Сборник задач по высшей математике 2 том

a IIoCKOJIbKY
j(x2 + y3) dl =	j(x2 + y3) dl + j(x2 + y3) dl + j(x2 + y3) dl,
L	AB	BO	OA

TO OCTaeTCH BbPIMCJIMTb KPMBOJIMHeil:Hblil: MHTerpaJI no KroK.n;OMY M3 OTpe3KOB

AB, BO M OA:

y

B 1

o

1

A x

Puc. 43

1) (AB): TaK KaK ypaBHeHMe npHMoil: AB MMeeT BM.n; y = 1 - X, TO dl = Jl + (y')2 dx = .;2 dx. OTCIO.n;a, Y9MTbIBaH, 9TO x MeHHeTCH OT 0 .n;o 1, nOJIY9MM

			1
j(x2 + y3) dl = j[x2 + (1 -					X)3].;2 dx =
AB			0			(1- X)4) 11 =.;2 (1 + 1) = 7.;2
			=.;2 (X3 _
					3		4	0	3	4	12 .
2) (BO): pacc)')K.n;aHaHaJIOm9HO, Haxo.n;MM x = 0, 0 ~ y ~ 1, dl = dy,
oTKy.n;a							1
			j	(x2 + y3) dl = j			y3 dy = ~.
3)			BO			0
	(OA): y = 0, 0 ~ x			~ 1, dl = dx.				i.
							1
			j	(x 2 + y3) dl = j			x 2 dx =
			OA			0
4) OKOH9aTeJIbHO
	j(	2 + 3) dl =		7.;2 + 1 + 1 = 7.;2 + 7 =					7(.;2 + 1)		•
		x	y	12	4	3	12		12·
	L
4.1.3.	BbI9MCJIMTb KPMBOJIMHeil:Hblil: MHTerpaJI
					jJX2 +y2 dl,
					L
	r.n;e L -		OKPf)KHOCTb x 2 + y2 = ax (a> 0).

190

a BBe,ll;eM rrOJlHpHhle KOOp,ll;HHaThI x = r cos cp, y = r sin cpo Tor,ll;a, rrOCKOJIh-

KY X2 + y2	= r2, ypaBHeHHe OKPY:lKHOCTH rrpHMeT BH,ll; r2 = ar cos cp,		T. e.
r = a cos cp,	a ,ll;H<p<pepeHIJ;HaJI ,1l;yrH
	dl = J r2 + r'2 dcp = va2cos2 cp + a2sin2 cp dcp = a dcp.
fIpH aTOM cp E [- ~, ~]. CJIe,ll;OBaTeJIhHO,
		11"	•
		2"
	jJX2 + y2 dl = a	j a cos cpdcp = 2a2.
	L	11"
		2"

4.1.4.Bhl'IHCJIHThKPHBOJIHHeitHhlit HHTerpaJI rrepBoro pO,ll;a OT <PYHKIJ;HH

		j(5Z -	2JX2 + y2) dl,
		L
r,ll;e L -	,1l;yra	KPHBOit,	3a,ll;aHHoit rrapaMeTpHQeCKH x			= t cos t,
Y = t sin t,	Z = t, 0 ~ t ~ 7r.
a nepeit,ll;eM B rrO,ll;hIHTerpaJIhHOM Bhlpa:lKeHHH K rrepeMeHHoii t. lIMeeM ,1l;JIfi
nO,ll;hIHTerpaJIhHOii <PYHKIJ;HH:
5z - 2JX2 +		y2 = 5t -		2Vt2(COS2 t + sin2 t) = 3t.
Terreph Bhlpa3HM Qepe3 t ,ll;H<p<pepeHIJ;HaJI dl:
dl = J(X')2 + (y')2 +	(Z')2 dt = J(cost - tsint)2 + (sint + tcost)2 + 1dt =
(cos2 -2tsin tcost + t 2 sin2 t) +				(sin2 t + 2tsintcos t + t 2 cos2 t) + 1dt =
	= V(cos2 t+ sin2 t) + t 2(sin2 t+ cos2 t) + 1 = J2 + t2dt.
TaKHM o6pa30M,
		11"			11"
j(5z - 2JX2 + y2) dl = j		3tv"2+t2 dt =			j ~v"2+t2d(2 + t2) =
L	0				0
				= (2 +	t 2)3/21: = J (2 + 7r2)3 -	2v'2. •

BU"I,uc.I!um'b c.I!eaY'IOw,ue 7CPU60.l!U1teiJ,1tue U1tme2pa.l!'bt nep6020 poaa:

4.1.5.j xy dl, r,ll;e L - KOHTYP KBa,ll;paTa Ixi + Iyl = a.

