Векторные функции

1.

Cfle/iaHO B
Flyvi

2.

838 III rilABA 13 BEKTOPHblE OYHKL4MM
^* *48. HawflMTe ypaBHeHun Kone6nK>mnxca OKpyxcHOCreM napaConbi
55. Hcnonb3yMTe cfcopMynw Opene-C^ppe. HTo6t>i AO*a3aTb Kawaoe M3 cneAytoiMMX
•"#•%
yTBepxfleHiM (flpocrwe MMcna o6c3Ha^aioT nport3BOAHye no t HaMH
y = 3x2 B TOMKax (0, 0) M (1, 3). H3o6pa3me Kax KOJie6nK>innecfl
AOKaGaTCJibCTBo TeopeMw 10.)
OKpyjKHOCTM, Tax M napa6ony Ha QAHOM 3KpaHe.
(a) r = s"T + K(S')’N (b) r* X r = k(s-)’B
(49. B KaKOM TOHKe KPMBOH x = t y = 3t, z = t HaxQAHTCfl HopManbHaa
(c) r - [s" - K2(S')']T + [3ks" + K,(S,)2]N + KT(S')’ B.
nnocKOCTb, napannenbuaa n/iocxocTM 6x ♦ 6y - 8z = 1 ?
KAC 50. ECTB nn Tosna Ha xpuBOM B ynpaxHeHMM 49. ra&
0
conpHKacawmaacfl ruiocxocTb napannenbHa nnocxocTvi x + y + z =
1?
56.
rioKa>KMTe, MTO xpyroBaa cnupanb r(t) = (a cos t, a sin t, bt), r/je a
[TlpuMeyaHMe: BaM noHaAODMTca CAS AJia
AM(fc(})epeHmipo3aHna. ynpomeHHH
M b - nono*MTejibHbie KOHCTamw. MMeeT nocTOHHHyio
M BbiuncneHna nepeKpecTHoro npon3BBABHM3.]
ynpaxHenMa 55(d).)
KpMBM3Hy M nocTOAHHoe Kpy^eHwe [Hcno/ib3ynTe pe3ynbiai
51. floKaxcuTe, MTO KpnBM3na cDfl3cina c Kacaie/ibHbiM M
57.
Mcnonb3y£rre (popMyny B ynpaxoneHUM 55(d), MTOGU Hati™
xpyMeHne
# •A• «
Hop*/anbHb*M BerropaMM ypaBHeHneM
AC
KpHBafl r(t) = (,t2.)
52. riOKa>KHTe. UTO KpnBH3H3 nnOCKOM KpMBOM P3BH3 K =
• (™
|Dc£/ds| . r^e 0 - yron Me>CAy T M i; TO ecTb <p - yron HaiaiOHS
KacaTeribHOfl /IMHMM. (3TO noKa3biBaeT , mo onpeAenemie
KpnBH3Hbi cornacyeTca c onpaaenenneM ana nnocKHx KPMDWX.
npMBCACHHUM B ynpax<HCHHH 69 B pasaene 10.2 )
53. (a) noxawMTe. MTO dB/ds ncpnoMAHKynapen B.
(b) noKa>KMTef MTO dB/ds nepneHflMKy/ispoH
•
T. (c) BbiBQAHTe M3 HacTGM (a) M (b), HTO dB/ ds = - r(s) N Ann
HexoToporo MMcna 7 (s), Ha3bJBaeMoro KpyMeHMesi M3
xpuBOM.
( KpyMeHne M3MepaeT
CTeneHb
3aKpyMMB3HMfl
KPMBOM.) (d)
54. CneflyMDiMne
(fropMynw,
Ha3biBaeMbie
(|X)pMynaMM
floKaxoue.
MTO WIOCKOM KPHBOM xpyMeHne paBHO 7(s) = 0.
OpeHe-Ceppe,
58. HaMAme KpnBM3Hy M KpyHeHnc KPMBOM x = sinh t, y =
59. Monoxyna fXHK MMeeT cfc>opMy ABOMHOM cnupann (CM. pwcyHOK 3
cosh t. z = t o TOHKe (0, 1, 0).
Ha dp. 819). Paanyc*a*AOM aiMpanw cociaanaeT OKOJIO 10 aurcTpeM (1
= 10 *CM). Ka*Aafl cnwpanb noAHnwaecfl npMMCpno na 34. A 3a
KaxcAWH nonHbiM o6opoT, M nonynaeTCfl npuMepHO 2.9 * 104 no/iHbix
BMTKOB. OucHMTe AAMHy xa>KAOM cnMpann.
60.
flaBartTe
paccMOTpuM
)Ke/ie3HOAopo)*woro
CyiMecrsyfOiyaa Aopoxxa Bflonb OTpuLiaie^bHOH OCM X flonxHa 6wTb
nnaBMO npMcoeaMHeHa K aopo)KKe BAonb HHHHH y = 1 X 1.
(a) HaMAme MHoronneH P = P (x) CTeneHM 5, TaxoM, HTO
<t>yHKUMfl F, onpeAeneHHa^
^ 0 SCTM X 0
F(x) = < P(x), ecnM 0 < x < 1
11 ecnM x > 1
1. dT/ds = KHI'
dN/ds - KT -H TB
npoeKTMpoBaHM^
nyrM o6ecn©MeHHfl nnaBHoro nepexcwa M©»wy ynacTKaMki np«Moro nyTM.
MMeiOT 0yHA3MeHTanbHoe SHaneHMe B AM04)epeML4ManbHoii reoMeTpuw:
2.
npoCneMy
aanaeTca HenpepbienuM M MMeeT HenpepbiBHbiM Ha<non u
Henpepb4BHbiM
3. dB/ds = - ml
(OopMy/ia 1 B3flTa M3 ynpa^eHHn 51. a OopMyna 3 B35?Ta M3
ynpax^eHun
53.) Hcnoiib3yMTe TOT 0aKT? HTO N B X T , HTOGW BbiBecTM 0opMyny
2 M3 0opMy/i 1 M3.
KpMBM3Ha. (6)
m Hcnojib3yMTe rpacpMMeaw^ lanbtyjiniop win Kounbioiep. MTOSU
Hapucosaib rpa$Mc M3 F.
I-1
13.4 flBM>KEHME B IIPOCTPAHCTBE: CKOPOCTb M
B 3TOM pa3Aene Mbi noKaweM. xax MAen KacaTenbHbix M HopManbHbix BeicropoB M KpnBH3Hbi Moiyr 6biTb
YCKOPEHME
•• m 9
0 0 9
•
^
0 9
9 0 9 9
9
9 9
HCn0Jlb30B3Hbl B C^H3HK6 AT1B M3yueHMB ABMXeHVifl o6beKT3, BKJ~1 K3M33 eTO CKOpOCTb M
ycKOpeHMe.
BAOJlb npOCTpaHCTBeHHOM KpMBOM. B HaCTHOCTM, Mbl
MA©M TO CTOnaM HbK)TOHa? MCnOnb3y9 3TM MeTOAbi An^
BbiBOAQ riepBoro 3aKOHa AsuxceHun nnaHeT Kennepa.
npeAnono?KHM MTO MacTnua ABn?Kera B npocTpaHCTBe TBK. MTO eeiaop ee nono?KeHMfl B MOMeHT BpeMeHM
t paseH r(t)
%«I A
m 9
m
0 m 9 m
OCpaTMTe BHMM3HH6 Ha pMC. 1. UTO npM Ma/lblX 3HaueHM^X
h BeKTOp
9 0
0
9
0
annpoKCMMupyeT HanpaaneHne
KpneoM r(t). Ero Ben^MMHa
• m
0
m m m
9
HacTuqbi,
ABM>KymeMC^
•• ■■• • m
n3MepseT paaMep Beiaopa CMeLi^eHHa B ea^HMi^y BPBMGHH. Beicrop (1) flaeTcpeaHse 3HaueHne
no

3.

PA3flEfl 13 4 flBH>KEHME B I1P0CTPAHCTBE: CKOPOCTb M YCKOPEHME II11 839
CKopocTb B TeMeHne BpeMeHHoro HHTepBana ATiMHbi h. n ee npe^enow ABnaeTCfl BeKTop CKOPOCTM v(t) B MOMeHT BpeweHM t:
# -i- /«\ — *•/ *\
Nx(t)
Mmr(t)
#i —• u n
TaKMM o6pa30M, BeKTop CKOPOCTM Taicre AB/iaera BeKTopoM KacaienbHOM M yKa3biBaeT B HanpaBneHnn
KacaienbHOM
DMHMR.
CKopocTb H3CTMI4W B MOMeHT BpeMeHH t paBHa BeniiHHHe BeKTOpa CKOpOCTM, TO eCTb | v(t)|.
3TO yMecTHO, noTOMy MTO M3 (2) M M3 ypaBHeHMH 13.3.7 Mbi MMeeM
|v(i) I = |r(i) \
6s
CKOPOCTb H3MeHeHH« paCCTO«HMfl no OTHOUJeHUK) KO BpeMeHH
dt
KaK M B cjiynae OAHOMepHoro ABMweHMfl, ycKopeHMe HacTML|bi onpeAermeTCfl KaK
npoM3BOAHaa OT CKOPOCTM:
a ( t ) = V ( t ) = r"(t)
nPMMEP I BeKTop nono>KeHMfl o6beKTa, ABHxcymeroca B nnocKOCTM, 3aAaeTc«
* m
nepe3 r (t) = tl + t2 j. HanAMTe ero CKopocTb M ycKopeHMe npH t = 1 M npoMJUUOCTpMpyMTe
reOMOTpMHeCKM.
PEUJEHME CKopocTb H yCKopeHHe B MOMGHT BpeMeHH t paBHw
v(t) = r'(t) = 3t2i -*- 2t j a(t) =
r"(t) = 6t i + 2 j
H CKOpOCTb TaKafl
|v(i) I = V(312)2 + (20)2 = + 412
TEC Visual 13.4 rKxasbsaer aHWHpOBaHHye Bexiopa CXOPOCTH
w yCKOpeHHfl ATlfl 06beKT0B, ABVD^yiU^XCfl npn t - 1, KOTOpwe Mbl
MMeeM
v(1) = 31 + 2j a(1) = 6i
eaonb pa3/IMUHb<X KpHBWX.
■ Ha pucyHK.e 3 noKa3an nyn»
M3CTV114BJ B npMMepe 2 c
BetcropaMM CKOPOCTM H ycxopeHMa
npH = 1. za
2j |v(1) | = / 1 3
3TM BeKTopw CKOPOCTM M ycKopeHMfl noKa3aHbi Ha pMcyHKe 2.
■
nPMMEP 2 HanAMTe cxopocrb, ycKopeHMe M pa3Hoab CKOpocreii nacTMi^bi c
BeKTopoM nojioweHMfl r(1) = (t2, e\ te').
peueme
v(t) = r'(t) = (21, e', (1 + t)e")
a(t) = v'(i) = (2, e\ (2 + t)e')
|v(t)| = v4t2 + e2t + (1 + t)2e2
■
BeirropHbie MHTerpa/ibi. KOTopbie SbinM Bee^eHu B pa3Aene 13 2 MoryT CbiTb Mcnonb3<?BaHbi
Afl* HaxcxAeHMfl BeKTopoB nono>t<eHHJi. Konaa HBBecTHbi Benropw CKOPOCTM MJIM ycKopeHMe. KaK B cneAyKXiiefc ip^iwepe

4.

840 III! rflABA 13 BEKTOPHblE OYHKLIMM
kUriPMMEP 3 flBM>Kymaflcn nacTMua HasuHaeTcn c HananbHOM no3Ht4MM r(0) = (1,0, 0) c HaManbHOM
CKOpocTbio v(0) = i - j + k. Ero ycKopeHMe paBHo a(t) = 4t j + 61 A* + K. HafiAMre ero CKopocTb
M nono*e*40 B MOf/sKT BDCM6HM t
PElilEHME nocKOfibicy a(t) = v*(t), MU MMeeM
v(i) = a(t) dt = (4ti + 6tj + k) dt
= 212i + 312j + tk + C
MTo6bi onpeflenuTb 3HaMenue nocTonnnoro Beicropa C, MU ncnont>3yeM TOT (JjaKT, HTO V(0) = i - j + k.
npeflbiAyii|ee ypaBHemie aaeT v (0) = C, noaTOMy C = i - j + k M
v(t) = 212i + 3r2j + 1k + i - j + k
= (2t2 + 1) i + (3t2 - 1)j + (t + 1) k
■BupaxeHwe AJIH r(r), KOTopoe m
nojiymuiM B
nocKO/ibKy v(t) = r'(t), Mbi MMeeM
r(t) = Jv(t) dt
rrpwtfepe 3.6urc wcncrbsceahcjir:* noc-pce-wa
TpaeicTcp*ii ^aCTMLlbl Ha pMCyHKO 4 flllH 03.
= [[(2r2 + 1)i + (3r2-1)j + (r+1)k]
dt = (r + r)i + (r2 - r)j + (3r + r)k
+D
nono)KMB t = 0, MU HaxQAUM, MTO D = r (0) = j, noaTOMy nonoweHMe B MOMem BpeweHM 13a£aercfl
y
r(t) = (3t2 + + 1) i + (t3 - t)j + (3t2 + t) k ■
PMCyHOK4
B oSmeM cnyMae BerropHue MHTerpanw ncwBonaiOT HaM BoccTaHaBnMBaTb CKopocTb. Kor^a ycKopeHMe
M3BecTHO M nonoweHMe, Kor^a CKopocTb M3BecTHa:
v(t) = v(1o) + a(u) du r(t) = r(tO) + v(u) du
•"o
Ecrm cuna, AeMCTBytomaR Ha HacTMLjy, M3BecTHa, TO ycKopeHMe MO>KHO HaiiTH H3l
Bioporo 3aKOHa ABHxemi* HbfOTOHa BeKTopnaa ocpcufl aioro aaKOHa macm. MTO BCJIM U JIK>6OM MOM6HT
BP«M«HM
t Ha oGbeicr Maccofi m AeflcreyeT cuna F(t), BbiabiBatOLqa^ ycKopeHMe a(t), TO
■YmoBaa CKopocTb oBbeKTa. ABn*ymerocfl B
nonoxeHMM P. paBHa w = dO/dt. me 0 - yran
naa3aHHbM na pwqivte 5
>4
F(t) = ma(t)
(1PMMEP 4 06beKT c MaccoM m, KOTOPUM ABMweTcn no KpyroBOM TpaeKTopnn c nocTonHHOM ymoBoft
CKopocTbK) v/, MMeeT Beiaop nonoweHMH r(z) = acos wti + a sin wt j. HaMAWTe cmiy, AewcTByKnuyjo Ha
oObeKT. M noKa>KHfe, MTO OHa HanpaBneHa K Hanany KOOpAHHai.
PeujeHne MTOOW HB\AT\A cuny, HaM CHanana Hy>KHO 3HaTb ycKopeHMe:
v(t) = r'(t) = -awsin wti + awcos wtj
«• *
a(t) = v'(t) = -aw2cos wti - aw'in wtj CneAoeaienbHO,
PMCyHOK5
Bropoii 33KOH HbtoTOHa AaeT
KaK = -mw’(a cos wt i + a sin w* j)
F(t)onny
= ma(t)

5.

PA3AEJ1 13.4 flBMWEHME B nPOCTPAHCTBE: CKOPOCTb M YCKOPEHME III! 841
06paTme BHMMaHwe. HTO F(t) = -mw2r(t). 3TO no»ca3biBaeT. HTO curia AeiicTByeT B HanpaeneHMH.
npoiMBononowHOM paAnyc-Beicropy r(t), M, cneAOBaTenbHO, yK33biBaeT Ha Hawano KOOPAMH3T (CM. PMC.
5). TaKaa curia Ha3biBaeTcn ueHTpocTpeMMTenbHOM (cTpeMamewcfl K ijempy) CMJIOM.
B
a nPMMEP 5 CHapflA BwcipenuBaeTCfl c yrnoM B03BbiLueHnn a n HananbHOM CKopocTbK) vo. (CM .
PucyHOK 6.) npeAnonaraa, MTO conpoTMBneHne B03Ayxa He3Ha4MTejibH0 M eflUHCTBeHHaB BHeiuHnn cuna
o6ycnoBneHa rpaBMTai4neti, Ha*vprre (pyHKL^mo nono>KeHMfl r (t) CHapnAa. KaKoe 3HaMeHne a MaKCHMH3npyeT
AanbHOCTb (npo^eHHoe paccTOAHne no ropM30HTa/in)?
PELUEHUE hta ycranaanuBaeM OCM Ta&iM oopa30M. MTO6W oapnA Ha^nancs c was ana KOoponHaT. (locxorbicy cwna. o6ycnoeneHHa«
rpaBmaunew. AewcTByeT BHM3, Mbi HMBBM
F = ma = -mgj
PMCYHOK 6
r^e g = | a | = 9,8 M/C2. TaKMM o6pa30M
a = -rA>K
nocKOJibKy v'(t)s a. MU nMeeM
V(t) = -gtj c
rae C = v(0) = vo. CneflOBaTenbHO
r'(t) = v(t) = -gtj + vo
l/lHTerpHpya CHOBa. Mbi nony^aeM
r(t) = " gt2j + tvo + D
Ho D = r (0) = 0, nooTOMy BeKTop nonoweHHH CHap^Aa 33AaeTCfl
in
r(1) = -3gtj + tvo
EC/IM MW 3annmeM | vo| = vO (HananbHaa CKopocTb CHapnaa), TO
Vo = BOKOC ai + BOCMH aj
M ypaBHeHMe 3 CTaHOBmca
r(1) = (wcos a)ri + [(%sin a)t - 1gr2] i
TaKMM oopa30M. napaMeTpuMecKne ypaBHeHun TpaeKTopnn ABnnioTCfl
■Eon* BU wcKnxwirre *3 ypaBHeHut* 4. BU
yBnayrre. yro y flBfweicfl ffiaapanwHOw (fcymjMein OT
s
X. Tamm o6paxM. TpaexTopitt nonera oapsqa ae/taeics
Macibio napa&oru
ropn30HTa/ibHoe paccTOBHMe d BBnaeTCB 3HaMeHMeM x npn y = 0. YcTaHaBJiMBaa y = 0T Mbi nonyMaeM
t = 0 MJIM t = (2v sin a)/g. 3TO BTopoe 3HaMeHMe 13aieM AaeT
d = x = (vo cos a)
2v0 rpex a _ vo (2 rpexa a cos a) _ npoTHB rpexa 2a ?
g
g
CKieBUAHO. MTO d nMeeT ceoe MaKCMManbHoe 3HaMeHne, Korfla sin 2a = 1, to ecTb a = T/4

6.

