Сборник задач по высшей математике 2 том

4.3.30. HaihH Koop,n:HHaTbI u.eHTpa TIDKeCTH O,n:HOPO,n:HOft nOBepXHOCTH

	z = Ja2 - x 2 - y2	(x ~ 0, y ~ 0,	x + y ~ a).
4.3.31.	HaftTH nOJUlpHblft MOMeHT HHepu.HH 10 nOBepxHocTH Ky6a Ixl ~ a,
	Iyl ~ a, Izl ~ a.
4.3.32.	HaftTH MOMeHTbI HHepu.HH TpeyroJIbHoft nJIacTHHbI x + y + z = 1
	(x ~ 0, y ~ 0, z ~ 0) OTHOCHTeJIbHO Koop,n:HHaTHblx nJIOCKocTeft.
4.3.33.	HaftTH nOJIHpHbIft MOMeHT HHepu.HH nOJIHoft nOBepXHOCTH U.HJIHH-
	,n:pa x 2 + y2 = R2, 0 ~ Z ~ H.
4.3.34.	BblqHCJIHTb nJIorn.Mb TOft qacTH napa60JIOH,n:a 2az = x 2 + y2, Ko-
	TOpM OrpaHHqeHa nJIOCKOCTHMH y = 0, z		= 0, z = ~, z = xtga,
	a - <l>HKcHpoBaHO, 0 < a < ~.
4.3.35.	BbIqHCJIHTb nJIorn.Mb TOft qacTH nOBepXHOCTH c<l>epbI X2+y2+z2 =

=a2, KOTOPM Bblpe3aHa U.HJIHH,n:POM x 2 + y2 = ay.

4.3.36.BblqHCJIHTb nJIorn.Mb TOft qacTH nOBepXHOCTH c<l>epbI x 2+y2 +Z2 =

=a2, KOTOPM Bblpe3aHa U.HJIHH,n:POM x 2 + y2 = b2, b < a.

KOHTponbHble BonpOCbl M 60nee CnO)l(Hbie 3aAaHMH

4.3.37.	HaftTH nJIorn.Mb nOBepXHOCTH z		x;, pacnOJIO:>KeHHoft BHyTpH
4.3.38.	U.HJIHH,n:pa x 2 + y2 = a2.
	HaftTH	!!zd(1,
		(T
	r,n:e (1 -	qacTb nOBepXHOCTH 2az = x 2 + Z2 (a > 0), Bblpe3aHHM
	KOHYCOM z = Jx2 + y2.
4.3.39.	HaftTH	f'r,
		j(x + y + z)d(1,
		(T
	r,n:e (1 -	BepXHHH nOJIOBHHa c<l>epbI x 2 + y2 + Z2 = a2.
4.3.40.	HaftTH
		(T
	r,n:e (1 -	rpaHHu.a TeJIa, 3MaHHoro HepaBeHCTBaMH Jx 2 + y2 ~

~z ~ 1.

4.3.41.BblqHCJIHTb

	!!zd(1,
	(T
r,n:e (1	- qacTb nOBepXHOCTH reJIHKOH,n:a x = U cos v, y = u sin v,
z = v	(0 ~ u ~ a, 0 ~ v ~ 211").

4.3.42. BblqHCJIHTb

(T

r,n:e (1 - qacTb nOBepxHocTH KOHyca

x = r cos cp sin a, y = rsincpsina, z = rcosa

(0 ~ r ~ a, 0 ~ cp ~ 211"), a - nOCTOHHHM (0 < a < ~).

230

4.3.43. BbIqHCJIHTb

I j(Xy + yz + xz) da, u

r,ll;e a - qacTb KOHHqeCKoit rroBepxHocTH z = Jx 2 + y2, BbIpe3aHHruI rroBepxHocTblO x 2 + y2 = 2ax.

4.3.44. BbIqHCJIHTb

IIx 2 dydz + y2 dzdx + Z2 dxdy, s

r,ll;e S - BHeWH.H.H cTopoHac<pepbI (x-a)2+(y-b)2+(z-c)2 = R2.

