =

,2.

3

V"":"

v' r'-z' r I(- 16,' r ,2_ 2 dx dy = 8 (,!-!) dx - -3-.r

'.

l

+ ' + z! =- ' ,

! + '

=

.

w,

"

9'.

3, 0 w, 1- = 11: "2 '

z

=

cos cos ,

V=

f

-

cos dx cos dy = : .

.

r

,l( . + + z

10.

,

4 2+ zl=4,

= 5,

V=

f

1

2}'I-

dy

(5-y-z) dz

=

10~-8z

110.

, .2

2

= . + 2 = 1, 2 + = 2 ,

v = JJXYdXdY =~ . 24.

111.

226.

.

-

t (, , z)

-

Oz

(. 157). .l. 6. 6.2"" 6.n N 1 (~j, 1), ~i) 6.

(1)

~ f(t,'1i,~t> 6.;;=1

=

,(e1J1)I'~l) 6.Pl+t(e2,1)2'~) 6.2 + ... +f(~",'1n,~") 6.n

, ~Pj, ~

(1),

6.; ;0 il1t

,

t (, , z)S

UHi11erpaJf

1t

i10 il S =

(2)

, , z) dp

n-+ 1=1

i;

'(~,,1),,~j) 6. =

t.Pi-+ =1

i;-f(ei'1)" ~ 6.p i .t)

t (, , z)= 1s

8 ,

. dp = . S

=

(, v),

-= (, v) ,

z

=

z (, v) ,

1

Quv,

,

1)

. . .

. n

45

(2)

(

224)

Jf(X, , z) dpS

.... I [ (, v), (, v), z (, v)]{

~Ea.,

;;>.

Oz

. .

S

,

81

8 , .;

(81)

Zl =

Vl-x 2 - yl ,

(82)

z:z

= - "l- 2 _ l.

T~a r

xyzdxdy = xyzdxdy + 1 xyzdxdy.8~

OCMQM ,

(5)

8

f

xyz dx dy =

8.

f

xyz dx dy -

f

xyz dx dy ,

-.

- 81

.

(5) ,

xyzdxdy = l x'J-'.dd - S

- : xl-'dd+ ==81

=

h

h

=2 xYYl-2 -.,.

.

1

\,.

8 .

h 8

38 cmehomx-=s9,

.v-p.sin6, '

ffXYZdXdY=2~

lsi6s6Vl-pldd6"",.

==

2 2sin26 l.' Vl-p1dp -15"

111.6",.

49

-

n :

10.

_ w n :

ffS

xdp- f f.

[+ q2+ 1 dxdy - R cos6d8 f V~~p2- ,

2#

R

,

;

S

nn

X2+y2+Z1 -. RJ

R l " dp = 1 2 Vl + q2+ 1 dxdy -4 s .

f

-2#

R

sin~29d8

p'dp 21tR' VRI_ p 2 -'

-

S

nn

2+,.+ZI_RI X11-)

_

Vxy;SJXYZdP= (1- S .

) Vp2+ ql+ 1 dd-V fXdxJ y(l-x-y}dy- :~,

S w x+y+z .. 1 .

20. w n :

!fXdYdZ~YdZdX+2dXdY- [dY-

1

!1

t

1

1

1

dz+ jd1dX+ [dxf dy -

S ... ', -,

1, = 1, z = 1 ;

z-o

JfZdXdY- !fZdXdY+ JJZdXdY88 ffZdXdY- ffZdXdYS

st-

.

8

-

--xyr - JJV l----'dxdya 1Jl l

v

-.

.

.

-

. 1 ----dxdyl '"

n

f JZdXdY-~: f "aI"'~~I-jiidds.

2

2.

-

: d9 l

(1JIcosJ&

+ l Siiil)pl dp - :

;

,@

S

8 !!. .JI + l + ZI ' 1.0'-

cz

."" IU8

50

228. 2

s

..

-

,

h

(.N'2 220),

h

. ~

Oz . 8

2

,

~,

1 2

w

zl=lt(x,y), ,

Za=12(X,y), L1

~ ,

Lz

(Q ).

.

L

~. ~

(.N'!! 225)

v=

ff 12 (, ) - ff 11 (, ) ff 12 (, ) dy - ff 11 (, ) dx =. , +~I -~I

LI

( ~ ~).

L)1)

1:1

(

in

~1 ( ~), h .

(6)

V=

ff 12 (, ) + ff 11 (, ) ff z dy ,=+~ ~

+~

~ , ~.