842 nil rnABA 13 BEKTOPHblE OYHKLIMM
B nPUMEPE 6 CHapnfl BwcTpenMBaeTcn c HananbHOM cKopocTbio 150 M/C M yrriOM
B03BbiLLi6HM8 45° c no3Mi4MM 10 M Hafl ypoBHeM 3eM/in l"Ae CHapafl yAapaeTca o
3eMJiK) H C KaKOfl CKOpOCTblO?
PEWEHUE Ecnn Mbi pa3MeciviM Hanaro KoopflMHaT Ha ypoBHe 3eMnn, TO Hana/ibHoe nonoxeHMe CHapafla
paBHO (0,10), M no3TOMy HaM HymHO CKoppeKTnpoB3Tb ypaBHenvin 4, Ao6aBHB 10 K BbipaweHwo y.
ripn vo = 150 M/C, a = 45° n g = 9,8 M/C2 Mbi nMeeM
x = 150 COS(T/4 )t = 75/2 t
y = 10 + 150 sin(T/4)t - 2(9,8)t2 = 10 + 75 /2 t - 4.9t2
YAap npoHCXOAMT, KorAa y = 0, TO ecrb 4.9t2 -75/21 -10 = 0. Pewaa 3TO KBaApaTHoe
ypaBHeHne (M ncnojib3ya TOJibKo no/ioxoiTenbHoe 3HaneHne t), Mbi nonynaeM
15yfl -I- V11.250 + 1%
21.74
9.8
Torfla x = 75/2 (21.74) = 2306, TBKMM o6pa30M, CHapaq nonaaaeT B 3eM/iio Ha paccTOflHMH OKOHO 2306 M.
CKopocTb CHap«A3 paBHa
v(1) = r(i) = 75/21 + ( 7 5 / 2 - 9 . 8 r ) j
T aw vSpisovi. e*o aopocrb np* yjjape pasna
|B(21,74)| - B(75 72)2 + (7512 - 9.8 • 21.74)2 = 151 M/C ■
TAHrEHUMAJlbHAfl H HOPMAJlbHAfl COCTABJ19lOll|ME YCKOPEHH9_
KorAa Mbi M3ynaeM ABn>KeHne HacTmjbi, nacTO ObiBaeT no/ie3HO pa3AeriMTb
ycKopeHne Ha flee cocTaB/ifliOLUMe, o,qHy B HanpaBneHMM KacaTenbHon. a ApyryK) B HanpaaneHMM
HOpMann. Ecnn Mbi 3anmueM v = v An* CKOPOCTM MacTMLibi, TO
T(t) =
r’(t)
v(i)
|r(] |v(>
v = vT
M TaK
ECHH MW AMcfe^epeHijHpyeM o6e nacTM aioro ypaBHeHtta oTHOCHTenbHO t, TO nonyMMM
s
a = v* = v't vT*
ECJIH MW ncno/ib3yeM BbipaweHMe AH* KpnBM3Hbi, 3a,qaHHoe ypaBHeHneM 13.3.9, TO Mbi MMeeM
s
K=
TaK
I T| = KB
EAUHMHHNM Beioop HopMa/iH 6bin onpeAenen B npeAbiAymeM pa3Aeoe xax N = T7| T’|,
nooTOMy (6) AaeT
T* = | T'|N = KBH
H ypaBHeHHe 5
CTaHOBWca
m
a = v't + kv2N i

7.

PA3flEJl 13.4 flBUWEHME B flPOCTPAHCTBE: CKOPOCTb H YCKOPEHUE III! 843
3anncbiBafl ar M ay f\r\n TaHreHunanbHOM M HopMa/ibHOM cocTaB/iHKXAMx ycKopeHna, Mbi MMeeM
a = a7T + axN
nae
JB] ar = V M an = kv2
3TO pa3pemeHne nponnfiKxrrpMpoBaHO Ha pwcyHKe 7.
flaBawTe nocMOTpwM. mo roBopnT<DopMyna 7. nepBoe. Ha mo cneAyeT oopaTMTb BHMMaHne. STO TO. mo
OMHopMaribHbiM BeKTop B OTcyTCTByeT. He3aBncnM0 OT Toro, KaK o6teKT ABMxeTca B npocipaHCTBe. ero
ycKopeHHe Bcer^a nexMT B ruiocKOCTM TMN (ocL^miJiMpyjOL^aa n/iocKocTb). (HanoMHMM, MTO T 3aAaeT
HanpaB/ieHMe ABMweHMfl , a N yKa3biBaeT HanpaBneHne noBopoTa KPMBOM.) flanee Mbi 3aMe^aeM, MTO
TaHreHUManbHa^ cocTaBJiaK>maa ycKopeHMfl paBHa v\ CKOPOCTM M3MeHeHMH CKOPOCTM. a HopManbHaa
cocTaBnafomafl ycKopeHMa paBHa kv2, KpMBH3He, yMHOweHHOfl Ha KBaApaT CKOPOCTM.
3TO MMeeT CMbicn, eciiM MU AyMaeM o naccaxwpe B aBTOMoOM/ie - pe3KMtt noBopoT
Ha Aopore 03HanaeT Oonbiuoe 3HaneHMe
Kpnen3Hbi k. noaTOMy cocTaBn^JOLqa^ ycKOpGHna. nepneH^nKynapHa^ flBMxeHMK). Be/in»ca . n nacca>*aipa
* * * W *
OTCpacbiBaeT K AsepM aBTOMo6u/ifl. Bbicoxafl acopocTb Ha noBopoTe MMeeT TOT xe atfxpeKT; cpamwecKM, ecriM
Bbi yABOMTe CBOK) CKopocTb, ay yBexiMHHTca B 4 pa3a.
XOTH B ypaBHeHMax 8 y Hac ecTb BbipaweHMfl TaHreHLjManbHOM M HopMa/ibHOM cocTaBJiaioiAMX yCKOpeHM« ,
>KeiiaTenbHO MMeTb Bbipawenkm, 3anncnmMe ToribKo OT r, r'M r". C DTO\A Menbio
Mbi OepeM ToneHHoe npoM3BeAeHMe v = vT Ha ar xax yxa3aH0 B ypaBHeHMM 7:
v • a = vT • (v’T + kv2N)
= BB'T • T +• KB'T • H
CreooBaTGfibMO
= w' (nocKonbKy T - T = 1 M T - N = 0 )
v- a _
0
r'(t)
r"(t)
v I r*() |
Mcnonb3yn cfeopMyny ana KpnBM3Hbi, npuBefleHHyio B teopeMe 13.3.10, MW wvieeM
a,\ — KV
r'(t) X r" ( / ) | . r'(/) X r''(/)
— I r (/) |- ” -
|r’(t)
nPMMEP 7 HacTnua flBM>KeTCH c c£yHKMneM nono*eHnn r(t) = (12,12, ). HawflMTe
KacaTenbHyKD n HopMaribHyto cocraBnaioaine ycKopeHMH.
peiLienne
r ( t ) = t 2 i + t2j + t2k r’(t) = 2ti
2tj + 3t2 k r"(t) = 2i + 2j + 61 k
| r*(1)| = 812 + 9t
C/iefloeatenbHo. ypaenenne 9 aaer Tanrenuna/ibKyio cocTaanflioinyio e mwe
0 -r"(t)
Si + 1 _
| r’(t)|
v/8t2 + 9t4

8.

844 III! ITIABA 13 BEKTOPHblE OVHKUMH
1 A>K K
9
r'(t) x r"(t) = 2t 2t 3t2 = 6t2 i - 6t2 j
I 2 2 6T
ypaBHeHwe 10 flaer HopManbHyio cocraanaioLuyio B BHae
r"0]
Ir’(t)|
V812 + 9t'
3AKOHbl flBl/l>KEHMfl nJIAHET KEnHEPAJ_
Tenepb Mbi onmueM OAHO M3 Bemwaflujnx AOCTiflKeHHfl MaTeMaTunecKoro aHann3a, noKa3aB, KaK
MaTepnanbi 3TOH rnaBbi Moryr ObiTb ncnonb30BaHbt ana flOKa3aienbCTBa 33KOHOB ABMweHMH
nnaHeT Kennepa. riocne 20 neT n3yHeHnn acTpoHOMMHecKMx Ha6nioAeHMM AaTCKoro
acipoHOMa TMXO Spare HeMeqKMfl MaieMaTMK M acipoHOM MoraHH Kennep (15711630) apopwiynnpoBan cneayioiuMe rpn 3aKOHa.
3AKOHbl KEIUIEPA
ri/iaHeTa BpamaeTca BOKpyr conHLja no annHmwecKOM op6nTe c conwneM B OAHOM
1.
epoxy CG.
JlMHMfl, coeflMHfliomafl conHue c nnaHeioiJi, oxBaibiBaei paBHbie ruioujaAM 3a paBHoe
Bpesia.
2.
3.
KBa/jpaT nepMOAa oGpaLAeHMn nnaHeTbi nponopuMOHaneH xy6y AnMHa
r/iaBHOM OCM ero op6nTbi.
B CBoeM KHMre Principia Mathematica 1687 roAa cap Mcaax HbiOTOH CMor noKa3aTb, MTO STM Tpn 3aKOHa
flBflflK)TC« CJieACTBM«MM AByX ero C06CTBeHHblX 3aKOHOB, BTOpOfO 3aKOHa ABHWeHMfl M
3aKOHa BceMupHoro TnroTeHMn. B AanbHeflLueM Mbi AOKa>KeM nepBbifl 33KOH Kennepa.
OcrajibHbie :a<ohbi ,aoKa3t.Ba-OTca s BMA© ynpattweHuu (c noAoeasKaMu).
nocKonbKy rpaBUTauMOHHaa awa conHLia Ha nnaHeie HaMHoro 6onbiue, MeM cw/ibi, AeHCTByK)mne Ha
Apyrwe HeOecHbie Tena, Mbi Mox<eM CMeno nmopnpoBaTb Bee Tena BO BceneHHOfl, xpoMe conHLja M OAHOM
nnaHeTbi, BpaLnawmeMcn BOKpyr Hero. Mbi ncnonb3yeM CMCTeMy KOopAMHaT c conHAeM B LjeHTpe. H
nycTb r = r(t) - BeKTop nonoweHMn nnaHeTbi. (C TBKMM >xe ycnexoM r MO>xeT ObiTb
•
M«
a
m
m
• m
m
m
t
BeKTopoM nonojKeHMB JlyHbi MJIM cnyTHMKS, ABM)KyiijerocR BOKpyr 3 eMnn. unn KOMOILM. .uBUJKymeMCTi BOKpyr 3 Beaat>i.) Beicrop
CKOPOCTM pa BOH v = r*. a BOKTOP yCKOpOHnn pa BOH a = r*\ MBI ncnonb3 yeM
cneAyK)ii4ne 3aKOHbi HbioTOHa:
BTopow 33KOH ABM>KeHnn: F = ma
oM
3 OKOH BceMupHoro T«roTeHw«:
u
rAe F - rpaBMTaLjMOHHan cwna Ha nnaHeTe, muM - Maccbi nnaneTbi M conHua, G rpaBMTauMOHHan nocTOHHHan, r = |r|, a u = (1/r)r - eAUHMMHbm Beicrop B HanpaeneHMHr
CHa^ana MU noKaxeM. mo nnaHeia ABMxeTca B OAHOM nnocxocTM nyTeM npMpaBHMBaHMB BbipaweHHM
Ana F B Asa aaxona HbtOTOna. Mbi HOXOAMM. MTO

9.

PA3flEJl 13.4 flBl4>KEHME B nPOCTPAHCTBE: CKOPOCTb M YCKOPEHME INI 845
M noaTOMy a
napannenbHa r. OTooaa cneAyeT. MTO r X a = 0. Mw ucno/ib3yeM cpopMyny 5 B TeopeMe 13.2.3
. %To6ki simcaTb ^(V*v) = r - X v *
rX
-vX:v+rX
a-0
+
0-
0
CneAoaine^Ho r X = hi
rAe h - nocTO^HHbiM BeKTop. (Mbi MOKeM npeAnorio)KMTb, MTO h # 0; TO ecTb r u v H e napa/irienbHbi.)
3TO 03HaHaeT, MTO BeKTop r = r(t) nepneHAHKyn«peH h ATO Bcex 3HaneHmi t. noaTOMy n/iaHeia BcerAa
newm
B ruiocKOCTM, npoxoAflmeii ^ePe3 Hanaro KOOPAHH3TT nepneHAMKyropHoe h. TaKMM oopa30M, op6ma
nnaHeTbi
■«
nOQArrairsar cata* rrccuyc ip»«/c
MTO6W A0K333Tb nepBwi»i 33KOH Kenjieps. Mbi nepenuujeM Beicrop h cneAyioiuMM oop330M:
h = rXv = rXr = ruX (ru)#
= ru X (ru* + r’u) = r2(u X u') + rr’(u X u)
= r2(u X u')
3areM
^ * ft
X (r2u X Lf) = -GMu X (u X uf)
^ r‘
=- GM[(u - u')u - (u u)u*] (no TeopeMe 12.4.8. CBOMCTBO 6)
Ho u - u = | u |2 = 1 n, nocKonbKy | u(t) | = 1, U3 npwwepa 4 B pa3flene 13.2 cneayeT,
HTO
u • u’ = 0. CneAOBaTenbHo
a x h = GMu*
MT3K (v X h)# = V- X h = a X h = GMu'
klHTerpupyB o6e Mac™ 3Toro ypaBHeww, MW nonynaeM
QT]
v X h = GMu + c
TA© e - nOCTOflHHblM BeKTop.
Ha 3TOM 3Tane YAO6HO Bb>i6paTt» OCM Koop^nHaT T3KMM o6pa30M. MTO6W CTaHAapTHbiM 6a3MCHbiM *
m
m
m
m
m
m
m
m
m
BeKTop k yKa3biean B HanpaeneHnn BeKTopa h. 3aTeM nnaHeTa ABM>KeTC* B rmocKOCTM xy. nocKO/ibKy
n v X h . n u nepneHAMKyropHbi h, ypaBHeHMe 1 1 noKa3biBaeT, MTO C newMT B nrocxocTM xy. 3TO
03HaHaeT, HTO MW MoweM BwOpaTb OCM x M y TBKMM oOpa30M, HTOOW Beiaop l newan B HanpasneHMM e, xaK
noKa3aHO Ha pucyHKe 8.
Ecnn 0 - yron Me>xny c M r, TO (r. 0) - nonnpHbie KOopflMHaTbi nnaHeTbi. M3
ypaBHeuMfl 11 MW MMGOM
r (v X h) = r (GMu o) = GMr - u r - c
= GMr u - u + | r|| c|cos 0 = GMr + rc cos 0

10.

846 Itll rflABA 13 BEKTOPHblE OYHKUMM
rfle c = | e |. 3aieM
r - (v X h) 1 r (v X h)
GM + c cos 0 GM 1 + e cos 0
rfle e = c/(GM). Ho
r (v X h) = (r X v) • h = h • h = | h2 = h2
rfle h = | h |. V\raK
h2/(GM)
r
eh*/c
1 •+• e cos O 1 e cos O'
3anncaB d = h2/c, nonynuM ypaBHeHne
pea*
r =-
1 e cos 0
CpaBHUBaa c TeopeMOM 10.6.6. MU BH/JMM, mo ypaBneHne 12 ABnaeTCfl nonapHUM ypaBHeHneM KomwecKoro
ceMeHMH c (jDOKycow B wana/ie KOOPAHH3T M 3KCi4eHTpnorreT0M e. Mu 3HaeM, mo op6ma nnaHeTbi npeACTaBnaeT
33MKHyTyto KpuByto. n nodTOMy KOHMKa flonxH3 6b»iTb arsnuncoM
COGOM
A
3TO 3aBepujaer BUIBOA nepBoro 3aKOHa Kennepa. Mbi npoBeASM Bac Mepe3 onpeaeneHne BToporo M
TpeTbero 33KOHOB B npwcnaflHOM npoeicre Ha CTpaHMLje 848. flOKa3aTe/ibCTBa 3TMX Tpex 33K0H0B
nOKa3blBaiOT MTO MeTOAbl 3TOM maBbl npeAOCTaBJlBKDT MOLMHbIM MHCTpyMeHT A-rm onucaHMfl
HeKOTopbix 3aKOHOB npMpoqbi.
-1
13.4 ynPA>KHEHHfl
■_I
I. B Ta6nnue npMBefleHu KoopAMHaTu Macmubi,
ABnwymertcfl B npodpaHCTBC no nnaBHOM KPMBOM (a)
HaiiAHTc cpeAHne CKOPOCTM 3a BpeMeHHbie
MmepBanbi [0, 1), [0.5, 1], [1. 2], M [1,
1.5]. (6) Ol^eHMTe CKOpOCTb HaCTMLJbl npM t = 1.
1
0
0.5
1.0
1.5
2.0
X
V
-
2.7
3.5
4.5
5.9
7.3
9.8
7.2
6.0
6.4
7.8
3.7
3.3
3.0
2.8
2.7
2. Ha pt^cyHKe noKasaH nyn> uacmt^bi. KOTopan aBMXSTCfl c ESKTOPOM
nono»;sHW3i r(1) B MOM6HT Bps MS HU t.
(a) Hapncyi4Te Betcrop, npeacTaB^ioii^MM cpe^Htoto CKopocTb uacTHL4a 3a MHTepsan
BpeweHM 2 = t = 2.4.
(b)
HapMcynTs Bsrrop. npeocTaenaioujMM cpsanioto CKopocTb 3a HHTspsan
BpeMeHM 1 . 5t2.
(c) HanmunTe BbipaweHMe A^H BeKTopa CKOPOCTM V(2).
(d) HapiicyMTe npn6nn>KeHMe K BeKTopy v(2) M oueHme
CKopocTb MacTMUbi npM t = 2.
3-8 HaPiAMTe CKopocTb. ycKopeHne M pa3HOCTb cxopocieM Macinubi c
33AaHH0M 4)yHKi4Meii nonoweHMR HapucyMTe TpaeKTopMK)
MacTHAbi
M HapMcywTe eSKTOpbl CKOPOCTM H ycKopeHH^ AJIH yK333HHoro 3HaueHna uacTo.
r(t)
■■■■■
4 r ( t > = ( 2 - 1 , 4/1 > . = 1
irl

11.