KOHTPOl1bHA" PA60TA

BaplilaHT 1

l -y2xdl,

L

r,ll;e L - ,rryra rrapa60JIbI y2 = 2x, 3aKJIIOqeHHruI MeJK,rry TOqKaMH (1, \1'2) H

(2,2).

1(4Y + 4) dx + (3x + 3y + 4) dy,

L

r,ll;e L - KOHTYP TpeyrOJIbHHKa x = 0, y rrpoBepHTb rrpH rrOMOm;H <POPMYJIbI rpHHa.

(4,9)

I ( 3X2 - 3y2 _1 m dx +

2V~)

(1,1)

= 0, 2x + 3y

6, H pe3YJIbTaT

(-6XY + _1_) dy.

2yxy

4. BbIqHCJIHTb rroBepxHocTHbIit HHTerpaJI rrepBoro pO,ll;a

2	2	2
r,ll;e S - 60KOBruI rrOBepXHOCTb KOHyca x2	+ Y2	= ~2' 0 ::; Z ::; h.
a	a	c-

5. BbIqHCJIHTb rroBepxHocTHbIit HHTerpaJI BToporo pO,ll;a

IIy2dxdz,

u

a - BHyTpeHH.H.H CTopOHa rroJIyc<pePbI

x 2 + y2 + Z2 = R2, Y ;;::: O.

231

BapMaHT 2

j(x + y)dl,

L

r,n,e L - KOHTYP TpeyrOJIbHHKa ABC C BepWHHaMH A(l, -1), B(-3,-1),

C(-3,2).

2. BbIqHCJIHTb MacCY .n.yrH qeTBepTH 9JIJIHIICa

x 2 y2_

a2 + b2 - 1,

pacIIOJIOlKeHHbIil: B IIepBoil: qeTBepTH, eCJIH JIHHeil:HM IIJIOTHOCTb B KaJK,n,Oil: TOqKe IIPOIIOPII,HOHaJIbHa a6cII,Hcca 9TOil: TOqKe, C K09<p<pHII,HeHTOM m.

3. BbIqHCJIHTb
	(2,2)	~) dy.
	j (6X - 3y - t) dx + ( -3x +
	(1,1)	Y
4. BbIqHCJIHTb IIoBepxHocTHbIil: HHTerpaJI IIepBoro po,n,a
	jjz4 dS,
	S
r,n,e S - 60KOBM IIOBepXHOCTb KOHyca 4(x2 + y2) = Z2, 0 ~ Z ~ 2.
5. BbIqHCJIHTb IIOBepXHOCTb HHTerpaJIa BToporo po,n,a
	jjz3 dxdy,
	u	= 10, pacIIOJIOlKeHHM B
a -	BHewmUI IIOBepXHOCTb IIJIOCKOCTH x + y + z
IIepBoM OKTaHTe (x ~ 0, y ~ 0, z ~ 0).
BapMaHT 3
	j(x2 + y2) dl,
	L
L -	OKPyx<:HOCTb (x + a)2 + y2 = a2, a > o.

2. BbIqHCJIHTb KPHBOJIHHeil:HbIil: HHTerpaJI BToporo po,n,a

j(x + 1) dx + xyzdy + y2 z dz,

L

r,n,e L - oTpe30K, coe,n,HHflIOIIIHil: TOqKY M(2, -1,3) C TOqKOil: N(7,4, 11).

232

3.

4.

BbI'IHCJIHTb

BbI'IHCJIHTbnOBepxHocTHbIit HHTerpaJI nepBoro po,n:a

jjz3 dS, s

r,n:e S - BepxmUI 'IacTbnOJIyc<pePbI x2 + y2 + z2 = R2, Z ~ o.

5. BbI'IHCJIHTbnOBepXHOCTb HHTerpaJIa BToporo po,n:a

BapillaHT 4

1. BbI'IHCJIHTb

j(x2 + y2 _ z) dl,

L

L - .n:yra u;enHoit JIHHHH x = a cos t, y = a sin t, z = bt, 0 ::; t ::; 1r.