(7)

v = xdydz,1:

v=

ff .v d.i ,1:

~.

(6)

(7),

h

(8)

V-

~f1:

fXdYdZ+YdZdx+ZdXdY,

1) 3 c~paHY 1: h '8.

111.

w Rn

51

'

~.

8

j~

,

.

,

Oz

(

)

8

1;

.

1;.1)

~, ~, ~ ,

(8)

.

(4').

= ~

JJXdYdZ+YdZdX+ZdXdY~

=

~ Jf(xCOSCX+YCOS~+ZCOSY)dP,

dp

.ftl1l u ~.

, n, . , ,Z

2 +

=

h, (8). .

r "'"

,

~ ="3 2 Vrl_yldYdZ+"3 2 Vrt-x 1 dzdx+ dxdy:=OyzOIVC '"

-+'

--,

- :229.(9)

"

dz ! -'

l dy +

:

dz ,.. - dx + : dx dy - , 11.-,

h

+r

+'

+y~",

'

-~,._".

~epaJl"

-

f(x,y,z)dxdys

-: JJf(X,y,Z)COSYdP s

.

S, dp 08 (9) .!!

s.

(10)Taqa (,

=

1 (, '1, ~), .

=

. (~J '1, t),

Z .... , . D (, v)

+ D (, ) D (, )D (;,7)

D

(, v)

, ,

; - Vl (, ) ,JUl u" S1

IJ -

VI (, ) ,

U

- . (, ) ; ,

h

(10),

S

" .. l (Vl' VIIt V.) - >"1 (, ) - ! (Vl' V2' V.) .;. At (, ),. z - , (Vl' VIIt V.) - (, ) 1) + - ) ~ m S, crpaHaa I.

S.

')

_ D(y,z) _ D(y;z)D(1J,b)D~~ D~JD~~

+ D(y;z)1

D(b,;)

+ D(y,z)

D(;,q)_

D~~D~~

D~~D~~

. D(y,z)

)(,

1

+ D(,;)

D(y,z)

+ D(;,q)

D(y,z) 1,

------

D~~

D~~D~b)

D~~

D~JD~~

D~~D~~ D~~D~~ ---+ -- - + - --.D~~D~~ D~~D~~ 1 +--

.-D (11,

D(z,x)

D(z,x) ,;)

1+--

D(z,x)

D (;, )

l'

111.

m

53

= D(~,y) 1 + D(x,y) 81 + D(x,y) ..D (1, ~) D (~, )(, 1)

cos ""2+8 2 + = D (, 1)

[D (!!) cosa;l + D(, ) cos ~2 + D (1, ~) D (~, ) .1 1 1 . 1,

+ D(x,y) cosy ]V +m+2 i

+,

nw

S

~OBapa

w

S ..

-,

w

S S1' , , z) du dv 2' Ouv, h

ff f(X,y,Z)COS(11)N

~2+2 +

dudv = ff'(X,y,Z)[DN

D

(, )

(1, ~)

cosa;l

+

+ D (, ) cos R + D (, y)cos .] 3++ ~ du dv D (~, ) 1"1 D (. ) 1 r 1 1 1 ., , ,

!t (s, , ,

'2 (~, , ), ,' (~, , ]

=

(~, /,

Q,

D ~f( ) '~P(I: ~ /, /, Z, V/ = ~ .." ;, y)I :,; D (q>I,'P2,q>s)l' (1:. -: ~.-).- V, ,=1 ,=1 .. " " ,

.

!, h 8 ,

(27)

f(x,y,z)dxdydz= fffp(~,YJ,QI D~'/i,~:'~3) /d,d'lid .. .:'

.

u

-,

, , ,

z

~,'1/, ,

dx dy dz

1l!.1 d~ dYJ d .

. WiI n

61wh 8'

. 1 ).

,

Oxyz

O~,. 1' '2 ', ,

~

3

'

(27);

Oxyz, ' h w ~ =-

const., (27) f(x,y,z)

7

(22) = sonst. = const. (!! 206).1,h

=

F(e,7J,bl =-1,

-.,

(28)

V = fffdXdYdZ =1.

(CP(,CPI' ') Id;d7d, (,7, I1.'

dv - dx dy dzD(,7,)

.l1l

V

. Itl1l.,

dv = D (1' '2 '.) d~ d7J dt .l1l . 'V ' ItltItl1l..

ds2 .- dx 2 +dyl+d"z2,

(22),

1

ds2 == Hl.del+H~ d7JI+Ha d 2 +2F1 d1d +2 F2 d d~+2 F. ded7J,

~ ()2 + (!)2 + (~')2 = ('1 )1 + (!)2 + (D'8 )21

~

'

I

7

7

1'

;" (1 )2 + (.)2 + (. )2, .