PA3flEJl 13.4 flBM>KEHME B nPOCTPAHCTBE: CKOPOCTb M yCKOPEHUE till 847
5.
26. riyiiJKa CTpenneT c yrnoM B03BbiujeHHH 30s. KaKOBa HananbHafl
r(t) = 3 cos ti 4* * 2 sin tj. t = m/3
cxopocTb CHapfl^a. GCJIW MaKCMManbMaa Bucoia cuapsjia cocTaanaeT 500 M?
6.
r(1) = e’i + e2*j, t = O
7.
r(t) = ti + t2j + 2k, t = 1
8.
r(t) = ti + 2 costj + sintk, t = 0>
9-14 HaiiAHTe
MacTMijbJ c
CKopocTb,
27. nyiuKa MMeeT HaMaiibHyio cxopoob 150 M/C HaMAMTe Asa yrna H3KJIOH3
aneBbiTMQK OQTOPbM Ua»«T 6b7b HCTOnb30&3M OTIS flOpaxeHXfl Lpnx na
paCCTQAHNM 800 M
ycKopeHne
u
pa3HocTb
CKopocreM
28.0T6Mea<xi*MM 6beT 6ewo5onbMWM MPMOM B 3 (feyiax naA 3eM;ieii no
HanpaeneHMO t
orpsautcHmo uetiipanbHoru nom. nofopoe nwee' XMOory 10 ityrutt M Kinx^rc* a 400
cpyiax 01 juwaujc*
3a,aaHH0M c|)yHKmieM nonotteHMfl.
nnomaAKM. MJW Bw/ieiaeT M3 6MTU CO cxopocTbio 115 (pyTOB/c noA
yrnoM 50s K ropn30HTa/in. 3TO xoyMpaH? (/HpyrnMn cnoear/n.
9. r(1) = (t2 + 1, 13, 12-1)
10.
r(t) = (2 cos t. 3t. 2 sin t)
11.
r(t) = /2ri + e’j + e'k
12.
r(t) = Pi + Intj tk
13.
r(1 ) = G'(COS / i ■+■ sin /j 4- k)
14.
r(z) = t sinti + tcostj + t2 k
nepencTad nw MAM ^epe3 3a6op?)
29. CpeflHeBeKOBbiM ropoA MMeeT cJjopMy KBaApaTa M 3aiAnu4CHl CTeHaMM
AAHHOM 500 M H BbicoTOM 15 M. Bbi KowaHAup aiaKyioiAeM apMMH, H
caMoe 6nn3Koe paccTOBHne, KOTopoe BW MO)KeTe noAOMTM K CTene.
cocTaBAJieT 100 M. Baiu nnan COCTOMT B TOM. HTo6bi noA^enb ropoA: 6pocan
paacaneHHbie K3MHH Mepe3 creHy (c HaMa/ibHOM
CKOpOCTbK) 80 M/c) riofl K3KMM yr/lOM Bb4 AO/DUMbl f1pMKG3dTb
CBOMM XIKV^IM
ycTaHOBMTb KaTanyribTy? (npeAnonoKMM, MTO nyTb KaMHeM
15-16 HaPiflUTe BeKTopu CKOPOCTM M nonoxceHMH sacrmjbi, KOTopaa nweeT 3aAaHHoe
nepneh^wcy.nxpew crene.)
ycKopeHne M 3aAaHHbie HaManbHbie CKopocTb M nono>KeHMe.
15.
16.
30. MHH Maccow 0,8 Kr noASpacbiBaeTCfl B B03Ayx Ha \or co
a(t) = i + 2 j, v(0) = k, r(0) = i
CKopocTbK) 30 M/C noA ymoM 30e K 3eMAe. 3anaAHbiM Beiep
a(t) = 2i 61 j + 12t2k, v(0) = i, r(O) = j - k
npmoiaAbiBaeT K MHMy nocTOAHHy»o cmiy 4 H B BOCTOMHOM
HanpaBneHHM. KyAa npn3eMjifleTCfl MAM M C KaKOM CKopocTbio?
I7-I8
031. BoAa, ABM>KyiAaBCH no npflMonMHewHOMy y^acTKy peKM, O6WMHO
TeneT SwcTpee Bcero B cepeAMHe a y 6eperoB CKopocTb 3aMeAnneTCfl
(a) HaMflme BCKTOP nonoc*eHHfl HacTMLiw. KOTopaa MMeeT 3a^aHHoe ycKopeHne-
nOMTH AO Hynn. PaCCMOTpMM AAMHHbIM npAMOM yMacroK
ynpaBneHne M 3aaaHHb*e HauaribHa^ CKopocTb M nono>KeHMe.
£ 2 (6) Mcnonb3yme nownbOTeo AHA nocrpooH*» rpa<$MKa rpacKTop#w MOCTHUW
peKH
17.
a(t) = 2ti + sintj + cos 2t kf v(0) = i, r(0) = j
. Te»yiu€*« Ha ceeep. c napannerb«»Mx SeperaMM Ha paccTOflWM 40 M ppyr OT
18.
a(r) = ti + e'j + e fk, v(0) = k, r(0) = j + k
waxotvai ibKOtH CKopocrb BCKlbJ cocia»n«eT 3 w.'c MU Moucew wcnonbaosaTb |
fipy^a Ecnn
0yHKL4Mo B KauecTse 5B30BOw Moaeribi ana CKOPOCTH noToo saiw x ezuiHHLj c
*
[19. OymojHfl nonoweHMn MacTML^w 3aflaeTCfl gepe3
3anaAHOro 6epera: f(x) = m x(40 - x).
(а) JlQAKa ABH)KeTCH C nOCTOHHHOM CKOpOCTbK) 5 M/C M3 TOHKM
r(t) = (t2. 5t. t2 - 16t). Kor^a CKopocTb MMHiiMajibHa?
A Ha 3anaAH0M 6epery, coxpaHnn Kypc, nepneHAMKynnpHWM
20. KanaH cnjia ipc6yciCH Ann Toro. HTO6BI nacTnua waccon m HMena
•*
%
%
cpyuKumo no/iO)Kemifl r(t) - t i ♦ t2j 4 t3k?
6epery.
Kax AaneKO BHM3 no pexe Ha npoTMBonono>woM 6epery noAKa
KOCHeTcn Oepera? HaHecMTe Ha rpacpMK TpaeKTopmo ABMxceHMfl
21. Cuna BeriMMMHOM 20 H fleMCTByeT HenocpeflCTBeHHO BBepx OT nnocKOCTw
nOAKM.
xy Ha o6beKT waccoM 4 Kr. 06ve»cr HaMMHaeicfl B Hanane KOOPAMH3T C
(б) npeAnono)+wM. MW xoTenH 6bi nunoinpoBaTb noAKy. mo6bi OHa
Hawa/ibHOM CKopocTbto v(0) = i - j. HaiiflMTe cpyHKUMfo ero
npM3eFAHMnacb B TOMKe B Ha BOCTOHHOM Gepery nfWMO
HanpoTMB A. ECTIM MU
nonoweHMH
M ero CKOpOCTb B MOMeHT BpeMeHU t.
noafiepxHBaeM nocro«HHy»o cxopocTb 5 M/C M nocroiiHHWM xypc, HanAHTe
22.
noKaxcvrre, MTO ecnn MacTMija ABHtteTcn c nocroflHHOM
CKopocTbK),
yron. noA KOTOPWM nc«Ka AO/DKna AOMraTbCfl. 3aTCM HaMOcmc na rpatpMK
0aKTi^ecxiM nyTb. no Koropowy cneflyer noftKa. Kaxeica nw 3TOT nyTb \
pearvCDwwM?
TO BeKTOpW CKOpOCTM M ycxopeHM* OpTOrOH3nbHbl.
23. CHapHQ BbicTpenMBaeTCJi c Ha^a/ibHOM acopocTbio 500 M/C M
yrnoM B03BWLUCHMfl 30s. HakanTc (a) Aank»HOCTb noneia
32. flpyroM pa3yMHOM MOAe/ibK) CKOPOCTM BOAbi B pexe B ynpa)KHeHMM
31 HBnneTcn CMHycoMAanbHan 4>yHKi4M«: f(x) = 3 sin(mrx/40). ECTIM
CHapnna,
JIQAOMHMK xoneT nepecenb peKy M3 nyHKTa A B nyHKT
(6) MaKCMMaJlbHyra AOCTMmyTyio BbicoTy M (B) CKopocTb npn yAape
24.
B c nocTOflHHbJM KypcoM
M/C,
yron. noA KOTOPUM nOAxa aon>3ia ABMraTbca.
BbICOTbl 200 M H3A 3eMHOM.
Ha paccTOHHHH 90 M, KaKOBa 6bma HananbHafl CKopocTb
Mana?
nocTORHHOM CKopocTbio 5
onpeAenMTe
flopa6oTaMTe ynpa>KHeHne 23, ec/iM CHapflA BbinymeH c
25.
MAM Gpocaeicn noA yrnoM 45° K 3eMne. ECTIM MAM
npn3eMJifleTCfl
M
33-38 HaMAMTe TaHreH4ManbHyK) M HopManbHyK) cociaenniounMe Beicropa
yCKOpOHUH.
33.
r(t) = (3t-t3) i + 312j

12.

848 III rflABA 13 BEKTOPHblE OYHKMHH
37.
r(t) = e'i + /2tj + e’ k
38.
r(t) = ti + cos2tj + sin2tk
41. OyxxyMfl no3MunoH*poeaHMR KOCMMSOCKOJO xopa6nR - 3TO
/ 4 \
r(t) = (3 + t) i + (2 + B t) j +(7 - —~J k
39. BenuMMHa Beicropa ycxopeHMfl a paBHa 10 CM/C2. Hcnonb3yMTe
pucyHOK Ana oueHKM TaHreHUMa/ibHOM M HopManbHOM
cocTaBnaiomMX a.
a KoopAMnaTbi KOCMMMCCKOM craMjMM paBHbi (6, 4. 9). KanuTaH xoneT.
HT06bJ KOCMMHeCKMki KOpa6/lb npUHajlHJ! K KOCMMMOCKOM CIclHUHH.
m
m
m
Korfla cneAyeT BWKJiiOMaTb ABMraTejin?
42.
Paxeia, OKuraKXuafl csoe 6opioooe
npocTpaHCTBe
TORJIMBO
npn
JUH>KBHMM
B
, MMeeT CKopocTb v (t) M Maccy m{l) B MOMCHT BPCMCHM t. ECJIM
BwxjionHwe ra3bi BWXOA^T co CKopocTbio v OTHOCMTejibHO paxeTw,
TO M3 BToporo 3aK0Ha ABMWeHMfl HbfOTOHa MOXHO BWBeCTM,
MTO
dv
ml- = -V,.
dt dt
40. ECJIM MacTMua c Maccoti m ABwweTca c BeKTopoM nono>KeHWH
r(z), TO cc yrnoBOM MOMCHT onpeAenneTcn KaK L(t) = mr(t) * v(t), a ee
KpyTamuM M0M6HT KaK T(t) = mr(t) X a(t). l"IOKa>KMTe, MTO L'(t)
(а) noKattWTe. MTO v(1) = v(0)/m(0) ^
v
m(t)
= T(t). BbiBeAMTe, MTO ecnM r (t) = 0 AHA Bcex t TO L (t) BB/iaeTca
(б)
nOCTOflHHbJM. (0TO 30KOM OOXpaMeHMfl ymoeoro MOMGHTa.)
cxopocTM, aaooei
MTO6W paxeTa paaomanacb no npflMOM OT TOMKM noxon no
npeBtuuawmcii cicopocTb oo co6crB6HHb4x Bbjxnonhtxx raaoe, xaxyio
aomo csoe* hanaribHOM Maco> paxeia AOJW^a Gun a 6u oxeMb 5
xaMecree Tonnuaa7
nPMKJIAQHOM
nPOEKT
3AK0Hbl KEnilEPA
MoraHH Kennep ccbopMyjinpoBa^ c/ieAytouiMe TPM 3aKOHa ABM>KeHMB nnaHeT Ha ocHOBe Maccbi
AaHHbix o no/ioxceHMRx nnaHeT B pa3Hoe BpeMR.
3AK0Hbl KEn/lEPA
1.
rbiaHeTa BpamaeTCfl BOKpyr conHija no annwTTMMecKOM op6nTe c conHi^eM B OAHOM
cf)OKyce.
2.
JIMHMH. coeAMHHioiAaa conHije c nnaHeTOM. oxBaibiBaei paBHwe nnomaAM 3a
paBHoe BpeMR
3. KBaApaT nepMOAa o6pameHnn nnaHeTbi nponopuiiOHaneH Ky6y AnHHbi
SonbiuoM OCH ee op6nTw.
Kennep
c0opMynnpoBan STM 3aK0Hw. noTOMy MTO OHM cooTBeTCTBOBaxiM acTpOHOMMMecKMM
ABHHUM. OH He Mor noHHTb , noMeMy OHM 6WJIM npaBAOM MJIM KSK OHM CBfl3aHw Apyr c ApyroM Ho
cap Mcaax HbfOTOH B CBOMX TIpHHUMnax MaTeMaTMKM" 1687 roAa no«a3ant KaK BueecTM TPM
3aK0Ha Kennepa M3 AByx coGcreeHHbix 33KOHOB HbKJTOHa, BToporo 33K0Ha ABMweHMH M 3aKOHa
BceMMpHoro TnroTeHMfl. B paaAene 13.4 MW AOKa3anM IlepBbiM 33KOH Kennepa, Mcnojib3yn
MCMMcneHMe BeicropHwx cpyHKLjMM. B STOM npoeicre MW no3H3KOMMM
m
m
uac c AOKa3aTenbCTB«iMM Bioporo M TpeTbero 3BKOHOB Keruiepa M MCcneayeM neKOTopwe M3 MX cjieacTBMM.
I. Hcnont»3yMTB cneay*omne luarw. uTo6bi ao*ca3aTb BTOPOM JHKOH Kennepa 06o3HaneHMB TB >KB. MTO M B
—
a
—
A0K323TenbCTBe llepBono 3a<OHa B paaaene 13.4. B uacTHOCT*. ncnojib3yMTe nonRpHbJe xoopflMH3Tb4 T3K. UTO6W
r = (r cos 0) i + (r sin 0) j.
(a) noxaxoiTe. MTO h = %
dt
o 6^
(c) ECJIM A = A(t) - 3To njioiAaAb.
oXBaTWeaeMas paAMyC-BeKTopoM r = r(t) B
(b) BbiBeAHTe,
MTO ^ -= h.
KiK no«a3aHO ha pwqrMiie, noxaxHTe, MTO
BpCMCHM [to, t]

13.

mABA 13 OB30P Mil 849
(<J) Caenaib BtJBOfl. MTO
•
cfe th - nocTOAHHaR
3TO roBopm o TOM. MTO CKopocTb. c KOTOPOW A Bb»MeTaeTCfl, nocroflMHa. A AOKa^oisaeT BTopoe KennepOBdcoe
3a«oH.
2.
nycTb T - nepnoA oopameHMfl nnaHeibi BOKpyr Co/iHua: TO ecTb T - BpeMa. HeoSxoAMMoe ei?i ann Toro.
MTOSW OAMH pa3
coBepiuMTb o6opoT no cBoew 3nruinTvmecKOM op6nTe (“IpeAnonoaoiM. MTO 6onbiiJOM H ManoM ocew annwnca
paBHbi 2a u 2b.
(a) Mcnonb3yMTe MacTb (d) 3aoami 1, mo6bi noKa3aib. mo T = 2 M6MT/M
X&
b* I. 2 3 4 5 *
UJU
a
(b) noKaaart, HTO= ed=2 —
_ 4TT: ,
(c)
Hcnonb3yvrre MSCTH (a) M (b), MTO6U noKaaaTb, MTO TZ = * —-— a .
CM
^#.
3TO A0K33bJBaeT Tpe™»i 33KOH Kennepa. [OGpaTime BHMMaHne. MTO KOHaaHTa nponopuMOHa/ibHocm
42/(GM) paBHa He3aBMCMMt>iM OT nnaHeTbi.]
3. riepMoa o6pameHnn 3eMnn no op6nTe cocTaan?ieT npn6nn3MTenbHO 365.25 cyTOK. Hcnonb3yMTe
•**«0*
3TOT cfcaKT M TpeTMM 33KOH Kennepa. uTo6bi Hawn* onHHy rnaeHOM OCM 3eMHOM op6nTbi. BaM noHaAo6MTC*
Macca co/iHqa M = 1,99 * 1030 KT M rpaBUTauHOHHaa nocTOHHHafl G = 6,67 * 10-1 H-M2/KT2.
4. MOXCHO BWBecTM cnyTHMK Ha op6nTy BOKpyr 3eMnn TaK, MTO6U OH ocTaBancB HenoABn>KHbiM HaA
3aAaHHWM MecTonono>KeHneM Ha OKBaTope. BbiMncnnTe BwcoTy. Heo6xoAMMyK) Ann TaKoro
cnyTHMKa. Macca 3eMfiH cocTaBJiaeT 5,98 * 1024 KT; ee paAnye cocTaB/raeT 6.37 * 10“ M. (3T3 op6nTa
Ha3bisaeTCH reooiHxpoHHOM OP6HTOM Knapxa B M©CTb Apiypa K. K/iapna. KOTOpuiii enepswe npeanoaoin 3Ty
HACK) O 1945 rofly. riepBUH iHKUA cnyTHMK. Syncom 2.6bin aanymeH B none 1963 rpoa.)
9
- 13 OB3QP t
_I
1P0BEPKA<0H4E1UHH
6. (a) KaKOBO onpeAeneHne KpnBM3Hbi?
I. MTO Taxoe BeKTopHaa (JyyHKUMfl? Kax Bbi HaxoAHTe ero
npoM3BOAHyK> H ero MHTerpan7
2.
KaKoea CB93b
npocTpaHCTBeHHWMH
Me>KAy
BOKTOPHWMM
cpyHKUMflMM
(6) HanmuMTe 4)opMyny ATIS KpnBM3Hbi B TepMHHax r*(t) H T’(t).
(c) HanmuMTe (J>opMyjiy AJifl KpnBH3Hbi B TepMHHax r'(t) M T(t).
M
(d) HanmuMTe 0opMyny AJifl KPHBH3HW nnocKon KPHBOH CI
ypaBHeHHeM y = f(x).
KpMBWMM?
3. Kax Bbi HaxoAHTe KacaTe/ibHbii/i BexTop K rnaAKOH KPMBOM B
TOMKe? Kax Bbi HaxoAHTe KacaTenbHyio JIHHHIO? EAMHMMHUM
KacaTeribHbiM
BeKTOp?
4.
ECJIH u H v - AH<t)<t)epeH4npyeMwe BeKTopHbie (J)yHK4HH, c CKa/wp, a f BemecrseHHOSHaMHafl CPYHICMKR hanmuwTe npaB*wa AIM? AH0$epeH4Mpo5aHMfl
cneAyioinux BeKTopHbJX cpyHKqHH.
(a) u(t) v(t) (b) cu(t)
(c) f(t) u(t)
7.
(a) HanmiMre(popMynweauHMMHoroHopManunoro M 6MHOpMajibHoro BGKTOPOB
rnsuKOM npocrpaHCTBeH HOM cpveoM r(Q.
(6) KaKOBa HopMa/ibHa^ nnocxocTb KPMBOM B TOMKe? MTO Taxoe
conpuKacaKKuawCH nnoCKOCTb? MTO Ta*oe KOne6ntoiufc«ca <pyr?
8.
(a) Kaic Bbi HaxoAHTe CKopocTb. o6opoTbi M ycKopeHHe MacTMAbi
KOTopaa
ABM>KeTC?T no npOCTpaHCTBeHHOM KpMBOM? (6)
3ailMUJMTC yCKOpOHMO B TOpMMHaX oro IdHrBHUMa/lbHOM M
5. KaK Bbi HaxoAHTe AnHHy npOCTpaHCTBeHHOH KpHBOH, 33A3HH0H BeKTOpHOH
HOpManbHOH cocTaBnatotuHX.
(d) u(z) • v(t)
cpyHKLiHen r(t)?
<e) u(t) X v(t) (f) u(f(t))

14.