4.1.6.	LJ	dl	, r,ll;e L - OTpe30K OA H 0(0,0), A(1,2).
	L Jx2	+ y2 + 4

191

4.1.7.	dl	L - OTpe30K AB, A(2,4), B(1,3).
	fx + Y' r,1l;e

L

4.1.8.f~, r,1l;e L - OTpe30K MN, M(O, -2), N(4,0).

x- y

4.1.9.	Lj y2 dl,	L -	)lyra			U:HKJIOH,1l;bI x = a(t -				sint), y = a(l - cost)
	L
	o:::;; t :::;; 211".
4.1.10.	j(x2 +y2 +Z2) dl, L -						)lyra u:eIIHofi JIHHHH x = acost, y == asint
	L		:::;;
	z = bt,	0 :::;; t		211".
4.1.11.	j(x + y) dl,		L	-	IIpaBblfi JIeIIeCTOK JIeMHHCKaTbI r2 = a2 cos 2<p.
	L
4.1.12.	j(x2 + y2)n dl, L -					OKPY:lKHOCTb x 2 + y2 = a2.
	L
4.1.13.	j xy dl, L -							X2	y2
			qeTBepTb 9JIJIHIICa 2" + 2" = 1, x ~ 0, y ~ O.
	Lab
4.1.14.	j Y dl,	L -	)lyra			IIapa60JIbI		y2 =	2px,	OTCeqeHHa5I IIapa60JIoit
	L
	x 2 = 2py.
4.1.15.	BblqHCJIHTb IIJIOIII.a,1l;b qacTH 60KOBOfi IIoBepxHocTH KpyroBOro U:H-
	JIHH,1l;pa x 2 + y2				=	R2, OrpaHHqeHHOfi CHH3Y IIJIOCKOCTbIO Oxy, a
	cBepxy IIOBepXHOCTbIO !(x, y)							= R +	x2
									R.
a I1CKOMa5I IIJIOIII.a,1l;b BblqHCJI~eTC~ IIO <popMYJIe
					8= j(R+~)dl,
							L
r,1l;e L -	OKPY:lKHOCTb x 2 + y2 = R2. IIoBepxHocTb U:HJIHH,1l;pa H IIOBepXHOCTb
! (x, y) =	R + ~2	CHMMeTpHqHbI OTHOCHTeJIbHO KOOp,1l;HHaTHbIX IIJIOCKOCTefi
Oxz H Oyz, II09TOMY MO:lKHO OrpaHHqHTbC~ BblqHCJIeHHeM HHTerpana IIpH
yCJIOBH~X Y ~ 0, x ~ 0,					T. e. BblqHCJIHTb qeTBepTb						HCKOMOfi IIJIOIII.a,1l;H H
pe3YJIbTaT YMHO:lKHTb Ha 4. IIMeeM
		y = VR2 -					X2, y'		X
								= - JR2 - x 2 '
	dl=V1 +(y')2dx=f1 +							x 2	dx=		Rdx .
							V	R2 - x 2			J R2 - x 2
CJIe,1l;OBaTeJIbHO,
	8=4 R(R+~2)						Rdx	=4	RR 2+x2 dx.
		j			R J R2 - x 2				J R2 - x 2
		o							0

192

IIOJIyqHJUI Onpe,Il;eJIeHHbIti HHTerpaJI, KOTOPbIti 6epeM nO,Il;CTaHoBKoti x =

0=; R sin cp, oTKY,Il;a

dx = Rcoscpdcp, 0 ~ cp ~~, JR2 - x 2 = Rcoscp.