F1 =

~CP1 ~1 + . alP2 f . '.,7 1 1

, =- ! !

+ .. + , '_ ~ '~ 7

F _ '1 !

~ ,

,

+ . ,! + ' 7

7'

~"

,..!. ,.[ 0('0'1' .)

ep'12

. .,

D(e,7J,

]2 -=

, . ,.

.

7

.

F1 -

"

,

!1 ,.

- ~

. 1 .

(28)

(29)1. 1) n . 38 8U .

62 dv ' d.e

~ =

dYj d~ . F1'~F2=F=0, const., 1 = const., ~ = const. , . ds 2 = 1 d;2 + 2 drj'l. + d~"

.

. (29)

V=

S fS v. de dYj d~ ,'

,

== sin cos ,

=

sin sin ,

z=

cos ,1)

(28)

v = pISi~edpdedcp,'

dv = 1'1 sin dp dO dq>206).IV.

V

u.

(2

234.(!!

.

(, ) 1

"

,

74)du = -- dx

(1) (., )

+dy.

Q (, )

" .

(2)

. P(x,YJdx+Q(x,y)dy

h (, ),

(3)h , ,

du

P(x,y)dx+Q(x,~)dy.

(1)

(3)

(,).

(2)

(4)

-=(,),

-=Q(x,y),2 Q

- - - ... - = - - ... dy 1)

"

(22).

JV.

63

(5) (, ),

.== Q __

_

12' llf

,

(2)

l

.l

'

(5)

12'

'

(5) , h (, ) (3). (4)

(6)

=

f (,)

dx+ q> ()' 12 ()

. '

(, )

(3),

.

= . > -dx+ ,

=

Q(x,y)

(5),=

Q dx + dq> f dy

Q(x,y),

Q(X,y)-Q(,)+ddq> ='

Q(x,y)

q>(y)

= fQ(Xo,Y)dY+C,'

"

' . ()

3h

(6),

h s

'.

.,

(7)

u(,)- (, ) dx+ fQ(Xo,Y)dY+C,

10

(3).

.

8

(3)

..

64

~ ~

~OM, .

(, ) = ~Q(x,Y)dY+ [" (, yo)dx+C.70

r - . h

lI.

.

-

.

.

(,, z) , Oxyz,

(8) (, , z)

du

= - dx + .,.- dy +- - dz

dz

Q(,, z)

l n .n

R(x, , z) Oxyz.

(9)

(, , z)dx

+ Q (, , z) dy + R(x, , z)dz

i (, , z),

(10)

du

= P(x,y,z) dx + Q(x,y, z)dY.+R(x,y, z) dz.

~8) (10) , (9) 8 (, , z). , ~

(11)

-

- P(x,y,z),

-

-

.... Q(x,y,z),

- -R(x,y,z)

z

(12) _; 1lll U (, ,

z

.,.

(9)

ui!,

z),aompe6Ho U (12) .lt ltallBO , , z ,.

U1

_

"li (-, ,

z)

h

(11).

.

(I)

..... (,,z)dx+cp(y,z),"

"

JV.

8 8n 8n8

65

,

fP (, z) z. (, , z) .i r

(10),

..

du ...

f11:

afP dx +

=

Q(x,y,z),

_ _ d+fP_R(,,z),ijz~z

"

az

..

(12),

--R(x.,y,z).z

ijfP

lDl

h

dy

dz

, h

-dy + --dz bz, (7)~

fP

==

Q-(,,z)d+R(",,z)dz

.

8 (,

z)

:

(., z) ...,r Zo

f (, ,Q)10 .

z) dy + R z.

(,"

z) dz

+,

tl

z..

(JI

fP (, z) (13), h

(13')

(, , z)

=

(, , z) dx + Q( , , z) dy + R ( ' . z) dz + ," '.

"

z

,

z.

'.

u..

(10).

. , , Zo

.

-

+

d

x~+y2

+

....

2 +2'

dy,]_-__2

(, } .;=~ .c..~""';'2-+"':-2

~~:+

Q (, ).

=

+ 2

5

66

ytJI08 (5) ,8 .Jl~fI.~I*JIUt,

.

(7).