850 llll ITIA BA 13 BEKTOPHblE OYHKUHM
TECT TIPABAA-nOWb"
OnpeflenwTe. aareeTca m yTBep*AeH*e MCTWHHWM MAM JXWWWM ECIM STO npaaaa ofoaoiMTe
5. ECJIM T(t) - eAMHMHHb4M KacaTenbMbJM BOKTOP rna/jKOM KPMBOM. TO
npMBM3Ha pauHa K — |dT/dt.
noMeMy Ecnn 3TO Hesepno. oObflCHKTe rK>evy MOM noMaeflMTe isuvep. onpoaepraKJmMM
3TO yT8ep*AeMne
1. KpuBaa c eeKTopHUM ypaBHCHMCM r(t) = t3l -♦* 2t3j ♦ 3t3k npeacTaenfleT CO6OM
•
6 BuHopManbHWM aeKTop paBeH B(t) = N(t) * T(t).
7. npeAnonoxMM. MTO fis ABa^u HenpepuBHo AH<M>epeHAnpyeMa Ha
nepentie
npflMy>o.
2. npoM3BO,QHaB B0KTopHOH (pyHKUMM nonywaeTcn rryreM
TOVIKaKPMBOM y = f(x). KpHBMSHap3BH3 O.
A
anctxJjepeHunpoBaHMP Ka>KAOM KOMnOHeHTHOM 0yHKI4MM.
3. ECHM u (l) M v (t) oerowTCH flwtxfeepeHijMpyeMbiMM eeicTopHWMM
(fcyHKUMHMM. TO
r •»
8 Ecnw k(t) = 0 Ana ao&x t, TO HJMMP npeocTaB/weT co€w nppuryw
JMHMCJ 9. ECJIM | r(t) | = 1 Ann BCGX t. TO | r*(t) | flanaeTCfl KOHCT3HTOM.
10. ECJIM | r(t) | = 1 pj\7\ Bcex t, TO r*(t) opToroHanbHo r(t) AJIH ecex t.
V(t)] = u’(t) v*(t)
11. Ko/ie6jiK)LnaflCfl OKpywHOCTb KPMBOM C B TOMKe MMeer TOT >Ke
4 Ecnn r (i) ABnaeTcs? Aw00epeMunpyeMOM Be<Topnow (^HKMMGM. TO
KacaTe/ibHbJM BeKTop, Beicrop HopManw n KpnBM3Hy. MTO M C’at. MTO
^•MOI-Ir' W I
TOMKa.
*
12.
Pa3JiMHHbie napaMeTpn3aunM
MfleHTMMHWM
CVIHOM M TOM
we
KPMBOM
npMBOAflT
K
KacaTenbHWM BeicropaM E ASHHOM TOMKe KPMBOM.
ynpa>KHeHMB
1. (a) HapwcywTe KpHByto cnoMoiubio BOKTOPMOM cfeymuMM
HOpManbHbM B0KTOP
(6) HaMAMTe r‘(t) M r"(t).
(c) KpMBM3Ha
2. nycTb r(t) = ( 2 -1, (e' - 1)/t, B(t + 1)).
HaMAMTe oGnacTb r.
(b)
HaiiAMTe lim,r(1).
12. HaMAMTe KpMBM3Hy an/iMnca x = 3 cos t, y = 4 sin t B
TOHKax (3, 0) n (0. 4).
13. HaMAMTe KpMBM3Hy KPMBOM y = X4 B TOHKO (1. 1).
(c) HaMflMTe r'(t).
3.
Haiw>Te BeKTopHyto cfcyHKUMK). npeACTaenfliomyK) KpusyK)
nepeceneHMfl
FR 4. HaMAMTe napaMeipMnecKMe ypaBHeHMB AJifl KacaTexibHOM K
KPMBOM x = 2 sin t, y = 2 sin 2t, z = 2 sin 3t B TOHKC (1. V3. 2).
M3o6pa3MTe KpMByK) M KacaTejibHyio JIMHMK) Ha o6ujeM 3KpaHe.
Ecnki r(t) = t2l +1 cos t + sin mt k, BbNMcnme r(t) dt.
A
6.
nycTb C - KpMBan c ypaBHeHMHMM x = 2 - t, y = 2t -1, z =
In t. HaMAMTe (a) TonKy, rAe C nepeceKaeT nnocKOCTb xz, (5)
15. HaMAMTe ypaBHeHMe conpMKacaKDLqeMC^ TVIOCKOCTM KPMBOM
^
m
x = sin 21, y = 1, z = cos 21 B Tonxe (0. TT, 1).
16.
Ha pMcyHKe nona3aHa KPMB3B C, npocne>KeHHaB nacTMueM c
BeicropoM
nonoxeHM^ r(z) e VOVEHT epeMenM
t. (a) HapMcyMTe Berrop. npeACTaBnnioiAMM cpeAHK>o CKopocTb uacTML^BJ nacTnua 3a
MH Top Ban BpcMonn 3 t = 3.2.
napaweTpHHecKMe ypaBHeHMH KacaTenbHOM JIMHMM B Tonne ( 1 , 1 , 0 )
(б) HanwajMTe BbipawenMO AJIH cKopocrw v(3). (c) HanuLuicre Bbipa>KGMM©
M
GAMHMHHoro KacaTonbHoro BeKTopa T(3) M
( B ) ypaBHeHMe HopManbHOM ruiocnocTM K C B Tonne (1,1, 0).
7.
14. HaMAMTe ypaBHeHMe Kone5nK)iAeMCB OKPYJKHOCTM KPMBOM y
= x4 - x2 B Hanane KoopAMHar M3o6pa3MTe KaK KpMByio, TaK M ee
ocuMnn»«py»oiuMM Kpyr.
L|nriMHApa x2 + y2 = 16 M n/iocKOCTM x + z = 5.
5.
An« KpiiBOPI. 3aAaHH0« r(1) = (3,32, 1), HafiTM
(а) eAMHMMHbiM KacaTenbMWM aerrop
r(t) = ti + cos npi/i j + sin npn k O
(a)
11.
/• • • •S
Hcnonb3yMTe npaBM/io CMMncoHa c n = 6, MTo6bi OLjeHMTb
AJiMHy AyrM KPMBOM C noMombio ypaBHeHMM x = t2, y = t3, z = t,
d> t3 _
8. HaMAMTe A^MHy KPMBOM r(t) = (2t3/2, cos 2t, sin 2t),
0< f < I.
9. Cnwpanb r (t) = cos ti + sin tj + tk nepeceKaeT KpuByjo r2(t) = (1
+ t)i + t2j + tk B TOMKe (1, 0, 0). HaMAMTe yron nepeceneHMH STMX
KpMBbix.
10. rioBTopHO M3MepbTe KpuByio r(t) = e’i + e'sin tj + e' cos t
•
k OTHOCMTenSHO ATIMHbl AyrM. M3M0P6HHOM OT TOHKM (1 ,0 . 1 ) B
HanpasneHMM yseriMueHM^ t.
H3p**cy»r© ero
V

15.

17.
18.
mABA 13 0B30P I «5
Hacruua ABMweTca c (ftymauieii nonoweHMH
(c) OnpeaenuTe ycxopeHne Kopnonuca yacTmjbi. KOTopaa ABM*cercm no BpaLqaKmjeMycH
r(t) = t B ti ♦ 4 e'k. MaMAnre cKopocrb, CKoponoAT>eMHOCTb n
awcxy E cooTBeTcTBMH C ypaBHemiet*
ycKopoHMe nacTMUbi.
r(1) = e ’cos wti 4- e ’sin wtj
Hacinua CTapTyeT B Hanane KOOPAHH«IT c HasajibHOH CKOpocTbK) i - j ♦
3 k. Ero ycKopeMMe paano a(t) = 6t i ♦ 12t2 j - 6t k. HaHAHTe oro cbynKumoi
>—Mowpon »«.
21.
22 npn npoeirrvipoBaHMM r»epea<rwb« KPMBWX AIW coefl^eHMfi ysacTKoe
npavwx xene3Hoaopo*nbix
nyTefi B3>KHO noHMMaTb, HTO ycKopeHMe
19. CnopTCMeH ,qenaeT OpocoK noA yrnoM 45° K ropn30HTaxin
20.
I
c H3ua.nbH0M CKopocTb*o 43 (fcyTa'C. 3TO ocTaanaeT ero pyxy Ha
BwcoTe 7 cfcyToe Has
A
3eMn«.
noeaaa AOJWCHO 6t>ub HenpepwBHbiM. HTo6bi peaKTviBHaa cmia.
0Ka3WBaewafl
noe3aoM Ha nyTH. Taioxe 6bina HenpepbiBHOM. H3-3a cpopMbi arm
KOMnoHSHTOB ycKopeHM* e paaaene 13.4 STO 6yaeT MMeTb MecTo. ecnn
*
KpHBM3H3 M3MeHBeTCn HenpepbIBHO. (a) JlorMMHbIM
(a) Hae MaxoAMTCfl CHMMOK 2 ceKynobJ cnycTH?
K an AH A3 T Ha KpHBytO nepOMOca AJW npncoeAHH«HHR K cy mecTayvoinei* |
(6) KaK BWCOKO JieTWT BbiCTpen?
AOpo>KKH. 3aAaHHbie y = 1 an» x = 0ny = V2-x Ana x >
(c) KyAa nonaAaoT BbiCTpen?
1//2, MoryT ObiTb (pyrnoinefl f(x) = V1 = x2,
HaiiAWTe TaHreMUManbuyio M MOpManbHyio cociannflK)u4MG BGKTOpa
0 < x < 1//2. rpa<t>HK KOToporo npeflCTao/iflei co6oii Ayry oRpyxcHocmi, i
ycKopeHna uacTm^bi c 4>/HKnneM nono>KeHM3
noK33aHHyK) Ha pvicyHKe. Ha nepuwii »3inHA0T0 Bbirn^iT pasyMno nonaxme. MTO
•
CpyHKMMH
r(t) = ti + 2tj + t2k
1 ecnM x O
v
I - x2, ecnn 0 < x < 1//2
/2 - x, ecnn x > 1//2
flucK pafluycoM 1 BpaujaeTCfl B HanpaB/ieHnn npoTMB nacoBovi
crpe/iKM c nocTOflHHOI4 ymoBOM CKopocTbK) w. HacTMua
HaMMHaeTca B qeHTpe AHCKa M ABM>KeTC« K Kpa»o no
HuriRercH HenpepwBHWM H MMCCT HenpepwBHWH HaicnoH, HO He HMeeT
cpMKCMpoBaHHOMy paAwycy, TaK MTO ee nono>KeHMe B MOMEHT
BpeMeHM t, t 0, 3aAaeTcn r(t) = tR(t), rAe
nenpepbiBHOH KPHBH3HW. CneflOBaTenbHO. STO He noAxoAfliAafl KpMBaa
nepeHoca.
(6) HawAKTe MHoroMPen fWTO*i CTencwM. noTopbiii 6yaoT cnyMHTb nepeaaTOHHoii
icpMBoii
R(1) = cos ot i sin wtj
Me)KAy cxieAytoiUMMM 0Tpe3KaMM npnMOM AMHHH: y = 0 A^I«
(a) rioKa>KMTe, HTO CKOpocTb v nacTHL^bi paBHa
x = O M y = x ann x > 1. MO>KHO an axo cae/iaTb c
v = cos wt i sin wt j 4- tva
noMOiubio nonnHOMa neTBepTOM crenenM? Hcnonb3yMTe
rpa(t>MHecxMM Ka^bKy^3T0p HJ1M KOMnbWTep. 4T06bJ Ha^pOCaTb rpa<t>MK
rae v4 = R'(t) - CKOpocTb TOOKM Ha Kpaio
▼
^
A
flMCKS.
"nOflWllOMeHHOH" 4)yHKi4MH n y6eQHTt>cfl. WTO OH Bbim^AHT Tax. KaK noKa3aHO na
piicyHKe.
(b) rioK0>KiiTe; HTO ycKopeHne a nacTMi^bi paBHO
a = 2B. 4- T B J
rae a = R"(t) - ycKopeHne TOHKH HS o6oae
1
flucica. flononHMtenbHbiM nneH 2 B Ha3biBaercfl
ycKopeHneM Kopno/iMca
; 3TO pe3ynbT3T B3anMOAe«4CTBn« BpameHMfl
AnCKa n ABn>KeHMfl nacTMLlbi. MO>KHO nonyHMTb
CpM3MMeCKyK) AeMOHCTpaUHIO 3TOrO yCKOpGHMfl,
noaoMAH K Kpaio ABH>KymeMCH Kapycenn.

16.

nPEHMYLHECTBAlUlKDC
I. HacTwua P ABHxera c nocroaMMOM yrnoeoa CKOpocrbK) @ no OKP/XCHOCTM. UGMTP KOTOPOM Haxoflmca e naMane KoopowHaT M
paanyc KOTOPOM paBeH R roBopax mo M3crnua Maxoxuuca o paoMOMepnoM KpyroaoM oBM*eHMM. npeanonoacHM. mo
flBMxeMwe npoMexoAWT npomB MacoBon CTpenKvi H MTO Macruqa HaxoAmca B TOM KG (R, 0), Kor^a t = 0. Be*rrop nonoxeHun B
MOMeHT BpeMeHM t 0 paBeH r (t) = R cos ot i + R sin ot j.
(a) Ha^AHTe Betcrop aopocrn v M noKaxme. mo v • r = 0. CAename BWBOA. mo v aBnaeica KacaTe/ibHoa K oxpyKHOCTM
M yKa3biBaeT B HanpaBneHwi ABMweHMn.
(b)
noKa^me, MTO CKopocTb | v | nacTMAbi RBJiaeTcn nocTORHHOfl wR. nepnoA T nacTMAbi -3T0
BpeMa. HeoGxoAMMoe Ana OAHOTO no/iHoro obopoTa. 3aicnK)HMTb. MTO
2 IT
CO
PMCYHOK 3AAAHM I
(c) HaMAHTe Bemop ycKopeHMR a. noKajwme, mo OHa nponopL^OHanbHa r n mo OHa yKa3biBaeT Ha
Hanano
KoopAHnaT. ycxopenne. oOnaaawmee CTMM CSOMCTBOM. Ha3WBaeica ueHTpocTpeMMTenbHWM
ycxopeHHeM. noKa)KHTe, MTO BenHMMHa Bemopa ycxopeHM* paBHa | a| = Rw2.
(d) ripeAnonoaiM. MTO MacTKua MMeeT Maccy m no*a»cMTe. mo BenMMMHa cunw F, KOTOpan Tpe6yeTcw AJIR
C03Q3HHB 3TOTO
H33b<B3eMOrO 4eHTpOCTpeMWTe/lbHCW CMJIOM. P3BH3
v/^2
f=\ rTn
2. KpyroBafl xpHBan paAnyca R Ha uiocce HaicnoHeHa noA yrnoM 0. MTObbi aBTOMo6nnb Mor 6e3onacHO
nepocoKaTb KpnBy»o 6G3 33Hoca, Kor^a HOT TpeHMn MeacAy Aoporoii M LUMHaMH. RoTepn TpeHMR Mower
npon30MTn. HanpHMep. econ Aopora noxpwTa nneHKOii BOAW HJIM nbAa. HoMMHanbHan CKopocTb vn KPHBOM 3TO MaKCMMaxibHaR CKopocTb, KOTOpyK) aBTOMo6nnb Mower pa3BMTb 6e3 3aHOca. npeAnono>KMM. MTO
aBTOMoGM/ib Maccoiii m nepeceKaeT KpwByio c HOMHHa/ibHOM cxopocTbio vg Ha aBT0M06niib AeMcreyiOT
ABe cunu : BepTHKanbHaR cmia mgT o6ycnoBneHHan BecoM aBTOMo5viriR, M cuna F, AewcTByiOLnan Ha Aopory H
nepneHAMKyrwpHaa ew (CMOTpme pncyHO»c)
BepTHKa/ibHaR cocTaBnn Ionian F ypaBHOBeuunBaeT Bee aBTOMo6wifl, TaK MTO | F | cos 0 = mg.
ropM30HT3JlbH3a COCT3BJl«K)U43a F COiASeT LjeHTpOCTpeMMTe/lbHyK) CMny H3 SBTOMO^MJie. T3K MTO.
COrnaCHO BTopoMy
• # * • 9
aaKOHy HbioTona M Mac™ (d) 3aoann 1,
3HCYH0K ,0/15=1 3AQAHM 2
..
mvi
I F I sin 0 = *-
R'
(a) noKa>KMTer MTO vf = Rg = 0.
(b) HaiiAMTe HOMMHanbHyio CKopocTb KpyroBOM KPHBOW paAnycoM 400 cfcyTOB, KOTopan HaicnoHeHa noA
yrnoM 12". (c) f1peAnono>KMM. MHweHepbi-KOHCTpyKTopbi XOTHT coxpaHHTb xpeH Ha 12®, HO xenaioT
yBenHMHTb
HOMHHaJlbHyK) CKOpOCTb H3 50%. KaKMM AOTDKeH 6blTb paAMyC KpMBOM?
3.
CHap^A BwcTpexiMBaeTCfl M3 MCXOAHOM TOMKM C ymoM B03BbJLueHMfl a M HaManbHOM cxopocTwo vo.
npeAnonarafl , MTO conpoTWBneHMe B03Ayxa npeHe6pe>KMM0 Mano M mo eAMHCTBeHHOM CWIOM.
A^MCTBywiAeM Ha cuap^A. BBnnera c«na TaxecTM, g. MU no*a3ann B npuMepe 5 B pa3Aene 13.4. MTO aenop
nono^eHun cnap^a pa3eH
r(t) = (vOcos a)ti + [(vsin a)t - 2gt2] j
Mbi Taioce noKa3ann, MTO MaKCMMa/ibHan ropM30HTanbHaa AanbHOCTb noneTa ewap^a AOCTwraera npH
*
a - 45 M B 3TOM Ciiynae oanbHocTb cocTaenaeT R - v8/g.
m
(а) noA K3KMM yrnoM cneAyeT 3anycxaTb CHap«A Ann AocTM)KeHMfl MaKCMManbHOM BbicoTbi M xaxoBa
MaKCMMaxibHan uucoTa?
(б) 3acpMKcnpyMTe HananbHyK) cxopocib v M paccMOipme napa6ony x2 ♦ 2Ry - R2 = 0, rpac})MK KOTOPOM
noKa3aH Ha pMcyHKe noKa>KMTe, MTO CHapnA Mower nopa3MTb xiK)6yK) uenb BHyTpH min Ha rpaHnije
o5^acrn. orpanuMenHOvi napa5onoM M ocb»o x. M MTO OH He woxeT nopa3MTb HM OAHY uerib 3a
3MCYHOK 0/15=1 3AflAMH 3
8S2
npeAenaMM OTOM o6nacm-
(c) ripeAnono*MM. MTO py)Kbe noAHmo Ha yron HaK/iOHa. MTOOU npm^enMTbca 8 i^ent. KOTopan noABemeHa
Ha
BbicoTe h HenocpeACTBeHHO HaA TOMKOM D Ha paccTOHHMM eAMHMu HM>Ke. Uenb ocBoGo)KAaeTcn B
TOT MOMeHT, Korqa nwcToneT BbicTpenMBaeT. noKawure. mo cnapnA Bceraa nonaAeer B uenb. He33BMCMMo
OT 3HaMeHHH vo. npH ycnoBHM, MTO CHapnA ne nona^aeT B 3eMn»o "no" D.