	R R2 + (Rsincp)2		11"
5 = 4 j		. R cos cpdcp = 4	j2	+ R2 sin2 cp) dcp =
	Rcoscp		(R2
o			0
			11"
		= 4R2	J(1 + 1-C;S2CP) dcp = 37rR2. •

o

4.1.16.HatiTH MacCY qeTBepTH aJIJIHnCa

					x2				y2
					a2 + b2 = 1,
		pacnOJIO)KeHHoti B nepBoti qeTBepTH, eCJIH JIHHetiHa5I nJIOTHOCTb B
		Ka)K,Il;Oti TOqKe nponOpU;HOHaJIbHa Op,Il;HHaTe aToti TOqKH c Koa<p-
a		<pHU;HeHToM k.
	IIoCKOJIbKY p(x, y) = ky, HMeeM
					m= jkydl,
									L
L -	qeTBepTb aJIJIHnCa
			x 2		y2				x ~ 0,	y ~ o.
			2"		+ "2 = 1,
			a		b
IIepexo,Il;HM K napaMeTpHqe~KHM KOOp,Il;HHaTaM aJIJIHnCa x =											a cos t, y =
=bsint. HanoMHHM, qTO C = v'a2 -								b2 - <PoKycHoe paccToHHHe aJIJIHnCa, a
c		aKcu;eHTpHCHTeT aJIJIHnca. HaxO,Il;HM
a = c -
dl = J(x l )2 + (y')2 dt = J a2 sin2 t + b2cos2 tdt =
		= va (1 -	cos	2	t) + b	cos	2		t = ';'a~2--~(;-a~2-_-=b"::;-2):-C-O-:s2::-t dt =
		2			2
										=aVl -	c 2 cos2 tdt.
IIepexo,Il;HM K BbIqHCJIeHHIO MacCbI
		11"								11"
		2			c2 cos2 t dt = - k~b					2
	m	= kab j sin tJl -								j VI - (c cos t)2 d(c cos t).
		o								0
BOCnOJIb3yeMcH <POPMYJIOti
		j VI - u2 du = ~(u~+ arcsin u),
7 C60pHHK 38,IUI~ no BLloweA .........&mI... 2 "YJ'C									193

f,1l;e U = e cos t. IIoJIY'IaeM		e2 COS2 t + arcsin(e cos t)] I:11"
m = - k~b . ~ [e cos tVl -					=
				=_~~b [-e~-		arCSine].
Y'IHTbIBaH,'ITOe =	v'a2 - b2		e2	b
	a	'v'l-		= a' nOJIY'IHMOKOH'IaTeJIbHO
m =	ka2b	(.2...Va2 _ b2 + arcsin v'a2 - b2 ).				•
2v'a2 _ b2		a2		a

4.1.17.BbI'IHCJIHTbMaccy H KooPMHaTbI II;eHTpa TIDKeCTH O,1l;HOPO,1l;HOti ,1l;y-

fH II;HKJIOH,1l;bI X		= a(t - sin t), y = a(l -	cos t), 0 ~ t ~ 211".
r'\.	My	Mx
'..I JIMeeM Xc =	rn:' Yc =	rn:' f,1l;e
	m= ! dl, My = ! xdl,
	L	L
HaxO,1l;HM x', y'	H dl no OT,1l;eJIbHOCTH: x' = a(l- cost), y' = asint,
dl = Ja2(1 -	2cost + cos2 t) + a2sin2 tdt =
=aJl - 2 cost + (cos2 t + sin2 t) = aV2(1 - cost) =
		= aJ2. 2sin2 ~ = 2asin~.
		211"	211"
m = !dl = 2a ! sin ~dt = -4a cos ~10 = 8a.
	L	0

y

o

27ra x

Puc. 44

JI3 pHC. 44 BH,1l;HO, 'ITOII;HKJIOH,1l;a CHMMeTpH'IHaOTHOCHTeJIbHO npHMoit x = 1I"a, nOSTOMY Xc = 1I"a. TaKHM 06Pa30M, My MO)KHO He BbI'IHCJIHTb,XOTH,

Y'IHTblBaHpaBeHcTBO

194

MO)KHO IIpe,Il;IIOJIO)KHTb, qTO My = 87ra2. npeMaraeM caMOCTOllTeJIbHO IIOJIy-

'-IHTbaTOT pe3YJIbTaT. BblqHCJIHM TeIIepb Mx:

	2~			2~
Mx = J y dl = J a(1 -		cos t)	. 2a sin ~ dt = 2a2 J 2 sin2 ~ • a sin ~dt =
L	0		0
	2~		2~
	=4a2 JSin3~dt=-8a2 J(I-cos2~) d(cos~) =
	o		0			2
			= -8a2 (cos 1.	_1 cos3 1.) I			~= 32a2
			2	3	2	0	3 .
OKOHqaTeJIbHO IIOJIyqaeM:							•
	=	32a2
m = 8a, M x		3 '
4.1.18.	HaihH KOOp,Il;HHaTbI u;eHTpa T1I)KeCTH ,IJ;yrH O,Il;HOPO,Il;HOfi KPHBOfi
			Y = ach~
	OT TOqKH A(O,a),Il;o TOqKH B(b,h).
4.1.19.	HafiTH KOOp,Il;HHaTbI u;eHTpa T1I)KeCTH ,IJ;yrH U;HKJIOH,Il;bI
	x = a(t -	sint),	y = a(l- cost),	0:::;;	t:::;; 7r.
4.1.20.	HafiTH MOMeHTbI HHepU;HH OTHOCHTeJIbHO KOOp,Il;HHaTHblx ocefi H Ha-
	qaJIa KOOp,Il;HHaT qeTBepTH OKPY)KHOCTH x 2 +y2 = a2, x ~ 0, y ~ O.

nJIOTHOCTb pacIIpe,Il;eJIeHHlI MacC ,IJ;yrH IIOCTOllHHa H paBHa k.

Q ~aHHM KpHBM (qeTBepTb OKP}')KHOCTH) CHMMeTpHqHa OTHOCHTeJIbHO 6HcceKTpHCbI y = x IIepBoro KOOp,Il;HHaTHoro yrJIa. OTCIO,Il;a 3aKJIIOqaeM, qTO

Jx H Jy O,Il;HHaKOBbI, T. e.
Jx = Jy = J	y2 dl = J x 2 dl.
L	L

Ilepexo,Il;1l K IIapaMeTpHqeCKHM ypaBHeHHlIM OKP}')KHOCTH x = a cos t, y = == a sin t, 0 :::;; t :::;; I' OTKY,Il;a dl = a dt, IIOJIyqaeM

	~		~
	2"	3	2"	3 (	. )	11!:	= 7r~	3
J x 2 dl = a3 J cos		2 t dt = ~	J (1	+ cos 2t) dt = ~t	+ Sl~2t	:		•
L	0		0
T		7ra3		7ra3				•
aKHM 06pMoM Jx		= Jy = 4'Jo = Jx + Jy = T·
4.1. 21.	BblqHCJIHTb Maccy qeTBepTH aJIJIHIICa x = 5 cos t, Y = 4 sin t, pac-
	IIOJIO)KeHHYIO B nepBofi qeTBepTH, eCJIH ee JIHHefiHM IIJIOTHOCTb P
	paBHa y.
4.1.22.	HafiTH MacCy KOHTypa aJIJIHIICa .
			x 2	y2
			a2	+ b2 = 1,

eCJIH ero JIHHefiHM IIJIOTHOCTb B KroK,Il;ofi TOqKe M(x,y) paBHa Iyl.

195

4.1.23.	RafiTH MacCY rrepBOI'O BHTKa BHHTOBOfi JIHHHH x = a cos t, y =
	=a sin t, Z = bt, eCJIH rrJIOTHOCTh B KroK,1l;ofi TOqKe paBHa Pa,1l;HYCY-
	BeKTOpy 9Tofi TOqKH.
4.1.24.	RafiTH MOMeHT HHeplI.HH OTHOCHTeJIhHO OCH Oz rrepBOI'OBHTKa BHH-
	TOBofi JIHHHH x = a cos t, y = a sin t, Z = bt.

BbI."tuc.I!um'b n.l!ow,aau 'qU.!!UHapu"teC'lCux n06epxHocmeii" oepaHU"teHHbl.X CHU3Y n.l!OC'lCocm'b'lO Oxy, a c6epxy n06epXHOCm'b'lO Z = f(x, y), npu YC.I!06UU, "tmo U36ecmHa Hanpa6.11J1'1Ow,aJl L 3moii, 'qU.I!UHapu"teC'lCoii, n06epXHocmu:

4.1.25.f(x,y) = J2x - 4x2, y2 = 2x.

4.1.26.f (x, y) = xy , x 2 + y2 = R2.