.

= .!~aY ..ax + .. + + . '

. .

l'

. u = ~ log ( + 2) + tg ~ + .2' "

6.

-

-:

10.

(2 l +2 + 2) +2'

(!

+2 +3 ) ..

.l 8.18.11 t}

.,,-~ + 2 + + +.

3.

20.

,

(3 l + 2 )'

+ 2 ( + )

. I ',.

"= (3'+2')+2 JYdY =x'f-2xy+yt+ .

30.

3

n= l' -('

'.

+ )'!

+

(2 +y2~1I

-'---

+.

h . . ll

.'

' -"'~' ,1Og(xJ -+ ') +" 19~;+.c.

40.

3 -=

-1

+22 .

_

2 + 2+ ' (

h " ,u=---+. +2-

,"

(2 + )!""

+)1,:

50.

"

- - Z,

-

'3

z

+ -:--.-,-dzZl

3-

JV . ll8.-n

IIJI ,1I.n

67

:-3 u=--+.

z

Q (,)

25. .

-

an

"

(, )

Gn " (.'

180).

n n

(14)

Pdx + ~dy.

.

,

(,)

(+4,)_

.

-----

.

.

.

160

161

!" n (x~, ) (,) " " ~1'

-

cpeloj JI

(!! 128) . h

.t

_(x+~x,y) - (,) = P(x+9L\x,y), 0 8.11. ,et8JlJt.:'+,r";"lli8iJ l)

,

20.

,J2xy~-:.~J~y= 1,2). = ~(.')

(i;H

. .....-'"

.

.

.

.

(3.4)

..

f

YdX

...

.~ .. = +-'

101),

.== -5.

(1;-:-2)

'.

'~d~'~~d-:-2 I I + I I '.

1

,,)."(.!)

.'

siny!x+~toS ydx= [ .$iL = sin -

"~;1"

..

,.'~

'.~

(\

~

.

.);. ~,~.

6o .rf\X- d.~-Xi~d~:'-'4) x~l,y- l.

W .( +}l)1

.

. ; 2& rpa.8. - .-':' . P(x,y.z).Q (x".z) ;R.(x,Y;ZYHenpeaAHe 8 .}' 11l.. . . w ~. I} =

+ ( -

~) ]= .. = f(x),,

f ()

'- 2 ,.,.,

,~n (, .) =

f [( - (1) + -f(x),.,. -n ) = t[(x- (3) + =- {[- ( -l)] .,., f ( 7=

f [~- (IJ -2) (1)1 = ,. ',=

{{),

. , f () li (1), h w ~ , w' ' ll.Atu1i (1) f1lUl1U ~' . ( ,

~~ 080, ~ , ~ ~:~, .,

..

IfHTepJ;lY ( ].

r nR~I(8 he lUljQ ..

n8 .

(1),2(1), . ;'

,.,., -2,

.. "

l:

10. .u uuu 'u () . ,_

-

,

= t () -

t ( +

(1)

,

. .

76

r n

20. l1uu uu l1udu uu lli.l1.

+ ) =~O

!(),

f(x + 1 lm -+

00+ h) - .f(x + ) = l'lm ! ( + h) - ! ()h h

-

!' ( + w) ~ !' ()..

30. iiuu u! ()

w ulliu,

(1)

! () dx = +

a+1D

f

f(x)dx.

! () dx =

f

+

f(x)dx

+ ! () dx

+

! (x)dx

=(1)

t

+ ,

h

(1).

(1)

.

,

= ).

x+h=y,

[!( + h)dx =- [1 () dy = [1 () dx =

Jt+w

h+

[f(x) dx,

h

.

40. u..ulliuu uu l1u uu . lliu l1epuo ull.

j~ F ()

t

(),

(2)

( + )1

- F

().= fl(t) dt -- 1(t) dt =

+

f

.

F(oo) - F(O),, ~

'F () = F~O), . F (

+ )

== ().

() = + cos F () = si. . , (2),

+

1 )

2 11:

x+a.r

2 11: =

( + cos ) dx =

f ( +

a.r

cos X),dX'=- 2a'lt,

furlr- ')

77

, , ,

(

+ 2 'It) + sin ( + 2 'It) =2a'lt+sin

+

2

+ sin 4= + sin

2a~

+

, ,

2o=2a'lt4=aO+sinO

2a'lt4=0,

F () =

,.: sin(

.