17.

PRO BLEMS PLUS
»
(a) CHapflA BucTpenHBaeicn M3 Havana KoopAMHaT BHM3 no HaKfiOHHOM nnocKocrvi, KOTopan
4.
cocraBnaeT
yron 0 c ropn30HTa/ibK) Yron B03BbiLueHnn nyujKM M HaManbHan acopocTb CHapnAa paBHbi a • M vo
cooTBeTCTBeHHO. HaMAMTe BeKTop nonoweHMn CHapnAa M napaMeTpuMecKMe ypaBHeHMn
TpaeKTopuM CHap^Aa B 33BMCMMOCTM OT upeweHn t. (MmopupyMTe conpoTusneHne B03Ayxa.)
(b) rioKajtcme. MTO yron nQA'beMa a, KOTopbiii MaKCMMM3MpyeT Anana30H cnycxa, nanneicn yrnoM
Ha nonnyTM Me>KAy nnocKOCTbto M BepTMKanbK).
(c)
np€Anono)KHM. mo CHapnA BwcrpenusaeTCfl Bsepx no HaK/iOHHOM nnocKOCTM. yron Hawiona KOTOPOM
paeen 0. no*a#nTe
PHCYHOK flJlfl 3AMHM 4
PMCYHOK flJlP 3A0AMH 5
, MTO Ann Toro, mo6u MaKCHwa/ibHO ysenumcrb Aanbuocrb noneia (B ropy), CHapaA cneAyeT sanycxaTb e
HanpaaneHnw ,i HaxoAnmeMca ha nonnyTH nie^AY nnoaocrbK) * aepTMtca/ibio.
(d)
B craTbe. npeAaasneHHOM 81686 roAy. SRMOHA Tanneii o6obmnn 3aKOHw
rpaBmauHn M ABHxenHa CHapaAa u npwMeHun KX
K apTHJinepmicKOMy Aeny. OAH3 143 nocrasjieHHwx AU aaflan 3aKJiKMa/iacb B TOM. mo6w awcrpenuTb oiapaAOM Tax. • UTO6W
nop33MTt» qanb wa paccToaHwn R BBepx no HBIC/IOHHCM nnocKOCTM. rio»caxnTe. UTO yron. non KOTOPWM
CHapnA flonxew 6biTb BunyujeH. uTo6b< nopa3*Tb 4erib. HO ncnonb30B3Tb HawMeHbUje« KorwuecTBO
OHeprwM. cosnapaeT c ymoM B U3CT14 (c). (Mcno/ib3yHTe TOT 03KT. UTO SHeprvw. Heo6xoAUM3a Arm
3anycxa cwapaAa. nponopquoHanbHa
A
S
r
\
V
V
A
»:E3ApaTy HauanbHOti CKOPOCTM. noaTOMy OHeprvm SKBHBaneHTHa MMHMMM3314MM HauanbHOii
CKOpOCTM.)
5. Man CKaTbiBaeTcn co crona co CKopocTbK) 2 cpyia/c. Bbicoia CTona cocTaanneT
3,5 4>yTa. (a) OnpcAenme TOHKy. B KOTOPOM MAM yAapneicn 06 non, M HawAme ero CKopocTb B MOMCHT
BnuflHne.
6. HaMAMTe
KpMBM3Hy
KPMBOM
c noMoiAbiononeia
napaMeTpMMecKidx
ypaBHeHMM
(6) HaiJiAHTe
yron 0 Mexgjy
TpaeKTopnei4
MAMS M BepTMKanbHOM
nnHnei4. npoBeAeHHOM Mepe3
TOMKy yAapa. (CMOTpme pucyHOK.)
x= |sin(mOa)dO y= cos(0202)d0
(c) npeAnono^oiM, mo MAM oiaoKHBaeT
OT nona
noA ieM we yrnoM, noA KOTOPUM OH yAapaeicn 06 non, HO
f
r
V
¥
\
f
TepneT 20% CBoeM CKOPOCTM M3-3a 3HeprMM. nornomaeMOM MHHOM npn y/jape. Hqe MHH yAapneTcn 06
non npM BTOPOM OTCKOKe?
ypaBHeHW?
7. ECHH cwapHQ BbiCTpenuBaeTcn c yrnoM B03BbiueHM3 a H HaManbHOM CKopocTbio v, TO napaMeipuMecxne
AT* era rpaeKTopwM nB.nRWTCB
x = (v cos a)t y = (vsin a)t - gt2
(CM. (IpMMep 5 B pa3Aene 13.4.) MbJ 3HaeM. MTO AanhwocTb (npOMAGHHOe paccTonHMe no
ropn30Hrann) MaKCMManbHa, KorAa a = 45s. Kaxoe 3HaMeHne a MaKcnMH3npyeT o6mee paccTonHMe,
npoMAeHHoe CHapflAOM? (yuarcHTe CBOH OTBeT npaBH/ibwwM c TOMHOCTbK) AO rpaAyca.)
8. KaOe/ib MMeeT paAMyc r M AiiMHy L M HaMaTbiBaeTcn Ha KaTymKy c paAMycoM R 6e3 Mpe3MepHOM
npMTvipKM. KaKOBa cawan KopoTxan A^MH3 KaiyuiKM, noKpbrroM Ka6eneM?

18.

HACTHblE
nP0H3B0flHblE
OyHKMMM AByx nepeMeHHux MO>KHO BH3yanw3npoBaTb cnoMombio KPMBWX ypoBneii. KOTopwe
coeAMHflioT
TOUKM. rae cpyHKijMfl n p M HM M3 ST 3aA3HHoe 3HaueHMe. ATMOcepepHoe ^aaneHne e ASHHUM MOM6HT
DpCMCMM 33DHCMT OT flOJirOTbJ H LLJMpOTbl M M3Mep«eTCfl D MMAflMOapaX. 3fleCb
KpMBbie ypOBH^ Hd3blB3IOTCfl H300apaMM H Te. MTO M300pa)K.eHbl Ha pHCyHKe. COeAHHHIOT
MeCTa , qae 7 Mapia 2007 roAa owno TaKoe xe AaBJieHwe (KpM3we. ooo3HaMeHHwe 1028,
Hanpw/ep. coeAUHR^T TOMKH CAasneHneM 1028 M6.) nosepxHOCTHwe Beip&i MiewT TeHAeHLiHto
Te^a »i3 o6nacieM 5b<coKoro flaB/ieHMfl nepe3 M30oapbi B oonacTH HM3Koro AasneHnn n flanaKncfl
cawwMM curibHWMM T9M, rfle nnoSapbi rmorHO ynaKOBaHbi.
flo CMX nop Mbi MMeriM Aeno c MCHwcneHMeM (pymcAMfl OAHOH nepeMeHHOfl.
Ho B peaxibHOM
m
*
m
MMpe 4>n3M^ecKne BS/IMMHHW uacTO aaBuc^T OT AByx MAM 6onee nepeMeHHbix. noaTOMy e STOM rnaBe
Mbi oopaTMM Hame BHMMaHne Ha (PYHKMHM HecKonbKHx nepeMeHHbix H pacnpocrpaHMM ocHOBHbie
MAen
AM(p(t)epeHUMaAbHoro MCMHCHCHHH Ha iaKne <pyH*i4HM.

19.

■I I
14.1 OYHKL4HM HECKOJlbKHX nEPEMEHHbIX
i_i
B 3TOM pa3Aene Mbi M3yMaeM cpyHKUMM AByx MJIM 6onee nepeMeHHbix c MeTbipex TOMeK 3peHna:
■ ycTHo j (no onncaHMK) B c/iOBax)
■ MMcneHHo (no Ta6nnue 3HaneHHii)
■ anre6panMecKM (no ABHOM cfcopMyne)
■ BM3yanbH0 (c noMOiubto fpa<fcm<a HHH KP*BWX ypOBH*)
OYHKUHH flBYX flEPEMEHHblX_
TeMnepaiypa T B TOHKe Ha noBepxHOCTM 3eM/in B xitoGofl MOMCHT BpeMeHM 3aBncnT
OT
AOJirOTbl X M LLIHpOTbl y TOM KM. Mbl MO)KeM AyMaTb O T KaK O Cpy HKI4MM AByX
nepeMeHHbix x M y min KBK 0 c^yHKLjMM napbi (x, y). Mbi o6o3HanaeM 3Ty
cfyHKLiMOHajibHyK) 3aBHCMM0CTb
, 3anMCbiBa« T = f(x, y).
06beM V Kpymoro L^nruiHApa 3aBncm OT ero paAnyca r M BbicoTbi h. Ha caMOM Aene, Mbi 3HaeM, MTO V
= mr2h. Mu roBopMM, MTO V ABnaeTCfl cpyHKAMefl r M h, M 3anMCbiBaeM V(r, h) = mr2h
OnPEflE/lEHME OyHKunfl AByx nepeMeHHbix - STO npaBwno. KOTopoe npwcBawBaeT Ka#AOM
ynopHAOHeHHOM nape BeniecTBeHHbix MHcen (x, y) B HaSope D yHMKanbHoe BeiAecTBeHHoe MMCJIO,
o6o3H3MaeMoe f(x, y). MHO^ecTBO D HBnaeTCB oPnacTbio f. a ero Anana30H - STO HaSop 3HaMeHMH.
xoTopwe npuHHMaeT ft. TO ecTb {f(x, y) I (x. y) € D}.
Mbi naCTO nmueM z = f(x, y), HTOGM HCTKO yKa3aTb 3HaHeHMe, npMHHMaeMoe fat B
oGu^ePi TOHKC (x, y). nepeMeHHbie x M y HBrwioTCfl He3aBMCnMbiMM nepeMeHHbiMH. a z 3aBMCMMOfl nepeMeHHOfl
[CpaBHMTe 3TO c oOo3HaHeHneM y = f(x) ana cpyrnojHfl OAHOH nepeMeHHofi]
_
<
%
OyHKuvin AByx nepeMeHHbix - STO npocTO cpyHKUHn. oonacTb ABMCTBMA KOTOPOH ABJineTcn noAMHOwecreoM R2,
a Anana30H
- noAMHO>KecTBOM
R. OAHMM
M3 OIOCOAGMCTBUB
SOB BM3yann3aunM
3KOM
nBJineTcn
AMarpaMMa
co
ECTIH 0yHKuna
f 3aAaeTcn cpopMy/iOM
M oGnacTb
He yKa3aHa,TTO
noA0yHKunn
oGnacTbio
ABHCTBMH
f
• • • • • •
/ % • • •
• mm
CTpenKOM (CM. puc. 1), rAe oGnacrb D npeACTaBneHa KBK noAMHorcecTBO nnocxocTM xy..
pewenwe
(a) J(3. 2 ) V 3 *
1 2
8SS
noHMMaercfl MMOWCCTBO BCex nap (x. y). Ana KOTOPUX AOHHO« Bbipa>K«Hne HBJIHOTCH H0TKO onpCAcnenHbiM
eeiuec 1 BeHHbJM HucnoM.
nPMMEP I fljin Ka>KqoM M3 cneAyioiUMx cfcyHKLjMfl BbiHMcnMTe f(3, 2) M HaiiAMTe

20.

856 llll r/lABA 14 MACTHblE nPOH3BOflHblE
x+y + 1 = 0
Bbipa;KeHne /yia f nMeeT cMbicn, eoin 3HaMeHaTenb He paBeH 0, a BenHHHHa nqq 3H3KOM KBaflpaTHoro
KopHfl HeoTpmiaTenbHa. TSKMM o6pa30M, 06/iacTb fls
D = {(x, y)| x + y+ 1 >0, x 1}
HepaseHCTBo x + y + 1 > 0 unii y > -x -1 onwcbiBaeT TOMKH. KOTopwe nexraT Ha npHMOM y =
-x -1 HJIH Bbiiue Hee, B TO BpeMa xax x # 1 03HaMaeT, MTO TOMKM Ha npaMOH x = 1
flo/i>KHbi 6biTb MCKJItOHeHbl M3 06/iaCTM. (CM. PMCyHOK 2.)
(b) f(3, 2) = 3 B(22 - 3) = 3 B 1 = 0»
PMCyHOK 2
06nacTb f(x, y) =
—
nocKonbKy In(y2 - x) onpeAenneTcn To/ibKO TorAa, KorAa y2 - x > 0, TO ecTb x < y2,
oOnacrb fis
«. i m
D = {(x, y) I x < y2}. 3TO Ha6op TOMex cneBa OT napa6onbi x = y2. (CM.
PMCyHOK 3.)
■
He Bee cpyHKLiMM 3a/jaK)TCfl BBHbiMM 0opMynaMM. (PyHKUMB B cneAYK)meM npuMepe onMcaHa
ycTHO M C noMOLLjbK) HMcnoBbix OLjeHOK ee 3HaneHMM.
nPMMEP 2 B pernoHax c cypoBOM 3MMHeM noroAOM MHAexc XOJIOAHOCTVI Beipa Macro ncno/ib3yeTCfl A^H
onucaHna OMeBMAHOM TH>xecTM xonoAa. 3TOT noKa3aienb npeAcraanneT COGOM cy 6i>eKTMBHyK) TeMnepaTypy.
PMCYHOK 3
KOTopaa 33BMCHT OT cfcaicTMHecKOM TeMnepaTypbi T M CKOPOCTM BeTpa v. TaKMM o6pa30M. 3TO c^ymcuM* T M v
OGnacTb f(x, y) = xln(y2 - x)
. M Mbi MOJKeM 3ariucaTb W = f (T. v). B Ta6nnue 1 npHBenenbi 3HaHCHMfl W. cocTaBncHHbie^ HaquoHanbHOM
MeTeopojiorwMecKoii cnywboM CUJA NOAAM MeTeoponorn^ecKOM cny*<6oM KaHaAbi
CKopocTb BeTpa (KM/M)
TAbHUMA B
MHA6KC XOflOAHOCTM BeTpa
B 33BMCMMOCTM oi
/X
leMriepaiypbi eoawyxa M CKOPOCTM
BeTpa
■ HOB* *fBCXOnQQHOCTH
BETPA Ha*
(•macc xonooHOCTH BeTpa 6un Beeper a
nofl6pe 2001 roaa w aenaeTCfl 5onee TOUHWM, ueu
CTapui
MKaecc, Toro Haaonwo xonoAHO. sorfla
Ann
K3yepeHMn
Ayer Beiep HO&WM KHA^KC OCHO6<IH Ha uoAe/v TOTO.
uenoBeoeacoe
cat ObCTpo rwqo Tep^er Tenno. OH 6UJI p 33 pa Doran B
xoqe WlHHSWea'MX HOlbTaHMM. B XOAe
KOTOpblX AOOpOBOnbUb nooEepramcb aoaeiiaaiiG
paim+tx TeMnepaw M aopocrew
aeTpa B oxna*jaeMOM aapcuHHaMwuecKOM Tpy6e.
u
O
2
&
£
"75
3
O
<
V
5
10
15
20
25
30
40
50
60
70
80
1
-5
1
-6
0
-6
-i
-1
-2
-2
-3
-7
-8
-9
-9
-10
5
4
3
0
-5
-2
-3
2
-4
-7
-9
-11
-12
-12
-13
-14
-15
-16
-16
-17
-10
-13
-15
-17
-18
-19
-20
-21
-22
-23
-23
-24
-15
-19
-21
-23
-24
-25
-26
-27
-29
-30
-30
-31
-20
-24
-27
-29
-30
-32
-33
-34
-35
-36
-37
-38
-25
-30
-33
-35
-37
-38
-39
-41
-42
-43
-44
-45
-30
-36
-39
-41
-43
-44
-46
-48
-49
-50
-51
-52
-35
-41
-45
-48
-49
-51
-52
-54
-56
-58
-60
-40
-47
-51
-54
-56
-57
-59
-61
-63
-57
—
64
-65
-67
HanpuMep, TaGnuija noKa3biBaeT, MTO ec/in TeMnepaTypa cocraBnneT -5°C, a CKopocTb BeTpa
cocTaBrmeT 50 KM/M, TO cy6teKTMBH0 6yAeT Taxwe xonoAHO, Kax npw TeMnepaType OKono - 15°C
6e3 BeTpa. Tan
f(-5, 50) = -15
nPMMEP 3 B 1928 roAy Hapnb3 KoOO M (Ion flyrnac ony6nnKOBann MccneAOBaHMe, B KOT
nepwoA B nepwoA 1899-1922 roflOB C m ocManb
CMOAennpoBann pocT aMepuKaHCKOM SKOHOMMKM

21.