2R

4.1.27.f(x,y) = 2 -.,fi, y2 = ~(x -1)3.

4.1.28.f(x,y) = x, y = iX2 (x E [0,4]).

4.1.29.BhIqHCJIHTh Maccy KOHTypa rrpllMOYI'OJIhHHKaco CTopOHaMH, JIeJKaIII.HMH Ha rrpllMhIX x = 0, x = 4, y = 0, y = 2, eCJIH p(x, y) = xy.

4.1.30.BhlqHCJIHTh Maccy ,1l;yI'Hrrapa60JIhI y2 = 2x, 3aKJIlOqeHHOfi MeJK,1l;y

TOQKaMH 0(0,0) H A(I, y'2), eCJIH p(x, y) = xy.

C nOMow,'b'lO 'lCPU60.l!UHeii,Hoeo UHmeepa.l!a I poaa 6b1."tUC.I!Um'b a.l!UHbI. 3aaaHHbl.3:, aye:

4.1.31.	ay2 = x 3 , 0 ~ x ~ 5a.				4.1.32.	r = asin3 <f!..
4.1.33.	a ~	+ e	-~	~ x ~ 4.		3
	y = 2(ea		a), 0

4.1.34.y = 1 -lncosx, 0 ~ x ~ i'

C nOMow,'b'lO 'lCPU60.l!UHeii,Hoeo UHmeepa.l!a I poaa Haii,mu 'lCOOpaUHambi. 'qeHmpa 17IJI:HCecmu 'lCPU6b1.X:

	2	2	2
4.1.35.	Xa	+ ya	= aa, y ~ O.
4.1.37.
4.1.38.

BbI."tuc.I!um'b aaHHbl.e UHmeepa.l!bI. I poaa:

4.1.39.	!vix 2 + y2 dl, I',1l;e L				3a,1l;aHa ypaBHeHHlIMH X = a(cos t + t sin t),
	L			tcost), 0 ~ t ~ 211'.
	y = a(sint -
4.1.40.	!	2	d~	2' I',1l;e	L - rrepBhlfi BHTOK BHHTOBOfi JIHHHH
	L	X	+y +Z

x = acost, y = asint, Z = bt, 0 ~ t ~ 211'.

196

4.1.41.	!(x + z) dl, r,Il;e L -	,rryra npOCTpaHCTBeHHoit KPHBOit, 3a,rr.aHHoit
	L
	_	_	3t2	_	3	,0:::;;	t :::;; 1.
	napaMeTpHqeCKH X -	t, Y -	J2'z - t
4.1.42.	HaitTH ,II,JIHHy ,rryrH	KOHHqecKOit		BHHTOBOit			JIHHHH X = aet COS t,
	Y = aet sin t, z = aet , 3aKJIIOqeHHoit Me)K,rry TOqKaMH 0(0,0,0) H
	A(a, 0, a).
4.1.43.	HaitTH ,Il;eKapTOBbI KOOp,Il;HHaTbI ~eHTpa TIDKecTH qmrYPbI, orpaHH-
	qeHHoit Kap,Il;HOH,Il;oit r = a(1 + cos cp).

4.1.44. HaitTH ,Il;eKapTOBbI KOOp)J;HHaTbI ~eHTpa TIDKeCTH ,rryrH JIorapH<p-
	MHqeCKoit CnHpaJIH r	= ae'P OT CPl = I ,Il;0 CP2 = 7r.
4.1.45.	BblqHCJIHTb	!
		Ix+yldl,
		L
	r,Il;e L - KOHTYP TpeyrOJIbHHKa ABC C BepmHHaMH A(O,O),
	B(l, 0), C(O, 1).
4.1.46.	BblqHCJIHTb HHTerpaJI !J2y2 + z2 dl, eCJIH L - OKPJ)KHOCTb

L

{X2 + y2 + z2 = a2, x=y.