2. IIPH nOMOrn;H KPHBOJIHHeitHoro HHTerpaJIa BTOpOro po,n:a BbI'IHCJIHTbnJIOrn;a,n:b <PHrYPbI, OrpaHH'IeHHoitKpHBbIMH x = 8 cos3 t, Y = 8 sin3 t, 0 ::; t ::; 27r.

3. BbI'IHCJIHTb

4. BbI'IHCJIHTbnJIOrn;a,n:h TOit 'IacTHnapa60JIOH,n:a Bparn;eHHH z = -a(x2+y2),

KOTOPM HaxO,n:HTCH B nHTOM OKTaHTe (x ~ 0, y ~ 0, z ::; 0) H OrpaHH'IeHa nJIOCKOCTblO z = -2a (a < 0).

5. BbI'IHCJIHTbnOBepxHocTHbIit HHTerpaJI BTOpOro po,n:a

jjyzdxdy,

q

r,n:e a - BHeWHHH CTopOHa nJIOCKOCTH 2x + 3y + 4z = 12, pacnOJIO)KeHbI B nepBoM OKTaHTe (x ~ 0, y ~ 0, z ~ 0).

233

BapillaHT 5

j(x - y)dl,

L

L - JIeBbIii JIerreCTOK JIeMHHCKaTbI r = aJcos 2cp, a> 0.

2. BbIqHCJIHTb pa60TY CHJIOBOrO rrOJIjI F = (x + y + 2)i + (8x + 7y + 6)j rro KOHTYPY TpeyrOJIbHHKa, CTOPOHbI KOToporo JIe:lKaT Ha rrpjlMbIX x = 0, y = 0,

3x + 2y = 6, H pe3YJIbTaT rrpOBepHTb rrpH rrOMOIII;H <POPMYJIbI rpHHa.

3. BbIqHCJIHTb
(0,0)	(
j		2	1 2	+ lOX2y) dx + (	1 2	+ lOX2y) dy.
1		x	+ 2xy + y	+ 1	(x + y)	+ 1
(0, 0)
4. BbIqHCJIHTb rroBepxHocTHbIii HHTerpan rrepBoro po,n,a
				jjx2 dS,
				s
r,n,e S -	HH:lKHjIjI qacTb rroJIyc<pePbI x 2 + y2 + z2 = R2, Z ~ 0.
5. BbIqHCJIHTb rroBepxHocTHbIii HHTerpan BToporo po,n,a
				jj(x + y) du,
				u
r,n,e u -	BHYTpeHHjIjI CTopOHa KOHHqeCKOii rrOBepXHOCTH x 2 + y2 = Z2, pac-

rrOJIO:lKeHbI B rrepBoM OKTaHTe (x ~ 0, y ~ 0, °~ z ~ h).

rnaBa 5. TEOPlll~ nO/1~

B 3TOit rJIaBe Bce paccMaTpHBaeMble <PYHKn:HH 6Y,IU'Tnpe,n:nOJIaraTbCH ,n:H<p<pe-

peHn:HpyeMblMH ,n:OCTaTO'lHOe'IHCJIOpa3.

§1. CKAJUIPHblE VI BEKTOPHblE nonH. nOBEPXHOCTb YPOBHH. BEKTOPHblE nVlHVIVI

CKallHpHoe nOlle

~ IJIYHKn:HH U(r) = U(x, y, z), r,n:e r = x . i + y . j + z . k -		pa,n:HYC-BeKTOp
npOH3BOJIbHoit TO'lKHnpOCTpaHCTBa 1R3 , Ha3bIBaeTCH C'lCaJlJlpH'btM nOAeM.			$:
	CKaJIHpHOe nOJIe MOJKHO paccMaTpHBaTb KaK 06b1KHOBeHHYIO <PYHKn:HIO U =
=U(x, y, z) Tpex nepeMeHHblX.
	HapH,IU'C onpe,n:eJIeHHblMH Bblille CKaJIHPHblMH nOJIHMH paccMaTpHBaIOT TaKJKe
nAOC'lCUe CKaJIHpHble nOJIH, T. e. <PYHKn:HH U = U(r) = U(x, y), r,n:e r		= x· i + y. j	-
pa,n:HYC-BeKTOp npOH3BOJIbHoit TO'lKHIIJIOCKOCTH.
~	IIo6epxHocm'b'lO yp06'HJ! CKaJIHpHOrO nOJIH U = U(x, y, z) Ha3b1BaeTCH MHOJKe-
CTBO TO'leK npOCTpaHCTBa 1R3 , y,n:OBJIeTBOpHIOID:HX ypaBHeHHIO U(x, y, z) = c,			r,n:e
C -	npOH3BOJIbHaH nOGTOHHHaH.		$:

AHaJIOrH'IHOonpe,n:eJIHeTCH nOHHTHe AUHUU yp06'HJ! nJIOCKOrO CKaJIHpHOrO nOJIH

U = U(x,y).

IIoHHTHe nOBepXHOCTH YPOBHH H JIHHHH ypOBHH CKaJIHpHOrO nOJIH TOJK,n:eCTBeHHbI nOHHTHHM, COOTBeTCTBeHHO, nOBepXHOCTH ypOBHH <PYHKn:HH U = U(x, y, z) Tpex nepeMeHHblX H JIHHHH ypOBHH <PYHKn:HH U = U(x, y) ,n:BYX nepeMeHHblX.

BeKTopHoe nOlle

~BeKTop-<PYHKn:HH F(r) = P(x, y, z) . i + Q(x, y, z)· j + R(x, y, z) . k Ha3bIBaeTCH

6e'ICmOpH'btM nOAeM.

$:

~BeKTop-<PYHKn:HH F(r) = P(x, y). i +Q(x, y). j, r,n:e r = x· i + y. j, Ha3bIBaeTCH

nAOC'lCUM BeKTopHblM nOJIeM. $:

~ JIHHHH ret) = (x(t), yet), z(t», KaCaTeJIbHble K KOTOPblM B KaJK,n:Oit HX TO'lKeco-

Bna,n:aIOT C HanpaBJIeHHeM BeKTopHoro nOJIH F(P, Q, R), Ha3b1BaIOTCH 6e'ICmOpH'btMU

AUHWlMU 3Toro nOJIH. $:

AHaJIOrH'IHblMo6pa30M onpe,n:eJIHeTCH nOHHTHe BeKTopHbIX JIHHHit nJIOCKOrO BeKTopHoro nOJIH.

235

Bellu"l.u'Hof1. zpaiJue'Hma

BeKTopHble JIHHHH BeKTOpHOrO nOJIH F (P, Q, R) MOrYT 6b1Tb Haii,n:eHbI H3 CHCTeMbI ,n:H<p<pepeHII;HaJIbHbIX ypaBHeHHii:

!~~ = P(X, y, Z),

dy

dt = Q(x, y, z),

~; = R(x,y, z)

HJIH H3 CHCTeMbI, 3anHCaHHOii B CHMMeTpH'IeCKOii<p0pMe:

BeKTopHble JIHHHH nOJIH Ha3b1BaIOT TaKlKe CHJIOBblMH JIHHHHMH HJIH JIHHHHMH TOKa.

~ rpaiJue'HmOM CKaJIHpHOrO nOJIH U = U(X, y, Z)									Ha3b1BaeTCH BeKTOpHOe nOJIe
{JU.	{JU	•	+	{JU k	=	U' •	U' •	+	U'	k	~
grad U = {Jx . 1+	{Jy	. J		{Jz'		x' 1+	y' J		z'	•	"F

rpa,n:HeHT CKaJIHpHOrO nOJIH B KalK,n:oii TO'lKenepneH,n:HKYJIHpeH nOBepXHOCTHM YPOBHH 9Toro CKaJIHpHOrO nOJIH. KpOMe Toro, rpa,n:HeHT CKaJIHpHOrO nOJIH U nOKa3b1BaeT HanpaBJIeHHe HaH6oJIbIllerO pOCTa <PYHKII;HH U = U(x, y, z).