PA3AEJ1 14.1 OyHKUMM HECKOJlbKHX flEPEMEHHblX lill 857
TABJ1MUA2
Toa
P
CTOPOHHMK ynpou4eHHoro npeACTaBJieHMfl 06 BKOHOMHKC, B KOTOPOM o6beM npomBOACTBa onpeAermeTca
■
K
1899
1900
1901
1902
1903
1904
1905
1906
100
101
112
122
124
122
143
152
100
105
110
117
122
121
125
134
100
107
114
122
131
138
149
163
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
151
126
155
159
153
177
184
169
189
225
140
123
143
147
148
155
156
152
156
183
176
185
198
1917
1918
1919
1920
1921
1922
227
223
218
231
179
240
198
201
196
194
146
335
366
387
407
417
431
208
216
226
236
244
266
298
161
KOJiMMecTBOM 3aAeMCTBOBaHHoro TpyAa M cyMMoft BJio)KeHHoro Kanmana. XOTB cymecTByeT MHOJKGCTBO
Apyrvix 4>aKTopoB. BTIMSWUHMX Ha SKOHOMnuecKne noK33aTenn. MX Mcwe/ib o*ca3ajiacB yAMBure/ibHO
W ft 4 I ft
TOMHOM. OyHKLina. KOTOpyio OHM Mcnonb30&ajiM Ana MOAenupoeaHUfl npoM3BOACTBa. Mwena BMAI
HI
P(L, K) = bL"K -
rAe P - OGIAUM o&beM npon3BOACTBa (AeHexHaa cTOMMOCTb Bcex TOBapoB, npon3BeAeHHbix 3a rOA), L - KOfiMMecTBO paOoHeft cunbi
(oOmee KOJiMMecTBO MenoBeKO-MacoB, oTpa6oTanHbix 3a TOA), a K - cyMMa BnoweHHoro Kanmana (AeHe>KHaa CTOMMOCTb Bcex
MamuH. o6opyAOBaHMfl M
B pa3Aene 14.3 MW HOKHMSM, K^K cJ>opMa ypaBHeHM* 1 Bt>ueKaeT M3 onpeAeneHHbix
3KOHOMMU6CKMX AOnyLHSHUM .
ft
Ko66 n flyrnac ncnonb30Bann SKOHOMn^ecKne AaHHbie. ony6nMKOBaHHbie npaBmenbCTBOM, A^I« nonyueHMfl
TaGnuubi 2. OHM B3flnM 1899 TOA B KaMecTBe Sa30Boro. M «a)KAOMy M3 P. L M K 3a 1899 TOA 6uno iipucuoGHO
3HaHGHne 100. 3HaMeH^m 3a /ipyrne rojat>i 6binn Bk>ipa>KeHbi B npoueHTax OT
nwaaaTenew 1899 roaa.
Ko56 M Ayrnac ncnonb30Bann MCTOA HanMeHbiiiMX KBaApaTOB. conocTaBmb AaHHwe TaSnuuw 2 c
c})yHKi4nn
|T]
P(L, K) = 1 01L0.75K0.25
(noApo6HOCTM CM. B ynpa>KHeHMM 75.)
Ecnn Mbi ncnonb3yeM MOAenb, 3aAam-ryK) (fcyHKLjMeft B ypaBHeHMM 2, Anfl BbinucneHUfl npomeoACTBa B 1910 M 1920
roAax, Mbi nonyHMM 3HaneHnn
n(147, 208)= 1.01(147)075(208)025 161.9
n(194, 407) = 1.01(194)075(407)025 = 235. 8
KOTOpbie AOBOJlbHO 6nM3KM K 4>aKTMMeCKMM 3HaMeHMflM, 159 M 231.
nPOM3B0ACTBeHH39 (pyHKLIMfl (1) BnOCneACTBMM MCn0Jlb30BanaCb B0
nPMMEP 4 CmyaLIMflX,
HaMAHTe AOMeH
M Anana30H g(x, y) = /9 - x2 - y2.
MH0FMX
HaHMHaa
OT OTAeribHbfX CpMpM M 23K3HUHE-3^ mo6anbHfalMM 3KOHOMMU6C KMUH BOnpOCaMM. 3TO CTa/IO M3BBCTHO K3K
PELUEHME OGnacTb g paBHa
W
f
t
npon3BOACTBeHHan cpyHKLinn KoOGa-flymaca. Ero oOnacTb paBHa {(L. K) | L = 0; K = 0},
noTOMy MTO LDM K npeACTaBnawT TpyA M Kanman M no3TOMy HMKorAa He GbrnaiOT oipMuaie/ibHUMM ■
= {(x, y) | 9 - x2 - y2 > 0} = {(x,y) | x2 + y2 < 9}
KOTopbm npeACTaBJiaeT co6oft AMCK C ueHTpOM ( 0 , 0 ) M paAnycoM 3 . (CM. pucyHOK 4 . ) flManaaoH g
paBeH
(z|z= /9 - x2 - y2, (x, y) E D}
.t
nocKonbKy z - nojioxMTe/ibHuft KBaApaTHwft Kopenb. z = 0. Tawe
Tso« cfcam >rax* xcisur
PMCYHOK 4
06nacTb ofg(x, y) =9 - x2 - * V ‘
9 - x2 - y2 9 - x2 - y2 *£ 3
{=|0 <3} =
[O, 3]

22.

858 III rilABA 14 MACTHblE nPOH3BOflHblE
TPAOMKH_
flpyrofl cnocoS
paccMOTpeTb ee
BM3yajiM3aLjMM
noBefleHun
cpyHKqmi
flByx
nepeMeHHbix
-
rpacpHK.
OnPEflEJIEHUE ECJIH f ae/meTca cpywajMeM flByx nepeMeHHbix c o6nacibio D, TO rpacpMK f - 3T0
MHO>KeCTBO BCeX TOMeK (X, y, z) B R3, T3KMX MTO Z = f(X, y) M (X, y) HaXOAMTCB B D.
I1QAO6HO TOMy. KaK rpacpMK 0yHKunn c OAHOM nepeMeHHOM npeACTaBnaeT COOOM KpuByio C c ypaBHeHMeM
y = f(x), TaK M rpa$MK 0yHKUMM c AByMfl nepeMeHHbiMn npeACTaBJiaeT COOOM noeepxHocib S c ypaBHemieM z
= f(x. y). Mu MOweM on3yann3npoBaTb rpacfwic S fas. sie>caLL4UM H^nocpcACTBCHMO HOA MAM noA ero o6nacib>o D B nnocKOCTu
xy
v
(C M . PucyHOK 5.)
PUCyHOK 5
nPMMEP 5 HapucyMTe rpacfcwK 4)yHKUnn f(x, y) = 6 - 3x - 2y.
PELUEHUE rpacfniK fh nMeeT ypaBHeHMe z = 6 - 3x - 2y, MJIM 3X + 2y + z = 6, KOTopoe
npeAcrasnaeT ruioaocrb. MTOOW nocrpowTb rpacpMx ruwaocrw, MU awa/ia HaxoflMM TOMKH nepecwenua noAcraanaa
y = z = 0 B ypaBHeHMe, Mbi noAyMaeM x = 2 B KanecTBe x-nepexBaTa AHanorviMHO, y-nepexBaT paBeH 3,
a z-nepexBaT paBeH 6. 3TO noMoraeT HSM Ha6pocaTb Ty nacTb rpacpuxa, KOTopaa neaoiT B nepBOM oKTaHTe. (CM.
PucyHOK 6.)
OyHKL|MP B npiiMepe 5 AB/iaeTCfl HacTHbiM cnyMaeM (pyHKHMH
f(x, y) = ax + by + c
KOTopan HaabiBaeTCfl HHHewHOH 0yHKL(Men. TpatpMK TaKOM 4>yHKunn MMeeT BMA ypaBHeHnn
z = ax -*• fc>y + c MJ~IM ax + t>y - z + c = Oi
HT3K.
3TO caMOneT. Bo MHorOM TaK we, KaK AMHeMHbie (PYHKUMM OAHOM nepeMeHHOM eawHbi B
ucHMcncHnn
A
c OAHOM nepeMeHHOM, Mbi YBMAMM, MTO AMHeMHbie c^yHKLjMM AByx nepeMeHHbix nrpaK)T L^eHTpanbHyK)
porib a MCMMCJ
C heoonwMi^i nepeMenHUMM
B npHMepe 6 HapMCyMTe rpacjDMK g(x, y) = 9 - x2 - y2.
■
PEUJEHUE rpa0HK MMeeT ypaBHeHMe z = V9 - x2 - y2. Mbi BO3BOAMM B KBaApaT o6e CTOpOHbi 3Toro ypaBHeHMH, MTo6bi
nonyMMTb z2 = 9 - x2 - y2, MAM x2 + y2 + z2 = 9. KOTopoe Mbi pacno3HaeM KaK ypaBHeHMe ccfjepbi c ijempoM B
PUCYHOK 7
HanaAe KoopAMHaT M paAMycoM 3. Ho, nocKOAbicy z 0, rpacJwK g - STO npocTO oepxHfla nonoBMHa 3TOM c4>epw (CM.
TpacpHK ofg(x, y) = v9 - x2 y2
puc. 7). H

23.

PA3AEH 14.1 cpyHKUMM HECKOJlbKHX flEPEMEHHblX III! 859
OOpaTMTe BHMMaHMe, mo BCA ccpepa He MOweT 6biTb npeACTaBneHa OAHOA
(pyHKLjnef» x M y. Kax Mbi BMAejiM B npwMepe 6, eepxHaa nonyccpepa ccpepbi x2 + y2 + z2 = 9
npeACTaBneHa
c^yHKqueM g(x, y) = 9 - x2 - y2. HM>KHHH nonyccjaepa npeACTaBneHa 4>yHK4neM h(x, y) =
- 9 - x2 - y2.
nPMMEP 7 Hcnonb3yMTe KOMHbiOTep ana nocTpoenMH rpacpMKa npoM3DOflCTDCHHOM cpyHKUMM Ko66a-flymaca
P(L, K) = 1.01L0.75 0.25
PELUEHME Ha pucyHKe 8 noKa33H rpa(J)wc P 3HaneHMM pa6oneM cnnbi L M Kanmana K, KOTopue JiewaT
Me>Kay 0 n 300. KoMnbtoTep napucoBan noBepxHOCib, haHeca BepTUKanbHbie C/IBA^. M3 3TMX
rpacJ)MKOB Mbi BMAVIM, MTO BexiMMMHa npon3BQflCTBa P yBenMMMBaeTCfl no Mepe yBenMMeHwn nnOo
L,T nn6o K , KaK n cneAoeano ownaaTb.
PUCYHOK 8
V nPMMEP 8 HawAMTe o6nacTb n ,qnana30H M HapucyMTe rpa<J)MK h(x, y) = 4x2 + y2.
PEIUEHME 06paTMTe BHMMaHne. MTO h (x, y) onpeaeneHo AHA BCBX B03M0)KHbix ynopflAoneHHbix
nap B8ii|0CTBeHHux Mwcen
(x. y), noaTOMy o6nacTb paBHa R2. Bce*i nnocKoan xy. flnana3on h - 3TO MHOwecTBO [0, %) BCBX
HeoipMuaienbHbix
AeMCTBUTe/ibHbix Hucen. [06paTme BHMMaHMe, MTO x2 > 0 M y2 > 0, rioaiOMy h(x, y) > 0 ocex x M y.J
^ «%
Tpa(pnK h MMeeT ypaBHeHwe z = 4x2 ♦ y2, KOTopoe npeflCTaBnaeT cohort annMrmmecKMM napaOononA,
KOTopbiM Mbi HapncoBann B npuMepe 4 B pa3flene 12.6. ropusoHTa/ibHbie Tpaccbi - STO 3juinncbi, a BepTMKanbHbie
Tpaccw - napaSonw (CM. pno. 9).
PHCYHOK 9
fpacfinK h(x, y) = 4x2 + y2
KoMnbioTepHwe nporpaMMbi ner«o AOCTynHu fljia nocrpoeHMfl rpatpuKos (pyHKuwrf
AByx nepeMeHHbix. B SonbLUMHCTBe TBKMX nporpaMM Tpaccbi B BepTvtKa/ibHbix nnocKOCTHx x = k M y = k piicyiOTca flna
oflunaKOBO
^
pd'JHBCBHHblO 3HdH6HMH k M M3CT M f pa$>MKa yCT p«3HHKM CH C nOMOLUbtO yfla/ieHMR CKpbITbIX nHHMM.
m

24.

360 III! rnABA 14 MACTHblE nPOM3BOflHblE
Ha pHcyHKe 10 noKa33Hw crewepupoBanMue KOMnworepOM rpa^rnw Mecxonbiwx ({jyHKLiHM. 06paTme BMUMawne. mo
MU nonyMaeu ocooeHHo xopoujee M3o5pa»eHMe (pyHKLjMM, Koma BpameHHe ncnonb3yeTca
ana nonyneHHs BMfla c pa3Ht>ix ToneK o63opa B Hacmx (a) M (b) rpacpMK f oneHb
n/iocKMfl M 6rn30K K nnocKOCTM xy , 3a MCK/iKweHMeM Manana KOopAMHaT; ato noTOMy, “•’’'MTO e'
onenb Ma/i, KOr,qa x nnn y aennKM
(a) f(x. y) = (x2 + 3y2)e
(b) f(x, y) = (x2 ♦ 3y2)e
(c) f(x. y) = sin x + sin y
PUCyHOK10
KPMBblE YPOBIHfl_
noKa y Hac ecTb ABa MeTOAa BM3yann3ai4MM cpyHKUMM: AwarpaMMbi co CTpemoMM M rpacpMKM.
TpeTMM MeTQA,
3anMCTBOBaHHui?i y KapTorpacpoe, npeACTaarmeT cofiov\ KowrypnyK) Kapiy, Ha KOTOPOM TOMKM
nocroHHHoro rnwbeMa
• • • • • « • coeAHHHKn-c*. o6pa3y« KOHTypHwe KpMBbie. win KpMBbic
ypOBHfl.
OnPEflEflEHME KpuBbie ypoBHH cfDyHKUnu n sy x nepeMeHHbix - STO xpuBbie c ypaBHeHM^MM f ( x ,
y ) = k . rae k - KOHCTaHTa (B flnana30He f).
KpMBaa ypoBHfl f(x, y) = k - STO MHOwecTBO Bcex Tonex B oCnacTM fat, KOTopbie
HaxoAflTca Ha 3aAaHHoe 3HaneHne k. flpyimiH cnoBaMM, STO noKa3b4Baei rAe rpacpHK fh MMeeT Bbicoiy k.
Ha pHcyHKe 11 Bbi Mo^eie BMAeTb CBH3b Me>KAy KPHBUMH ypoBHH H
ropn30HTajibHbiMM TpaccaMH. KpuBbie
ypoBHfl f (x, y) = k - 3TO Bcero /lMiub cneAbi rpa<t>nxa fin ropn30HTaribHow nnocKOCTM z = k ,
cnpoeunpoBaHHOM BHM3 Ha nnocxocTb xy. Miax, ecnw Bbi napMcyeie KpwBbie ypoBHfl
(pyHKUMM M
BH3yanM3npyeTe. K B K OHH noAHHMaioTca Ha noBepxHOCTb Ha yK333HHy*o ewcory. TO Bbi f.*oxre MbieneHHO cocTaeHTb

25.

PA3flEJl 14.1 OyHKLlHH HECKO/lbKMX IIEPEMEHHblX INI 861
x
PMCyHOK 12
PUCYHOKI
TEC Visual 14.1 AaHMMupyeT pucyHOK II BMecre cocTaBbie M300pa>KeHMe rpa<J)MKa. noBepxHOCTb KpyTan TaM. rAe KpMBbie ypoBHn pacnono>KeHbi 6JIM3KO
nOK33o«aa KpM6b»e ypoe**e*. KOTopfee rooHHMawrca apyr K apyry TaM. me OHM HaxoaaTCfl .aaribiue apyr OT apyra. ona HecKoribico 5onee nnoccan.
^ * m
no rpa0MKOB ctyHKUMM. OAMH M3 pacnpocTpaHeHHbix npuMepoB KpMBbix ypoBHB BCTpenaeTCB Ha Tonorpac})MMecKMX xapiax
ropHbix perMOHOB. T3KMX KaK KapTa Ha pMcyHKe 12. KpMBbie ypoBHfl - STO KpMBbie nocTonHHOM BbicoTbi
HaA ypoBHeM Mopa. ECJIM Bbi MAeie no OAHOM M3 STMX KOwrypHbix nnHMM, BW HM noAHMMaeTecb, HM
cnycxaeTecb. flpyniM pacnpocTpaHeHHbiM npnMepoM HB^aeTCH cpyHKUMH TeMnepaTypw, npeACTaBJieHHaa
B nepeoMi a63aue STOTO pa3Aena 3Aecb KpMBbie ypoBHn Ha3biBa»OTCB M30TepMMMecKMMM M
coeAMHHioT MecTa c OAMHaKOBoPi TeMnepaTypoti. Ha pMcyHKe 13 noKa3aHa KapTa
noroAbi MMpa c yKa3aHMeM cpeAHiix TeMnepaTyp BHBapn. H30TepMMMecKne TOMKM - 3TO
KpMBbie. KOTopbie pa3AejinioT LjBeTHbie nonocbi. M3o6apbi Ha KapTe aiMoccpepHoro AaBJieHMn
Ha CTp. 854 npeACTaBnaiOT coGofl e^e OAMH npMMep
KPMBMX ypoBHA
PMCYHOK 13
Cpe^HMe MHpoewe TewnepaiypbJ wa ypoBHe Mop« B
BHBape B rpaaycax I4enbcnn
Tapoan. AiwoaDepa Beeaewie e Meipojiorno
4-e G
1969 nepeoe^Tano c pajpetuen*} Pearson Education. Inc.

26.