4.1.47.BblqHCJIHTb nJIO~a,rr.b 60KOBOit nOBepxHocTH napa60JIHqeCKOrO ~H­

	JIHH,Il;pa y = x 2 , OrpaHHqeHHOrO nJIOCKOCTHMH z = 0, z = 2x, X						= 0,
	x=1.			= In(l + t 2 ),
4.1.48.	BblqHCJIHTb MacCY KPHBOit		x		Y =	2 arctgt -	t Ha
	yqacTKe OT t = 0 ,Il;0 t = 1, eCJIH ee JIHHeitHM nJIOTHOCTb pasHa
	p(x, y) = e-Xy.
4.1.49. BblqHCJIHTb MacCY qeTBepToit qacTH 9JIJIHnCa
	x 2	y2	x ~ 0, y:::;; 0,
	a2	+ b2 = 1,
	eCJIH JIHHeitHM nJIOTHOCTb p(x, y) = xy.				a x	x
4.1.50.	BblqHCJIHTb MacCy Bceit ~enHoit JIHHHH y =
					'2 (ea + e- a), eCJIH ee
	JIHHeitHM nJIOTHOCTb p(x, y) =			\.
4.1.51.	BblqHCJIHTb			Y
		!(x -	y) dl,
		L
	r,Il;e L: x 2 + y2 = ax.
4.1.52.	BblqHCJIHTb	! xJx2 -		y2 dl,

L

r,Il;e L - JIHHHH, 3a,rr.aHHM ypaBHeHHeM (x2 + y2)2 = a2(x 2 _ y2),

X ~ 0 (nOJIOBHHa JIeMHHCKaTbI).

197

4.1.53.				j arctg ~ dl,
				L
	r,ll;e L -	qaCTh CIIHpa.nH ApXHMe,ll;a r = 2<p, 3aKJIIOqeHHruI BHyTpH
	Kpyra pa,n;Hyca R C IJ;eHTpOM B Haqa.ne KOOp,ll;HHaT.
KOHTponbHble BOnpOCbl III 60nee CnO)l(Hbie 3aACIHIIISI
				4			4
				j (x"3 +y"3)dl,
				L
	r,ll;e L -					2	2	2
		,ll;yra aCTpOH,ll;hI			x"3		+ y"3	= a"3, JIe:lImIIIruI B		IIepBoil: qe-
	TBepTH.
4.1.55.	BhlqHCJIHTh			jlyldl,
				L					y2), X ~ o.
	r,ll;e L -	,ll;yra JIeMHHCKaThI (X2 + y2)2 = a2(x2 -
4.1.56.	BhlqHCJIHTh			j. /	2 + y2 dl,
				V X
				L
4.1.57.	r,ll;e L -	IIOJIYOKPY)KHOCTh X 2 + y2 = ax, y ~ o.
	Hail:TH MHHy IIpOCTpaHCTBeHHoil: KPHBOil:
		y =		. x				4-x
			arCSIn -4 '				z = In --
								4+x
	OT TOqKH 0(0,0,0) ,ll;O TOqKH A(2, 3, 4).
4.1.58.	BhlqHCJIHTh			j	zdl,
				L
	r,ll;e L -	KOHHqeCKruI BHHTOBruI JIHHH5I x = t cos t, Y = t sin t, z = t,
	0~t~1r.
4.1.59.	Hail:TH	Maccy ,ll;yrH		IIapa60JIhI y2				= 2px, 3aKJIlOqeHHoil: Me)K,ll;y
	TOQKaMH 0 (0, 0)		H	A ( ~,p), eCJIH ee JIHHeil:HruI IIJIOTHOCTh paB-
	Ha p(x, y) = y.
4.1.60.	Hail:TH Maccy ,ll;yrH KPHBOil:					x = at, y = 2'z = 3'
								at2	at3	3aKJIlOqeH-
	HOil: Me)K,ll;y TOqKaMH 0(0,0,0) H							A (a, ~, ~), eCJIH ee JIHHeil:HruI
	IIJIOTHOCTh paBHa p(x, y) = /'f..
4.1.61.	BhlqHCJIHTh			j	xyzdl,
				L						R2
	r,ll;e L -
		qeTBepTh OKPY)KHOCTH x 2 + y2 + Z2 = R2, x 2 + y2 = -;[,

z ~ 0, JIe)KaIIIa51 B IIepBOM OKTaHTe.