Ha3bIBaIOT CKaJIHpHOe nOJIe

5.1.1.	HaiiTH BeJIHqHHY H HanpaBJIeHHe rpa,Il;HeHTa CKaJI1IpHOrO nOJI1I U =
	= x2 - y2 + yz - X B TOqKe A(l,O, -1).
a Haxo,n:HM qaCTHble npOH3Bo,n:Hble <PYHKII,HH U:
	U~ = 2x - 1,	U~ = -2y + z,	U; = y.
TaKHM 06pa30M, grad U = (2x -		1) . i + (-2y + z) . j + y. k. IIo,n:cTaBJI1I1I B
nOCJIe,n:Hee paBeHCTBO Koop,n:HHaTbI TOqKH A, nOJIyqHM:
	grad U(A) = i - j = (1, -1,0).
	IgradU(A)1 = J12 + (-1)2 + 02 =../2.			•
Ha1J.mu apaaue'Hm c7l:aJlJlp'Hoao nOJIJI U(x,y,z):
5.1.2.	U = xyz.	5.1.3.	U = x2 + y2 -	Z2.
5.1.4.	U=eXY _ yz2.	5.1.5.	U = In(x2 + y2 + z2).

236

HaiJ.mu apaaue'ltm CfGaMp'ltOaO nOM U(x, y, z) 6 mO,,{fGe A(xo, Yo, zo):

5.1.6.	U = 3x2 - xy3 + XZ - Z2, A(l, 2, 3).
5.1.7.	U = zsin(x - y), A (~,~,1).
5.1.8.	x+y
	U = -z-, A(2,0, 1).
5.1.9.	U = arctg(x + 2y + Z2), A(l, 1,0).
5.1.10.	B KaKHX TOqKax rpa.n;HeHT CKaJUIPHOro rroJUI U = xy + 2z - Z2:
	a) K03131HHeapeH BeKTOpy a(l, -1, 2)j
	6) rreprreH,n;HKY31jfpeH OCH OXj

B) rreprreH.D:HKY31jfpeH rr31OCKOCTH 2x + y + z = I?

Q BbIqHC31HM rpa.n;HeHT rr031jf:

grad U = y . i + x . j + (2 - 2z) . k = (y, x, 2 - 2z).

a) ,lVIjf OTBeTa Ha 3TOT Borrpoc HarrHmeM YC310BHe K03131HHeapHOCTH BeKTOPOB grad U H a:

¥.. ---.: ~ _	2 -	2z ¢} ~ _ ¥.. _	z - 1
1 - -1 -	2	-1 - 1 -	-1 .

2ho ypaBHeHHe rrpjfMoil:, rrpoxo,n:gm:eil: qepe3 TOqKY Mo(O, 0,1) C HarrpaB31jfIQ- m:HM BeKTopOM l( -1,1, -1).

6) i(l, 0, 0) - Harrpaq,lmlOm:Hil: BeKTOp OCH OX. YC310BHeM rreprreH,n:HKY31jfpHOCTH BeKTopOB grad U °H a jfB31jfeTCjf paBeHCTBO HY°3110 HX CKaJIjfpHOrO rrpOH3Be,n;eHHjf: grad U . i = ¢} Y = 0. YpaBHeHHe y = 3a.n;aeT rr310CKOCTb

Oxz. Bo Bcex TOqKaX 3TOil: rr310CKOCTH grad U rreprreH.D:HKY31jfpeH OCH OX.

B) IIoCK031bKY grad U rreprreH,IJ;HKY31jfpeH rr310CKOCTH 2x+y+z = 1, TO OH K03131HHeapeH ee HOPMaJIbHOMY BeKTOpy n(2, 1, 1), T. e • .D:OJDKHbI BbIII031HjfTbCjf