)

= sin

+

),

f () = cos

(

+ )

2'1t -,

f ( + (&1) = sin{(38

[ ( [ (

+ ) + =+CI)

sin

( (

+

CI)

=

cos.

+ ]

= cos

+- + ) = sin (.( + ), +- + aCl) = cos (( + ),

w = 2 'It,

W

2'1t = - .

.

{() ~

tg (

+ )~

I

{()

= cotg ( + )

-,

f ( + CI) f ( + )

= tg [ (

)

=

+ ) + ] = tg ( + + ) =tg ( + ), cotg [ ( + ) + ] = cotg ( + + : cot& ( + ") BCI)

='It,

.

W

=

'It

{()

CI),

.h f (kx) Q ; ,

/[k239.(3)1)

(~+ :)]='/(kX+CI)):=f(kX).1)

'

OpTOOH&JIe .

[, , (

'1

< )

- . . -

(), /2'() , , /,. (),. '"

kx ...

,

f(k)

-1() - '( + 01) ==

f(kx

+ ) -

f

lk ( + ~)]

'78 .~

-.

(3) (filfl, .,..}, .

(4)

'm ({) dx = , =F n7{m,lJ ... l,2, . ~)

. ,,,() ( :=-1'2' .. ) , n [, .8~ii , 8 (3)

(5)

'~ () dx =

kn "

( ":'1.2, ...),

k,.

> .

kn - 1 = 1~2, .. I .

,

{3 HOPAfrUPDH. , () -,

DD .., ' ... "

, (3) h lJlJ.1l' (, ], (6)

!tn ) ( ) d

{

=,

{

=Fn,I=

("", "1 , 2',.; .. )" , ' =

,. r Jl 1

. , lI (3) .8 (4)' (5), 8 . (5)

(4)

11

() ,,-/,. () dx -=-=Jlm {x)fn (x)dx = ,,-Im , kn ,km Vkm kn=~ '

1

1

1

'

, =F 11,'

,f [v:,. '"

'

(X)JdX .~;,.J [1 () =:1,

( = 1,2, ...).

"

(3)

lllt

-=

1

Vk1

11 (),

....... I.,,(x), ... , ; " , kti V ks

1,

1

',.

(), ...

' 4 , u (, ).

1:9~VlkwrOt,Ut8fl (

~

k, -

V!t., ) ~x..,i"

1n (;i''~l1;',

11f.II;

.(~, ~ 1, ,,,,;',.. '

IIfnll = 1 ("",'.

1,2,3, ...) () j{'~pilio~opliup~H~:':'.

u.Atu. '""'7.1~~ "" .,,' ~":"

,~.

'.

'':.'

',:

\

-1"

~".

i:'-

1. sin , sin 2, ... , sin n) . ..".~;::, ,.:,(~II:-t!1.g.',20}i ,::Q..

(4)., '

,'~

,: ;.>'.';:

,',/_ " ':-=: $~ ; ,-/_ 6:~ X~ : ; " .::,..... I n, ..

" .("', ',,,,'.;'

1

1.

1.

t)

1.

,2

3'

'

''V3t

"(6).

~ ~ [, {l

2'.

~*';' !.' .... ';"

t

21], fC.JI~e'i,'.,! (--, ,. ';

'!

'

" :(:n.,:,

,; ,.,' ,,'\

")!;",~qf:~x~cR~~x"~:i:,;;~8~'~i~"U,,,;,;. ~ . . \ $'!','; . . . "-\ ';.: 't- ;', ~. ':~ ~ . '.'>~ OpmOHOp.AtupaH [, 4 +2r.), (6). ;;:-;:.'!

-==, ,,_ cos ,V23t,

.

1)li ',.\ ,.,'\--

,,_ cos 2 , .. , ,,_ cos_.

':"1';'

",." " n, ; :'.\ ,', _) . ;

8" (4):

.; ~:

~

,

.;

1~

.

:1.,: ", ,-

"." ",

:';1

:~";,; ~),6~.;,It~1.,,4'.:),~..:;,:',(8)l ()

-

dx dx' .");:~ 4.:(~;-':4)n, " .,." )~ , " ';': ,:':,. '(' ,. ..,... n dx

d (x~:';'lj:"'H; i:f".::. , () =

J(j2":(xi':.2l)t' ~;ti:,:,; '1 : " ", ... ,

;'.1': (,"' :"

80 u

[- 1,

+ l.

(1 -

2) "

- 2 ' +'"11 (

+ 1) "'"

(9)

dx

-[(I- 2 )'+(+l)=,

d

"",,().