862 III! rflABA 14 HACTHblE nPOM3BOflHblE
nPMMEP 9 KoHTypHafl KapTa (pyHKMMM fis r»OKa3aHa Ha pwcyHKe 14. Hcnonb3yMTe ero A^H OLieHKM 3H3MeHMM f(1, 3)
H f(4, 5).
PELUEHME ToHKa (1,3) HaxoAMTCfl Ha nonnyTM Me>KAy KPMBHMM ypoBHfl c z3HaHeHMflMM 70 M 80. Mbi oueHMBaeM, MTO
f(1,3) = 73
A»<anawMHbiM O6C>330M. MW oueHMBaeM, MTO
f(4, 5) = 56 ■
nPMMEP 10 Hapucytrre KpuBbie ypoBHH (JjyHKLjMM f(x, y) = 6 - 3x - 2y i\r\n 3HaHeHMfl
k = -6, 0, 6, 12.
PEUJEHME KpMBbie ypoBHfl BBnflfOTca
6 - 3x - 2y = k MJ~IM 3x + 2y ■+■ (k - 6) = 0>
*
3TO ceMeflCTBO DHHHM C HaxriOHOM - 5. HeTbipe KOHKpeTHbie xpuBbie ypoBHH c k = - 6, 0, 6 M 12 paBHbi 3x +
2y - 12 = 0, 3x + 2y - 6 = 0, 3x + 2y = 0 M 3x + 2y + 6 =
0. OHM Ha6poc3Mt>i na pMcynice 15 KpMBwe ypoenfl npeacTaanaoT CO<5OM napanncjibHwu
Ha paBMOM
m
m
pacnoncwwHHuc
m
paocroflHMM APyr OT Apyra. ntxxonbicy rpa$MK f npeAcraBJiaeT COGOM nnocxocrb (CM. pwc 6). ■
PMCYHOK I5
(I I1PMMEP II HapucywTe KpMObie ypoBHfl cpyHKquM
KoHTypHa^ Kapia
f(x, y) = 6 - 3x - 2y
g(x, y) = 9 - x2 - y2 A-rm k = O, 1,2,3 PeiueHMe KpMBbie
ypoBHR ABJIHIOTCS?
n 9 - x2 - y2 = k nnn x2 y2 = 9 - k2
3TO CGMCMCTBO KOHLJCHTPMHCCKMX oKpyxcHOCTeM c i^eHTpoM (0: 0) M paAnycoM 19 - k2.
CnyMaM
k = 0, 1, 2, 3 noKa33Hbi Ha pucyHKe 16. nonpo6yMTe BM3yajiM3MpoBaTb STM Kpnewe ypOBHa, noAHaTbie
BBepx, moObi apopMwpoBaTb noBepxHocrb, H cpaBHme c rpacjMKOM g (nonyapepa) Ha pucyHKe 7. (CM.
TEC ’ Visual 1 4 . 1 A . ;
PHCyH0K16
f
KoHTypHan KapTa ofg(x. y) = 9 - x2 - y2
nPMMEP 12 HapwcyMTe HeKOTopbie KpMBbie ypoBHfl (JjyHKLjMM h(x, y)
= 4x2 + y2.
PEUJEHME KpwBbie ypoBHH HBnflKrrcfl
4x2
y^ k n/i^i
l

27.

PA3AEJ1 14.1 OYHKL\\A\A HECKOJlbKMX flEPEMEHHblX Mil 863
KOTopuM ripn k > 0 onucbiBaeT ceMeiicTBO annuncoB c noiiyocflMM IYJ 2 M /k. Ha pMcyHKe 17(a)
noKa3aHa
KOHTypHa^ KapTa h, HapMCOBaHHaa KOMflblOTGpOM, C KpMBbIMM ypOBHfl, COOTBeTCTByK)U4MMM k =
0.25, 0.5, 0.75, .... 4. Ha pucyHKe 17(b) noKa3aHbi STM KpMBbie ypoBH*, nOflHflTwe AO rpacJ)MKa h
(annunTMHecKMM napa6onoMA). rae OHM ciaHOBATCH ropM30HTanbHbiMM TpaccaMM. Ha pMcyHKe
17 MW BMAMM , KaK rpaC^MK h COCTaB/lfleTCfl M3 KpMBbIX ypOBHfl.
TEC Visual 14.IB aeMOHCTpMpyeT
B33b Me>Kiiy nOBepXHOCTflMM M MX OHTVpt*
b>MM OpTaftM
PMCYHOK17
Tpa0MK h(x, y) = 4x2 + y2 (JxjpMupyeTCfl
(a) KoHTypwaa opra
nyreM ncwHJmifl KpwBbix ypoBHfl.
(til rcQKEftantMbe lpccu xcji'aarwcr CU6CM
TOBMuetiMCfO *--oe»*n
nPMMEP 13 nocrpoMTe KpuBwe ypoBHH arm npoM3BOACTB6HHOiii (JjyHKUMM Ko66a-flymaca M3
npuMepa 3.
PEIUEHME Ha pMcyHKe 18 MW ncnonb3yeM KOMnbiorep. mo6u HapwcoBaTb KOHTypHwn r pa t f w x Ko 6 6 a -
flymaca
npOM3BOflCTBOWHaB (JjyHKUMB
A
P(L, K) = 1.01L0.75K0.25
PHCyH0K18
KpMBbie ypoBH^ oGo3HaMeHbi 3HaqeHiieM npon3BOACTBa P HanpMMep, KpMBaa ypoBHfl
, o6o3HaMeHHaa KaK 140, noica3biBaeT ece 3HaneHMB pa6oneP! c*mw L M KanmanbHbix BnotfenHii K.
KOTopbie npMBOflflT K npoM3BOflCTBy P = 140. Mw BM/IMM. HTO npn (J)MKCMpoBaHHOM 3H3MeHMM P no
wepe yBenwMeHMfl L K yMeHbiuaeTCfl.^ M Hao6opOT.
• • III*
fljia HeKOTopwx i\eneu KOHTypHaa xapia 6o/iee none3Ha, ^eM rpac^MK. 3 TO, 6e3yaiOBHO, BepHO B
npMMepe 13. (CpaBHme pncyHOK 18 c pMcyHKOM 8.) 3TO TaioKe BepHO npw OLjeHKe 3HaneHMM
c^yHKUMM, Kax B npMMepe 9.

28.

864 III! rflABA 14 HACTHblE nPOM3BOflHblE
Ha pucyHKO 19 noK33aHbi HeKOiopwe cieHepMpOBaHHwe KOMribK^epOM Kpuswe ypoenw BMecie c
cooTBeiCTBywinuMM aenepiipoBaHHWMn KownbwiepoM rpa$MKaMM. OSpaTHTe BHMwaHne. MTO Kpuswe
ypoBHfl 3 Macro (c) aoflara Bweae BOJIM3M
Hana/ia KcxipA^HaT. 3TO cooTBeicTuyeT TOMy cfcaKTy, HTO rpatfcMK B nactM (d) onenb KpyTOrt o6nn3M Havana KoopannaT
(a) KpMBbie ypoBHfl f(x, y) = -xye
PMCYHOK 19
(b) flea BHfla f(x. y) = -xye*
(c) KpMBbIG ypOBHW f(x, y) =
-3v
(d) f(x, y) -f-y2
x2 + y2 + I
OYHKMMU TPEX M/1H EOJ1EE nEPEMEHHbIX_
OyHKuna Tpex nepeMeHHbix, f, - STO npaBuno, KOTopoe npucBanBaeT xa>KAOM ynopflAOMeHHOM TpoiiKe (x, y, z) B
o6nacTM D C R3 yHMKa/ibHoe BemecTBeHHoe HMC/IO. o6o3HanaeMoe f(x, y, z). HanpuMep. TeMnepaTypa
T B TOHKe Ha nOBGpXHOCTM 3GMJ1M 3aBMCMT OT AOJirOTbl X M LUMpOTbl y
TOMKH n OT BpeMeHM t, noaTOMy My Momu 6y 3anncaTb T = f(x, y, t).
nPMMEP 14 HaMAMTe AOMeH fif
f(x, y, z) = B(Z - y) + xysin z
PELUEHME Bbipa^KeHne A^n f(x, y, z) onpeAeneno AO Tex nop, noxa z - y > 0, noaTOMy o6na
D = {(x, y, z) € R’| z > y}
3TO nonynpocTpaHCTBO, cocToamee H3 Bcex ToneK. KOTopwe /ie*aT waa nnocKocTbo z = y ■

29.

PA3flEn 14.1 OYHKUMM HECKOJlbKHX flEPEMEHHblX INI 865
OMeHb TPYAHO BH3yajiM3npoBaTb cfcyHKLimo Tpex nepeMeHHbix no ee rpacfcMKy nocKonbKy OHa
nexana 6bi B MeTbipexMepHOM npocipaHCTBe. OAHaKO Mbi no/iyMaew HeKOTopoe npeACTaBneHne o
• • » • • • « « « * ^ % • • • « f. MccneAya ero noBepxHOCTM ypoBHa. KOTopbie aBnntoTca noBepxHocTOMM c ypaBHe«M^Mn f (x. y. z) = k. rae k - KOHCTSHTS
ECOM TOHK3 (x, y, z) nepeMemaeTcn no poBHOfl noBepxHOCTM, 3HaneHMe f(x, y, z) ocTaeTca
0HQPMHUH
9 nPMMEP 15 HaMflUTe noBepxHOCTM ypOBHH <£yHKUHM
f(x, y, z) = x2 + y2 + 22
PELUEHME noBepxHOCTM ypoBHfl paBHbi x2 + y2 + z2 = k, rae k > 0. OHM o6pa3yiOT ceMeMCTBO
KOHLieHTpMnecKMx ccfeep c paawycoM Vk. (CM. PncynoK 20.) TaKMM o6pa30M. nocxojibKy (x. y. z) M3MGMHeTCn no
OK)6OH
ccpepe c qeHTpoM 0, 3HaseHtie f(x, y. z) ocraeTca (puKCupoBaHHbiM ■
MoryTObiTb paccMOTpeHbi (pywaiMM moOoro HMcna nepeMeHHbix. CDyHKi4Mn n nepeMeHHbix
• • • • • •« * • *
%
m
- 3TO npauMJio. KOiopoe npHCBanaaGT nucno z = f(x1, x2.....x.) n-KopTexcy (x1. x2.....x,) AeMCTBMTent*Hbix
PMCYHOK 20
Hwcen. 06o3HanMM nepe3 RH MHOwecTBO ecex TaKnx n-KOpTeweirf. HanpMMep. ecnn KOMnaHUH ncno/ib3yeT n
pa3JiMMHbix MHrpeAueHTOB npn M3roTOBJieHMM nmneBoro npoAyKTa. c - 3TO CTOMMOCTb eAMHMLibi j-ro
MHrpeAneHTa.
a x - x2
Mcnonb3y»OTC^
i-ro MHrpeAneHTa, TorAa ooman cTOMMOCTb C MHrpeAneHTOB
IT] C = f(x1,
x) = c1x1 +eAHHMLibi
c2x2 + + cnxn
f
nBnaeTcn c^yHKMueM n nepeMeHHbix x1, x2, x:
CtyHKUMH f - 3TO BeinecTBeHH03HaMHa^
riOAMHOxecTBOM R". MHorAa Mbi
0yHKmifl,
o6nacTb
AeMCTBM^
KOTOPOM
aB/wera
•*
6yaeM McnonbaoeaTb BeKTopHyio HOTauwo A^ 6once KOMnaxTHoro HanwcaHHfl Taicwx cpyHKUMii: ecnM x = (x1,
x2 ; .... xn), Mbi Macro nmueM f(x) BMecTO f(x1, x 2 , x , ) .
C noMombio 3T0tt 3anMCM Mbi
MoxeM nepenncaTb 0yHKiiMK> onpe^enewHyx) B ypaBHeHMM 3. KaK
f(x) =« c • X
•
rAe c = (c1, c2, , cn) M c - x o6o3HanaeT TonenHoe npon3BeAeHne BexTopoB C M X B Vn.
BBMAy 0AH03HaHH0r0 COOTBeTCTBMH Me>KAy TOHKaMM (x1, X2.....X ,) B R" M
MX BeKTopaMM nonoweHMfl x = (x1, x2,.. x,) B V. y Hac ecTb TpM cnoco6a B3mnHyTb Ha cpyHKUMio,
onpeAeneHHyio Ha noAMHOweciBe P":
1. Kax cJjyHKUMH OT n BeuiecTBeHHwx nepeMeHHbix x1? X2. - x*
m
m
m
0
%
2. KaK cpyHKUMfl eAMHCTBeHHOM TOHesHOM nepeMeHHOM (x1, x2,.... xn)
3. KaxcfcyHKiAMH OAHOM BeKTopnoM nepeMeHHOM x = (x1, x2, , xn)
Mbi yBMAHM, HTO Bee Tpn TOHKM 3peHV\9\ no/ie3Hbi.
14.1 ynPAWHEHMfl
I_I
I. B npMMepe 2 Mbi paccMOTpenM cftyHKUMio W = f(T, v), rAe W -MH^exc
xonoAHOCTVi BeTpa, T - cpaKTHMecKan TeMnopaTypa, a v -CKopocTb Beipa.
MucnoBoe npeACTaBneHMe npuBeAeHo B Ta6nnue 1. (a) KaKOBO
3HaMeHMe f(-15, 40)? B MeM ero 3HaMeHMe? (6) OnMLUMTe
cnoBaMM 3HaMeHMe Bonpoca KaKoro 3HaMeHMfl
v psBHo f(-20. v) = -30?" Toraa oTBeTbTe Ha Bonpoc.
(c) OnmiJMTe cnoBaMM 3HaseHne Bonpoca "fl/ifi KaKoro 3naMeHHB
Tis f(T, 20) = -49?" Torfla OTBeTbTe Ha aonpoc.
(d) B neM CMbicn (pyHKL)MM W = f(-5, v)?
Onmome noBefleHMe OTO« (pyHKLiMM
(e) B neM CMWCJI cpyHKUHM W = f(T, 50)?
OnMLUMTe no
3T0fl (pyHKL(MM.

30.

866 III! mABA 14 HACTHblE nPOM3BOflHblE
TABJ1HUA 4
2. HH^eKC TeMnepaTypbJ M onaxwocrM I (nnn axpaiueHHO humklex) - STO BOcnpviHnviaewa?i
TeMnepaiypa
soaayxa.
Koraa
<$>aKTMNeo<afl
TeMnepaiypa
paBna
T.
a
oTHOcmenbHafl l BiiajKHOdb paeHa h, noaiOMy MU MOWGM 3anncaTb I -
\/
v X.
f(T. h) Cneaywma* Ta6nnua SHaMeHMM I npeAcraejiaeT coSoiA Bwflepxay M3
Ta6jiMLibi. cocraBneHHOM HaL|MOMajibH&JM ynpasjreHHeu OKeaHvmecKMx H
aTMOC0epnbix MCcneAoaaHMH.
TABDULIA 3 KaxcyinancH leMnepaTypa B MBMCMMOCTM
<A
OT TeMnepaTyptJ n ena>CHOCTvi
c
M
W
OTMOCKTonbHan anawocTb <%)
U.
o
1>
E
I
a
E
v
y
<
\h
T\
80
20
30
77
78
82
84
90
87
95
100
85
40
79
50
81
60
70
82
83
90
93
86
88
90
93
%
KM)
106
93
96
101
107
114
124
99
104
110
120
132
144
■
3
SL
&
c
5
30
40
50
10
2
2
2
2
2
2
2
15
4
4
5
5
5
5
5
5
7
9
9
9
8
8
30
9
13
16
17
18
19
19
40
14
21
25
28
31
33
33
50
19
29
36
40
45
48
50
60
24
37
47
54
62
67
69
HaiiflMTe ,qnana30H f.
(b) HaMAme obnacTb f.
(a) BbiHMcnaeT f(2, 0).
(c)
Anana3QH f.
HaMAMTe
8. HaMAMTe M HapMcyMTe o6nacTb aeMCTBMfl <J>yHKL4MM
f(x, y) = VI + x - y2. KaxoB Anana30H f?
9. nycTb f(x. y, z) = ev:'** r ‘. (a)
BbiHMcnaeT f(2, -1, 6). (c)
HaMAMTe Anana3QH f.
B npHMepe 3 o6cy>KAanocb. HTO npon3BOACTB0 YABOMTCB, ec/iM
OnpeaenMTe. BepHO JIM STO TaioKe Ana o6men
20
7. riycTb f(x, y) = x'e3fr.
npoBepbTe npoM3BOtacTBCHwyK) cpyHiajMKD Ko66a-Ayrnaca
YABOMTb Kax KonuMeCTBO paSoneM CM/ibi. Tax M KonunecTBO KariMTa/ia .
15
(c) Haflame M HapucyflTe oSnacTb f.
(d) KaKOBbi 3HaMeHMB 0YHKUMM I = f(80r h)
P(L. K) = 1.01L0.75K0.25
10
6. nycTb f(x, y) = ln(x + y -1).
(a) OUQHMTO f(1. 1). (b) OLteHMTe f(e, 1).
(a) KaKOBO 3HaMeHMe f(95t 70)? B MeM ero
3HaMeHMe? (b)fljia KaKoro 3HaneHM« h paaHO f(90, h) = 100? (c) flrm
KaKoro aHaneHM* T paeHO f(T, 50)« 88?
3.
5
20
(d)
n I = f(100: h)? CpaBHMTe noBeA^Hne DTMX AByx 0YHKL4MM h.
ripocaniwTwibHocTb
I'wacw)
(b) HaMAMTe o6nacTb f.
10. IlycTb g(x. y. z) = ln(25 - x2 - y2 — r2).
(a) BbiHMcrmeT g(2, -2, 4).
obnacTb g.
npOM3BOACT6eHHOM (pyWCJp*;
(b) HaMAMte
(c) HaMAMTe AManasoH 3H*4CHMM.
P(L, K) = bL°K\ °
4.
MHfleKC xonoflHOCTM BeTpa W, o6cy*AaeMbiM B npMMepe 2, 6bin
CMO^en^poBdH cnoMOuibiO cne~yom< $yHKu/n
W(T, v) = 13.12 + 0.6215T - 11.37b016 + 0.3965Tb016
npoBepbTe. HacKonbKO 6JIM3KO 3Ta MOAe/ib comacyeica co
3HaMeHMflMM B Ta6nnue 1 fl/ifl HGCKOJibKnx 3HaneHMM T M v.
5. Bbicoia BonH h B oncpbiTOM Mope 3aBncm OT CKOPOCTM V
o BeTpe M npoAon)KMTenbHOCTM BPSMSHM t. B TeueHne KOToporo
□eTep nyn c TSKOM CKOPOCTLMO. 3HaMGHMH CPYHKIAMM h = f(v, t)
11-20 HaMflMTe M wapMcyMTe o6nacTb ACMCTBMH cpyHKUMM.
11.
f(x, y) = Vx + y
12.
f(x, y) = V^"
13.
f(x, y) = ln(9 - x2 - 9y2)
14. f(x. y)
= Vy - x B(y
+ X)
15. f(x. y)
= V1 - x2 -
V-y2
16. f(x. y)
= Vy + v25 - x2 - y2
f(x, y')
y - x2
l —x2
3anMCaHb4 B cpyiax a Ta6nMue
4. (a) KaKOBO 3HaMeHne f(40,15)? B neM ero 3HaHeHMe?
18. f(x. y) - arcsin(x2 + y2 - 2)
(b) B MeM cMbicn cpyHKUMM h = f(30, t)? OnmuMTe
19. f(x, y. z ) = V1 r x2 - y2 - z 2
nooeiieHMe 3TOM cpyuKunn.
(c) B MeM CMbicn 4)yHKisy\v\ h = f(v, 30)? OnuiuHTe 20. f(x, y, z) = ln(16 - 4x2 - 4y2 - z2
nOBGA«HH« 3TOM CpyHKUMM.