paBeHCTBa
	¥.. = ~ = 2 - 2z ¢} ~ = ¥.. = z - 1
	2	1	1		2	4	-1'
C31e,n;oBaTe31bHO, grad U rreprreH.D:HKY31jfpeH HCXQ.D:HOil: rr310CKOCTH									B	TOqKaX
rrpjfMoil:,	rrpoxo,n:gm:eil: qepe3 TOqKY			Mo(O, 0,1) C HarrpaB31jflOIII.HM BeKTopOM
l(2,4,-1).										•
5.1.11.	B KaKHX TOqKax rr310CKOCTH ]R2					rpa.n;HeHT rr310CKoro CKaJIjfpHOrO
	rr031jf U =	1 2	1	2					.
		2x	- x + 2Y		rreprreH.D:HKY31jfpeH			pa.n;HycY-BeKToPY
	r=x·i+y·j?						U =
5.1.12.	Rail:TH yro31	Mex<,IJ;y rpa.n;HeHTaMH rr031jf						2 X 2	B	ToqKax
	A(3,0,1) H B(l,-l,O).							y +z
5.1.13.	B KaKHX TOqKax rpa.n;HeHT CKaJIjfpHOrO rr031jf
	U = r2		(r = x . i + y . j + z . k,				r = Irl) :

a)rrapaJI31e31eH OCH OXj

6)rreprreH,n:HKY31jfpeH OCH OXj

B) rrapaJI31e31eH rr31OCKOCTH x - y + z = 3j

r) rreprreH.D:HKY31jfpeH rr310CKOCTH x -	3y + 2z - 1 = OJ
.n.) paBeH OJ	.
e) HMeeT Be31HqHHy, paBHYIO 4?

237

= y dy,

(xo, Yo, Zo -

5.1.14. Rail:TH JIHHHH YPOBH.H nJIOCKHX CKaJI.HPHbIX nOJIeil::

a)U = xy;

6)U = r2, r,ll;e r = x . i + y . j, r = Irl;

0)U = arcsin(x + y).

5.1.15.Rail:TH nOBepXHOCTH ypOBH.H CKaJI.HPHbIX nOJIeil::

a) U = r, r,ll;e r	= X· i + y. j + z· k 1'1 r = Irl;
6) U = x 2 + y2;
x2	+y2
0) U = arctg	z2

5.1.16. Rail:TH nOBepxHocTH ypOBH.H CKaJI.HpHOrO nOJI.H U = f(r), r,ll;e r = = X· i + y. j + z· k 1'1 r = Irl, <PYHKU;H.H f(t) onpe,ll;eJIeHa,ll;JI.H Bcex

t~ 0 1'1:

a)f(t) - MOHOTOHHa;

6) f(t) - He MOHOTOHHa.

5.1.17.Rail:TH BeKTopHbIe JIHHHH BeKTopHoro nOJI.H F(P, Q, R):

a)F(3, 1,2);

6)F(y, x, z);

0)F(z - y,x - z,y - x).

a a) CocTaBHM HOPMaJIbHYIO CHCTeMY ,ll;H<p<pepeHU;HaJIbHbIX ypaBHeHHil:, onpe,ll;eJI.H101ll;y1O BeKTopHbIe JIHHHH nOJI.H:

dx	3
dt	= ,
dy = 1
dt	'
dz	= 2.
dt

Ee perneHH.HMH 6y,nyT np.HMbIe JIHHHH, 3a,n,aBaeMbIe B napaMeTpHqeCKoil: <pop-

Me CHcTeMoil: ypaBHeHHil::

X = 3t+xo, { y = t+yo,

z = 2t + zo,

npOH3BOJIbHbIe nOCTO.HHHbIe). TaKHM 06pa30M, BeKTopHbIMH JIHHH.HMH 6y,nyT np.HMbIe C HanpaBJI.HIOIll;HM BeKTopOM l(3, 1,2).

6) B 9TOM CJIyqae MO:llCHO nocTynHTb TaK :lICe, KaK B nyHKTe a), COCTaBHB CHCTeMY JIHHeil:HbIx O,ll;HOPO,ll;HbIX ,ll;H<p<pepeHU;HaJIbHbIX ypaBHeHHil: 1'1 pernHB ee (CM. § 7, rJIaBa 2). O,ll;HaKO BeKTopHbIe JIHHHH HaXO,ll;.HTC.H npOIu;e C nOMOIll;blO MeTO,ll;a COCTaBJIeHH.H HHTerpHpyeMbIx KOM6HHall;Hil:. IIPOHJIJIIOCTpHpyeM ero B 9TOM (1'1 B CJIe,nylOIll;eM) nyHKTe.

3anHrneM ypaBHeHH.H BeKTopHbIX JIHHHil: B BH,ll;e CHCTeMbI

d:=~=d:.