() " () Lgd-

I(

(9)

- [(1 - 2 ) '() + ( dxd

d

.

+

1) PIlI(x) =0, .= .

dx [(1 - 2 ) ,,' ()] + ( + 1) . ()

" () () , , h

" ()

d ' d - [(1- 2) ' ()] - () -[(1 - 2) ,,' ()] + d-c dx,

+ (

-) (

+

+ 1) m () " () = .- 1

.

1

( - n)(~ + +

1)

f

+1

111

() " () dx = , , -

-1

.

f () P,,(x)dx == ,-1

+1

9= ,

. Lgd- u. h , 1 ) -

Furlr-

81

f PS,. ().-1

+1

dx =

,. () II () dx-1

+1

-

f z Jl '[u;-

t ()

.ll

~e . ;-

t ()

II k. .

f ().

f(x) ' '

f (),

.ll

t (),...., -.! + ~211=1

( cos n

+ ,. si n). Poje-o~

.

f

()

.

.,...,,"

.ll

==...

.

.~ f(x).

~Qll

rllI

(10)

1I0

' .ll . = - n. B.IIII

sin

,.

=

n

cos'n

... '

cos n't

,. = ,.

sin ',

,

.-

(10)

.-!

2

+~

[" sin (n

11=1

- )] .ll .-! + ~ [t" cos (n - ,/).,2n

11=1

, 1)

f

()

cos

f

()

sin n

qe Jl.

84

r

r -

.,1

2

'

+ 2

tg

.

= _

" '

tg~' = -.!!.

TaK~,

COS

....

"i +-".

22

Sl lx

= ----

-

- l

(10)

t ()=

lll1lu U1D.ep8tUy

(14)

limk-+oo

rf ()f

..

sin kx dx

, ! () cos kx dx - .k-+ao

1) .u

() (, ,

h

~je

If ()\

(JIF.!

125,(!

50.).

(, .

If () I

f ()

190).

Fourler-O.8 ~.

85

.n n [, ) .n , , n ,

- "

t ()

' [, ).

(15.). ;

, ~ ( 1 - ;) (I - I_I)

< -.

;=1

(i > ). 2" (() n

..

;

.

[Xt-l, {. l )

(16)

Jt(X) sin kx dx ==

1=1%1-1{

t

{

! () sin kxdx1(I-I) 8in

=

1=1

I (~I-I) '%1-1

%

sin kx

+

+ f'f[t(x)1=1{_I

-

kx

;

n ,!

(k

> (1=1

If It 1(%1 - 1) sin kx dX"' = ~ k[=1I-I,

1(-':) [08 kX;-1

-

08 kXt] , () sin

U.

(31)

fui- sinus-~Ma

u .

.

-

h

Fui- -

f (), (- ~, + ~)

( -

.1-) = (), ("" - ) = (),

. =..~ ..;.:~

:

[1 + (- 1)"] q> () cos nx dx, "

2

furlr-

'

111

211

4-

7t

f () cos

2

.

2 dx.

.() , ~oaJe-OB

~inus-,

. (r..- ) ='- (),

(- ) = - (), ,

(41),. 2

N = ~ () si n dx =

:

[ '()

sin n dx

+ () sin n dx =!..2

!..=

.

~ [1 + (-I)n] ()

, 2

sin n dx, .

2

2n

=

4 ' () sl 2 n dx, -' ..

.

,

b2ll +1

= 6.

( -) -= - (),~ 2

cp(s -

) ~.. cp (),

N = oane(43")

:

s[1 ~ ( - l)nJ () sin n

dx,

"

n = ,

blll +1 =

~ ()

s2

sin (2!1

+ 1) dx ), . 8

- () .. 8 . 8.11 ( 1;,

(-

,

+ ).

.

lLAt. - 141. r ,() = (42), . cosinaS-8 -8.11 [, ]. , (40),

112

r

, = ~ dx = f, = ~ cos n dx =

--[(-1)11-1]n 2 .~

2

I

, , 4 _ - - , u,n

(44)

-

_ ~2

-

~ ( ~ cos (2n - 1) ) . 0- _ ~ , ""'" ""'" 1t 11=1 (2n - 1)2

'/1;.

u. ii1. Jl

( -1t,

) h

-=---}:

1t

4 ""

2

1t 11=1

cos (2 - 1) , (2 - 1)2 II

-1t~ , un.i1Jl& lin

" () I&

=

+~k=1

11

(

cos kx +

~I

sin kx) ,

(66)

,!()-,,()

+-

+

83 f ( dt .'I "li-

148

.. (!!+

2] 3)

t (t) cos (t (2 214)

-

) dt.+

~

!. Jda. ! (t) cos . (t - ) dt = ~ ! ) dtJ cos . (f - ) da 11: 11:

+

_

~CI)

-CI)

8.11,

1.