198

r 2

F = mJsinadl

COrJIacHo 3axOHY BHo-Casapa 9JIeMeHT TOKa ,!I;eil:cTBYeT Ha MarHHTHYIO Maccy m C CRJIOil:

'

r,o;e J - TOK, dl - 9JIeMeHT ,!I;J1HHbl npOBO,!l;HHKa, r - paCCTOHHHe OT 9JIeMeHTa TOKa ,!I;O MarHHTHoil: MacChI, a - yrOJI MeJKJJ;y HanpaBJIeHHeM npHMoil:, COe,!l;HHHIOmeil: MarHHTHYIO MacCy H 9JIeMeHT TOKa, H HanpaBJIeHHeM caMoro 9JIeMeHTa TOKa. 2ha CHJIa F HanpaBJIeHa no HOPMaJIH K nJIOCKOCTH, CO,!l;epJKaIIIeil: 9JIeMeHT TOKa H TOqKY, B KOTOPYIO nOMeIIIeHa MarHHTHax Macca, HanpaBJIeHHe CRJIbl onpe,o;eJIHeTCH no npasHJIy 6ypasQHKa.

OnUpMC'b 'Ita 3a1W'It Buo-Caeapa, pewum'b c'//'eay'lO'I11,ue 3aaa"tu:

4.1.62. Hatiul CHJIY, C KOTOPOti TOK J B 6eCKOHe'lHOM IIP.HMOJIHHetiHoM "POBO,lI.HHKe ,ll;eticTByeT Ha TO'le'lHYIOMarHHTHYIO Maccy m, Haxo-

,ll;.H~IOC.H OT "POBO,ll;HHKa Ha paCCTO.HHHH a.

4.1.63. ITo KOHTYPY, HMeIOm;eMY <POpMy KBa,rr.paTa co CTOPOHOti a, Te'leT

TOK J. C KaKOti CHJIOti 9TOT TOK ,ll;eticTByeT Ha TO'le'lHYIOMarHHTHYIO Maccy m, HaxO,ll;.H~IOC.H B IJ;eHTpe KBa,rr.paTa?

4.1.64.C KaKoti CHJIOti TOK J, TeKym;Hti IIO KpyroBoMY KOHTYPY pa,rr.Hyca

R, ,ll;eticTByeT Ha TO'le'lHYIOMarHHTHYIO MacCY m, IIoMem;eHHYIO

B TO'lKY P, JIe:lKa~IO Ha "ep"eH,ll;HKYJI.Hpe, BOCCTaHOBJIeHHOM B

IJ;eHTpe Kpyra Ha pacCTO.HHHH h OT 9Toro Kpyra?

4.1.65.C KaKoti CHJIOti TOK J, TeKym;Hti IIO 3aMKHYToMY 9JIJIHIITH'IeCKOMY

KOHTYPY, ,ll;eticTByeT Ha TO'le'lHYIOMarHHTHYIO Maccy m, HaxO,ll;.H-

~IOC.H B <poKyce 9JIJIHIIca?

4.1.66.C KaKoti CHJIOti TOK J, TeKym;Hti IIO 6ecKoHe'lHoMYIIapa60JIH'Ie-

CKOMY KOHTypy, ,ll;eticTByeT Ha TO'le'lHYIOMarHHTHYIO Maccy m, IIoMem;eHHYIO B <poKyce IIapa60JIbI? PaccTO.HHHe OT BepwHHbI ,ll;0 <poKyca pasHO ~.

B'bt"tuc.//,um'b n'//'o'l11,aa'b 'qU.l/,u'ltapu"tecICux noeepx'ltocmeti, 02pa'ltu"te'lt'lt'btX C'ltU3Y n.//,oc,,"ocm'b'lO Oxy, ceepxy aa'lt'ltoti noeepx'ltocm'b'lO Z = !(x, y), npu yc.//,oeuu, "tma 'ltanpae.I/JI'lO'I11,M 3aaa'lta ICpueoti L:

4.1.67.

!(x, y) = xy, L - '1eTBepTb9JIJIHIICa

x 2 y2

a2 + b2 = 1,

JIe:lKam;a.H B IIepBoti '1eTBepTH(x ~ 0, y ~ 0).

4.1.68.!(x, y) = y, L - Y'lacToKIIapa60JIbI

y2 = 2px

OT Ha'laJIaKOOp,ll;HHaT ,ll;0 TO'lKH(xo, Yo).

4.1.69.!(x,y) = x 2 + y2, L - IIP.HMOJIHHetiHblti oTpe30K, COe,ll;HH.HIOm;Hti

TO'lKHA(a, a) H B(b,b).

199