PaBeHcTBo d: = ~ 06pa3yeT nepBylO HHTerpHpyeMylO KOM6HHall;HlO. 2ho ypaBHeHHe JIerKO pernaeTC.H MeTO,ll;OM pa3,ll;eJIeHH.H nepeMeHHbIX: x dx

238

1, k3

= 0),

oTKy,ll;a X2 = y2 + CI • ,nJUI rrOJlyqeHHH ellle O,ll;HOil: HHTerpHpyeMoil: KOM6HHaUHH HCrrOJlb3YlOT 06bIqHO H3BeCTHOe CBOil:CTBO rrpOrrOpUHH:

kl al + k2a2 + k3 a3 klbl + k2b2 + k3 b3 '

rrpHqeM MHO)l{HTeJlH ki rrO,ll;6Hpa1OT TaK, qT06bI B qHCJlHTeJle 06pa3oBaJlcH

AH<p<pepeHUHaJl 3HaMeHaTeJlH HJlH qTo6b1 B 3HaMeHaTeJle rrOJlyqaJlCH HyJlb,

a B qHCJlHTeJle - rrOJlHbIil: AH<p<pepeHUHaJl. B HameM CJlyqae, CKJla,D;bIBM qHCJlHTeJlH H 3HaMeHaTeJlH rrepBbIX AByX APo6eil: (3AeCb kl = k2 =

rrOJlyqHM cJleAYlOlllYlO HHTerpHpyeMylO KOM6HHaUHIO

d(x+Y)=dzz. x+y

IIHTerpHpYH ,ll;aHHOe paBeHCTBO, rrOJlyqaeM x + y = C2 z. TaKHM o6pa30M, BeKTopHbIe JlHHHH 3a,ll;aIOTCH CHcTeMoil:

x2 - y2 = CI ,

{

x+y=C2z,

T. e. HBJlHIOTCH JlHHHHMH rrepeCeqeHHH rHrrep60JlHqeCKHX UHJlHHAPOB x 2_y2 = = CI C rrJlOCKOCTHMH x + y - C2 z = o.

B) 3arrHmeM, KaK H B rryHKTe 6), ypaBHeHHH BeKTopHbIX JlHHHil: rrOJlH B

cHMMeTpHqeCKoil: <POpMe:

z - y x - z y - x

COCTaBHM rrepBylO HHTerpHpyeMylO KOM6HHaUHIO:

dx	dy	dz	dx + dy + d~	d(x + y + z)
z -	y = x -	z = y -	x = -:-z( ---y"")+----:-(x----'-'-z-:-)-+'-("-y---x-'-) =	0

QTo6bI rrOCJleAHee OTHomeHHe paBHHJlOCb rrpe,ll;bIAYlllHM, He06xo,ll;HMO, qTo6bI BbInOJlHHJlOCb d(x + y + z) = 0, T. e. x + y + z = C. BTOPM HHTerpHpyeMM KOM6HHauHH MO)l{eT 6bITb rrOJlyqeHa CJleAYIOIIIlfM o6pa30M:

dx	dy	dz
z - y = x - z y - x =
		xdx + ydy + zdz	1	2	+ y2	+ Z2))
			d (-2 (x
		= ~--~--~~~~--~--~=			0
		x(z-y)+y(x-z)+z(y-x)

CJle,ll;OBaTeJlbHO, d (~(X2 + y2 + z2)) = 0, T. e. X2+y2+Z2 = R2. TaKHM 06pa30M, BeKTopHbIe JlHHHH rrOJlyqalOTCH KaK rrepeCeqeHHe rrJlOCKocTeil: x+y+z =

= C co C<pepaMH x 2 + y2 + Z2 = R2 H, CJle,ll;OBaT€JlbHO, HBJlHIOTCH OKP~Ho-

CTHMH C ueHTpaMH Ha rrpHMoil: I = ¥= I H pacrrOJlO)l{eHHbIe B rrJlOCKOCTHX,

OTceKalOlllHX Ha OCHX KOOp,ll;HHaT paBHbIe OTpe3KH. •

239