. . -

n :

10.,

n(105) u

)

= e-kx , k>O,

>,

Furlr-

151

h

-2

xd ' e- kt cos tdt ... -2 k ~. "

fcos

f cos

d ~+~

=e-k:&, .

>.

20.

(106)

f(x)

= e- k ,

k>O,

>,

-2 ~

sd

.

' 2 sin -ktstdt=.. d =e- kx ,~ ~+~

>.

255. 8 Furir- . . ~Jl 1 ) . (107)

.

",.(,), '1'1('),

... , ",,,(,), ... . 3

2 (

< < ; < efOBe tes-

()

DescarOxyz. 80

(9)

, .- , =- ,

= xi+yj+zk,

,

= (x,y,z),

z

=

! t ()

OXy'z. h , Q Q. 8ll,

(12)

=- (ll .. ) i

+ ( ) +(, bz) k , , ,

' ll . ,

. . . h , , (12), 'i++zk = i+"+zk,. a~ =, ! = , ! =b~. . }. u MN lJ (. 180). ltu,

MN= P+Q -2

1.

8

163

=

AD,_ q

-+

;=

.

-+

MADN-+

MN 8

-+

==

-+

+p+DN ,-+

MBCN

MN-+q+h

-+

-+

CN.-+-+

h

2MN ~ ++ +q +DN+CN = p+q,

-+

-+

-+

D180

AIi---:--.L---~,

,

181

-+....

+

=

,

DN + CN-+ -+

-+-+

= , Jl .

20.

, !' ""?

u

AD+BE+CF =, .

(.

181).

-+-,

-+

2

CP=a+-, 2

-+

,-+

=='+ - .

.

2

, h

"

-+ -+ -+ 1 3 ++= a+b+c+-(++)= -( ++) =

2

2

++ = . .

t1.

-

~

:

10.

1Ulu,

' (.

182),

164

. .

(13)

{

- -

DE--2'-+ -+ ..,.

-+ -

-:+~

81:= -+=--

-+

b~C'2 '...

-+

=

---

-: '2

-

, h

AB+BC+CD+DE+EF+fA =. ll . 80Jl " .

-+

-+

, ,ll .

'

(13) cJleAyje--.

-+

-+

-+

-+ MN

= ,

...

""" h 1

...

-+

-+ -+ NP-CD (CJl. 183).

.

...

,

...

I8, .

-+ CD

,llJl 1

ACD 1 . - , poyrao.MNP 80

...

...

CD

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enopea

a.llre6pa

165

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.

.

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,

.

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. .

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. , , , .

Il"

-+

+ + + + - 3 .+ AF =-+-+

-+

-+

-+

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.

,

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+

== , +

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=-

-+ -+ -+ -BD+AE==AD, .

40.

,

.

- (3, 4, 5)

""

Belllllop

ll lI1.

(

= 3, 1

==

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a=3i+4j+5k,

'' = )'9+ 16 + 25 '"'" 5V2, = 3i+4j+5k,==

3V2 cos = -10 ' cos ~

5 ' cos

2V2

=-

-2"

V2

- 45' .

. 5'.

. ,

.

. Iloje cellY mIlU .

ll, ++ -

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2 .-+ -+ -+ -+ -+ -+ ++--+

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+= ,

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=

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CTora

, ,

-+ -+ -+ -+ -+ -+ +++++ = , ,

3 .

++ == 2.

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6'.

." " IIlllJp lJ.

" S, , , 06'JlC. peClJellmUBHo -+ -+ -+ -+ -+ ... , , , "' AS + BS + CS - DS + BS + FS. j~ "8 lI ,

-+ -+ -+ ... -+ -+ AD+DS==AS, BE+ES-BS,8 ,

... -+ CF+FS

=

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... -+ ... ... -+ -+ -+ -+ -+ AD+BE+CF+DS+BS+FS= AS+BS+CS,

166, +

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+ ,= ,-+=

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AS+BS+CS70.u uu.

-+

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DS +ES+PS.

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= (5,-2,4)1,

ii m.

I= \12-5 + 4+6- = "45,5

=

~ I!=

=

~~~~1-1- ~ !tV45

cos = ~45'

cos ~

=

-2 45'

cos =

4 "45 . (.

80.

-+

Oxyz. 1 il m~

184).

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-+

8

=r2 -r1 , .

r1

r2

r1 r2O~______~______

= =

1 i

k. x 2 i+Y2j+Z2 k .Zl

+ yt t

/

a=(X 2 -)

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+ (Z2-Z1) k,

: I =sis-

\

a~ +a;+-a~II - 1

= \

( 2

-

1):)-- - (y;=y-;Y~ -+

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ZJ2 ,

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,

90. ll ,1 li 3, li li.4t ii}' . li. .I/. Heraili'IBHlI.U . Oz .

)

= i + + , k, (-

=

cos =

-305 ,

: = -

! ! cos (

= : : cos (% -= = - 3 5 0:,

COS:x,

1.

167

= 3 cos IX (

-

;-" - k),

3

=

v3 . 9 cosk) ..

J

,

COSIX =

-=,

1

V3

=

v"3 ( - i + -

10.

.

uu

(2,-5,3)

u r 1 (4, 1, 2) u r 2 (3, - 2, 5), -+ ... -+-+ u . . r, ~ , r 2 = . Hahu -+},

u

.

u n , -i>

.

= + = 2i-5j+3k+ 4f+j+2k -~ 6f-4j+5k, (6,-4,5),

-+

-+

= + ~ 6f-4j+5k+3i-~j+5k = 9f-+l0k,

...

...

...

(9,-6,10),.-

-.

=-AB-8C=-4i-j-2k-3i+2j-5k = -7i+j-8k.

-+-+

262.

.

u ii.110U li

cosinusa

u.4t , .

4

. =

I I I cos (,

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.

,

, , r .

Bel{-

D

u

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-

....

....

=

-+

=

08

...

('l. 185). I

1.

8 n

177

(27) , , .

< 0.1)

(27)

> .

n .

~

(27)

, uulll .,

50. ullllll .." , f1l lllj.(+) =+. = ,

(28)

-+

= 08,

-+

==

-+

R

-

= (.

-+

189).

D

.

189

: ,

..

==

-+

+, ....

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R h 2)

1

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. =

001

== Ib+cl cos(Dj OD) =Ib+cl si(+, ), =(+),

(30)

...

=R,

... -+ , .::::;: , OD. 8n

~

~s ,I l 80,

DOA.1)3

< , 08., .,' OD. n JUl4R

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...

<

(27)

.

ra eroB .

1) nn

OCD8 R n . D. 81'12

178~ ~ ~

~ ~ ~

8 1 , 1 ,~

OD 1 ~

081'...

!,

OD 1

=

08,

= ,

+ 1~

=

~

-+

R,

.

== 081 + 1 (29),

(30),

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=

Il sin (, ) = 08 1 ll

2 = Icxal-Icllal sin (, al= 1 Il,

2 =

1 (

+ )l

=

'+ cllal sin (+,)= 1 'lal,

082 D 1 Cz 081 D 1 , , 081 D J 1 081 1 1 I 1 900. 082 2 2 ~ ~

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-+

,

(30),(

2 =

082 + OCz=

+)

+ ,=

(28).

(+) - +:(a+b)x(c+d)~=

(+)

+.

axc+bxc+axd+bxd.

~ 8 . ~

(31)

=-

+oyj+ozk,

=

l++zk ,

Oxyz.

, hII=,

iXj=k,=,

iXk--,

jxl= .... k, kXI= ,

jXk=l, kXk ... 0.bzk)=

k

=

-1,

,

(32)

= (l+

+ozk)

(.. + +

= .. () + .. tjXil + . b.. (kX )+ (I) ++ (}+.

(kxjl +0.. (iXk)+oy z (jXk)+d" 1 +dyj +dz k

+. (k k) = ( - . ) '+(. - bz)j +

+ ( -

)

k

=

1.

l.II

179

i

k

(32')

-

" :

.

h

IJl

h. , , , , . ,

.

(32),

: " , -, -,

:

,

. =

....

:

:

(O.II) ..

-

U

a ..... 2f+3j-k,

==

-31-j +k.i

k

=

2

3-1

-1

=2f+j+7k.

-3 08

+1

laxbl- V4~1+49-V54266. m .

(.

=

, ==

-+

08,

--+

--+ ==

190).

'.

() .- []

ulii

.

.'

= -= d,

-+