31.

PA3flEn 14.1 OyHKUHH HECKOJlbKHX nEPEMEHHbIX llll 867
21-29 HapucynTe rpa<$)MK c^yHKLjMM.
32. noKa3aHw <qBe KornypHue KapTbJ. OAMH M3 HMX npe£Ha3HaMeH
ATO f.
21. f(x, y) = 3 22. f(x. y) - y
(oTopoiii npeAaaBnj?ei COSOH KOHYC ^Ipyrow - (JjyHKUMM g. rpatfjmc
KOTOPOM npeftaa&rtReT cooo/ napaSonoaA HTO ecr0 mo M noMeuy?
23. f(x, y) = 10 - 4x - 5y 24. f(x, y) = cos x
I
25. f(x, y) = y2 + 1 26. f(x, y) = 3 - x2 - y2
27. f(x, y) = 4x2 + y2 + 1
28. f(x, y) = /16 - x2 - 16y2
29. f(x. y) = Vx2 + y2
30.ConoaaBbTe cJjyHia^io ceerpa<J*iK0M (noMenenHWM l-Vl).npMBeflHTe npuMMHbi
3a Barn Bbioop
33. HUMAMTO I OH KM A M B HU KUpTO OflUHOKOM ropbl
(a) f(x, y) = |x| + |y| [6) f(x, y) = |xy|
f) f(x, y) = , ,
2 +-yy2)2
2
(d) f(x,Iy)+-x(x2
(e) f(x. y) = (x - y)2
• A
(puc. 12). KaK 6bi Bbi onncann MecTHocTb B6HH3M A?
[ f(x, y) = sin(|x| + [y])
PHAQM C B?
34. CflenaMTe rpy6bifl Ha6pocoK KOHTypHOM KapTbi 0yHKL4MMt
rpaipwc KOTopcii nc*a3an
ZA
35-38 noK33aHa KomypHafl KapTa 4>YHKUMM. Mcnonb3yirfTe ero. MTO6W cflenaTb
rpy6bm HaGpocoK rpacfrMKa f.
31. floKa3aHa KOHTypHa* *capTa Ana cfcyHKLiMM f. Hcnonb3yMTe ero Arm
oueHKM 3HaMeHMM f(-3r 3) M f(3, -2). MTO Bbi MO>Keie CKa3aTb
o $opMa rpa0MKa?
37. v

32.

868 III ITIABA 14 HACTHblE nP0H3B0flHblE
39-46 HapucyMTe KOHTypHyto KapTy cpywajiM, noica3faBaiou(yio
HeacanbKO KpuBbix
ypOBH«.
61-64 OnucbiBaioT
cfiyHKijHM
noBepxHocm
ypoBHfl
61. f(x, y, z) = x + 3y + 5z
39. f(x, y) = (y - 2x)2 40. f(x, y) = x3 - y
•41. f(x,
f(x, y)
-Bx
[43.
y) =
= yye*
45. f(x, y) = Vy2 - x2
42. f(x, y) = ©*
44. f(x, y) = y ceK x 46. f(x,
y) = y/(x2 + y2)
47-48 HapucyPiTe KOHTypHyto KapTy n rpacpHK cpyHKunn
M CpaBHMTC MX.
62.
f(x, y. z) x2 + 3y2 + 5z2
63.
f(x, y. z) = x2 - y2 + z2
64.
f(x, y. z) = x2 - y2
65-66 Onmume, Kax rpatfcMK g nonyHaeTca M3 rpacpMKa
f.
[65 ] (a) g(x. y) = f(x, y) + 2 (b) g(x, y) = 2f(x, y)
47.
f(x. y) = x2 + 9p2
48.
f(x. y) = V36 - 9x2 - 4y2
(c) g(x, y) = -f(x, y) (d) g(x, y) = 2 - f(x, y)
66. (a) g(x, y) = f(x2, y)(b)g(x, y) = f(x, y + 2) (c) g(x. y) = f(x +
3. y - 4)
49. ToHKafl MeTanriMMecxafl nnacTMua. pacnonowennan o nnocKOCTM xy, HMGGT
TeMnepaiypy T(x, y) B TOMKe (x, y). KpwBbie ypoBHa T Ha3UBaK)TCfl ra -- - - - -
Tryvjtfn TPMneDaTvpa 67-68 KnonbJytowunwDiepnoapcaHiw rp*t*.a c^crb^eaH^M par****
M30TGpMWHeCXMMM. nOTOMy MTO BO BCex To^^tax M30TepMMMeCK0M TOMKM TewnepdiyMO V' _ n _ _
ozjMHaxoBa HapucyviTe neKOTopwe M3orepMMMecKne 3naMem«i. ecnw
0yHKunfl TeMnepaiypw 3aAaeTca
T(x, y) = 100/(1 + x2 + 2y2)
50. ECJIM V (x. y) - 3/ieKTpnnecxMM noTeHunan B TOMKe (x, y) B
nnocKOCTM
xy, TO KpMBbie ypOBHfl V Ha3blBa»OTCfl 3KBMnOTeHUMaj1bHblMM KpHBblMH,
noTOMy HTO BO Bcex TOHKax TaKOH KPMBOM 3neKTpnMecKMM noTeHijiian
AOMeHoe M TOMe* 3penMfl nonyMHTe pacneMaixy Aawmyio xopotumi
M AonnH*. Bw 6w CK33ann, HTO (pyHiojun MMeeT MaKCWManbHoe
3HaneHne? MoweTe JIM Bbi onpeAenuTb KaKne-jin6o TOMKM Ha
rpacjwxe. KOTopbie BW Momu 6bi CMecTb "TOMKaMM jioKa/ibHoro
-OHacneT "jiOKanbHbix
MMMMManbHbJX
6annoe
MaKCMMyMa"?
Kax
67.
f(x, y) = 3x - x4 - 4y2 - 10xy
68.
f(x, y) = xye
OAMHaKOB HapMcynTe HexoTopwe 3KewnoTeHunajibHbie upMBue.
earn V (x, y) = c / Vr2 - x2 - y2, r^e c - nono>KMTenbHa« nocTOflHMan.
51*54 Hcnojib3yiiTe HOMnuoTep Ann noapoe*^ rpact^xa
0yHKunn c *cnonb30B3H*ev p33/in«-iHbJx AOMenoB M TOM ex 3peHHA.
nonyMHTe pacnenaTxy Toro, KOTOPWM. no Baweuy MHemiK). AaeT xopowwii
o63op. Earn &awe nporpaMUHoe ooecneMeHHe Tac*;e C03AaeT
xpwewe ypoenn. TO HanecMTe HecxonbKO KOHTypHtix TIMHMM TOM ace {ftyHKLjMM M Cp^BHMTe C I
51.
y) * e * + ©2*’'
pa0Mf(x.
K O M.
52.
53.
69-70 Wcnofb3>we rawrbOTep jyw nocTpoewoi rpatpitKa c
Knorb»BaHHe« paan**ba AOMeHOB M TOMex 3peHna
(pyHKUMM HTO npowcxoam.
xorfla
M X. M y CTanoeflTCfl 6o/ibiuMMM? HTO nponcxQanT. KOTfla
npoKOMMeHTwpyMTe
orpa
HMMH
BaK>mee noBeAeHMe
(x. y) npn6nn)KaeTCH K Hanarry KOOpaMnaT?
69. f(x. y)
X+V
x3 + y'■
7° f(x. y, .*
f(x. y) = (1 - 3x2 + y2)e
£071. Hcnonb3yi4Te KOMnbiOTep AHA nccneAOBaHnn ceMeucTBa
f(x, y) = xy2 - x3 (o6e3bflHbe ceflno)
ct>yHKi4MM f(x, y) = e. KaK 3aBncnr c^>opMa rpa0MKa
na c?
• A
54. f(x, y) = xy3 - yx3 (coOanbe ceA^o)
B*72. Hcnojib3yvfre KOMnwoTep ana nccneAOBaHnn cer^ewcTBa n
55-60 ConocTaBbTe (pyHKUMio (a) c ee rpac^MKOM (o6o3HaMeHHbJM
A-F Ha i CTp. 869) M (6) c ee xonTypHOM xapTOM (o6o3naHeHHOM l-VI).
z = (ax2 + by2)e
06bncHMTe npnuMHbi BaLuero Bbi6opa.
Kax 4>opMa rpa4)MKa 3aBncnT OT Mucen a
m
55.
z = sin(xy)
57. z = sin(x - y)
56. z = e' cos y*
M b?
58. z = sin x - sin y
59 z = (1 - x2)(1 - y2)
60. z =-—^—7
1 +x2 + y‘
73. Mcnonb3yHTe KOMnwoTep HocneAOEahHn cewencrsa noBepxhocreM
z = x2 + y2 + cxy. B nacTHOCTH, Bbi AOn>KHbi onpeAenMTb
nepexcAHwe 3HaMeHn« c. zm« KOTopbix noBepxHocrrb M3MenfleTCfl OT
on
OAHOTO I Tuna HeTwpexyronbHOM noBepxhOCTM K Apyrm.

33.

PA3,QEn 14.1 OYHKI4MM HECKOJlbKHX flEPEMEHHblX III! 869
fpacfcHKM M KOHTypHbie KapTbi ynpa>KHeHm?i 55-60
111

34.

870 III rflABA 14 MACTHblE nPOH3BOflHblE
^74. noCTpOMTe rpa4>MK cfyHKunM
ffl75. (a) noKa*HTe, HTO, npuHUMas noraput^Mbi, o6maa (Jjymcijmi
Ko56a- Ayrnaca P = bL °K " MOJKOT 6biTb BupaxtCHa KaK
f ( x , y) = e^V
f(x, y) = Vx2 + y2
P* L
In — =B b +• Bln ~
f(x, y) - sin(vx2 + y2 )
f(x, y) = B /x2 + y2
(b) Ecnn MU AonycTMM x = ln(L / K) v\ y = ln(P / K). ypaBHeHne B
Macro (a) CTaHOBUTCB nnHePiHWM ypaBweHweM y = ax + B b.
" fo.v>
B o6u4eM c/iysae. ecrin g aanaeTCfl 0yHKune^» AQHOVI nepeMeHwOM. ro Kate BwmflAMT
Mcnonb3ytiTe
TaGmmy 2 (B npMMCpe 3), mo6b< cociaBHTb tabJiMuy 3HaseHHM
f(x,y) = 9(vx2 + y2)
In(LTK) M ln(P/K) 3a 1899-1922 rpflbi. 3aieM
ncno/ib3yMTe
nonyneHO M3 rpacJjMKa g?
rpa<pMMecKMM Ka/ibKynnTop unw KOMnbtoTep. HTo6bi H3MTM /WHMKJ
perpeccvin nanMeHbiuMx icaaApaToe sepe3 TOMKM (B (L / K). B (P /
K)).
(c)
BbibcuMTe. MTO npon3BOflCTBeHHa« c&yHKL^in Ko66a-Ayrnaca
pafiHa P = 1,0170.75g-0.25
=14.2 nPEflEJlbl M HEnPEPbIBHOCTb
L
J
flaBaMTe cpaBHMM noBeAeHMe cfcyHKUMM
f(x, y) si . 2
si*, v) =
H
x 2+ty
TL2 - y 2
nocKonbKy x M y oOa npnbnn?KaK)TCJi K 0 [M. c/ieflOBaienbHO. TOHK3 (x. y) npnOnn^aera K Hanany KoopflMHaT].
TABJ1MLIA 2 3HaneHHfl g(x, y)
3H9^eHMfl TABni^qw I Bbiicn»ONGHbi(x. y)
>sv v
-1.0
-0.5
-0.2
0
-1.0
-0.5
-0.2
0
0.2
0.5
1.0
-1.0
0455
0.759
0.829
0.841
0.829
0.759
0455
-1.0
-0.5
0.759
0.959
0.986
0.990
0 986
0.959
0.759
-0.5
- 0.600
-0.2
0.829
0.986
0.999
1.000
0.999
0.986
0.829
-0.2
-0 923
-0.724
0
0.841
0.990
1.000
1.000
0.990
0.841
0
-1.000
-1.000
-1.000
0.2
0.829
0.986
0999
1.000
0 999
0.986
0829
0.2
- 0.923
-0.724
0.000
0.5
0.759
0.959
0.986
0.990
0.986
0.959
0.759
0.5
-0.600
1.0
0455
0.759
0.829
0.841
0.829
0.759
0455
1.0
0.000
X
X
0.000
0.600
0000
0.000
0.600
0.923
0.724
0.000
0.724
0.923
1.000
1.000
1.000
0.2
0.923
0.724
0.000
-1.000
1.000
1.000
1.000
0.000
0.724
0.923
0.5
1.0
0.600
0.000
0.000
—0.600
-0.724
-0.923
1.000
-0.724
0.000
0.600
- 1.000
-0 923
-0.600
0.000
B TaGjiML|ax 1 n 2 npMBeAeHbi 3HaHeHMfi f(x, y) M g(x, y) c TOMHOCTbio AO Tpex 3H3KOB nocne 3anaTOM QJ\I\
ToneK (x, y) B6JIM3M Hana/ia KoopAMHaT. (06paTMTe BHMMaHne, HTO HM OAHa M3 (pyHKLpiM He onpeAeneHa B
HaMane KOOfWMHaT.) noxoxe. MTO no wepe Toro, xax (x. y) npMOmwaera K (0. 0), 3HaNeHna f (x. y) npuGmxaiOTCfl K
1. TorAa K3K 3HaMeHnn g (x. y) He npn6nn)KaK)TCH HH K xaKOMy HMCJiy. OKa3biBaeicn, MTO 3TH AoraAKH,
ocHOBaHHbie Ha MMC/ioBbix AOKa3aTe/ibCTBax. BepHu, H MU nuiueM
sin(x2
^
•imn -:-i * ) 1 M
limi
(x. y) ->(0, 0) x y~
* - y
(,y)(0.0) x2 42 He
B obujew cnyyae MU ncnonb3yew ooo3HaMenwe
(,. ,7-W,(x'y)
cymecTByeT

35.

PA3flEJl 14.2 nPEflEJlbl M HEflPEPblBHOCTb Mil 871
MTo6bi yKa3aTb, mo 3HaMeHMa f (x, y) npn6nn>KaK)TCfl K Mucny L no Mepe
npn6jin>KeHMn TOMKM (X, y) K TOHKe (a, b) no ntoOoMy nyTH, KOTopbin ociaeTcn B
npeaenax obnacTM f. flpyrnMM cnoBaMM, Mbi MoweM
CAenaTb SHaneHHfl
f (x, y) HacTonbKO 6nM3KMMM K L, HacKOJibKO HaM MpaBMTCfl, B3flB TOHKy (X, y)
AOCTaTOHHO 6nM3K0fl K
TOHKe (a; b), HO He paBHOfl (a: b). flanee cneAyeT bonee TOHHoe
[TonPEAEnEHME riycTb f- cfcyHKqMH AByx nepeweHHwx, oOnacTb D KOTOpOM BKnKwaeT TOHKM,
onpeAeneHMe.
CKonb yroAHO 6nn3Kne K (a, b). 3aieM Mbi roBopuM, HTO npeflen f (x, y) no Mepe npn6nn*eHHfl (x,
y) K (a, b) paBeH L, M Mbi 3armcbiBaeM'
»«»■»> * L
ecjin Ann Ka>KAoro MMcna e > 0 cymecTByeT cooTBeTCTByiomee MMcno 8 > 0 Taxoe, HTO ecnn (x, y)
D M O <(x - a)2 + (y - b)2 < 8 TorAa |f(x, y) - L|
ApyrmiH o6o3ia^eh/isMn f\nn npe^e.na s onpefle/ieHHM 1 aejiaicrca
lim f(x, y) = L M f(x, y) -> L KSK (x, y) -> (a, b)
y->b
Q6paime BHMMaHMe, HTO | f (x? y) - L| - STO paccTOflHMe Me>KAy HMcnaMM f (x, y) M L, a V
(x - a) 2 + (y - b)2 - 3TO paccTonHMe Me>KAy TOHKOM (X: y) M TOHKOM (a, b). TaKMM o6pa30M , B
onpeAeneHMM 1 roBopmca, HTO paccTOflHMe Me>KAy f (x, y) M L MO>KHO CAenaTb CKO/ib yroAHO
ManbiM. CAenaB paccTOflHne OT (X, y) AO (a. b) AociaTOHHO wa/ibiM (HO He 0). PucyHOK 1
Mn/iKxrrpMpyeT onpeAeneHMe 1 c noMOLiibK) AHarpaMMbi co crpe/iKaMM. ECJIM 3aAaH KaKOM-nn6o
HeCo/ibLUOM MHTepBan (L - e, L + e) BOKpyr L, TO Mbi MoweM HSMTH AMCK DS C qeHTpoM (a, b) M
paAMycoM 8 > 0, TaKOM, KOTOPWM;
noMeiAaeT Bee TOHKM B D8 [KpoMer BO3MO>KHO, (a, b)] B MHTepBan (L - e, L +
e).
PMCYHOKI
Apyraa nnmocTpaun« onpeAenenwA 1 npMBeneHa Ha pMcyHKe 2, rAe noeepxHoerb S npeACTaBnaeT
CO6OM rpa0MK
• • • - • • • - • • • • « % « • • •• • • •• • f. ECHM 3aA3HO e > 0. Mbi MOweM HartTM 8 > 0 TDKUM oGpaaow. MTO BCJIM (x. y) or paHUHBHu
re^HneM B Ancxe
D3 M (x, y) (a. b), TO cooTBeTCTByiomaH nacTb S ne>KMT Me>KAy ropM30HTanbHbiMM nnocKocTHMM
2=L-klZ=L-*-e.
flnn cpyHKqMfl OAHOA nepeMeHHOfl, KorAa Mbi no3BonneM x
npMbnMwaTbcn K a, cyiAecTByeT TonbKo ABa B03M0>KHbix HanpaBneHMn
npMbnMweHMn: cneBa mv\ cnpaBa. Mbi IIOMHMM M3 rnaBbi 2, HTO ecnM limf(x)
lim,f(x), TO limf(x) He cyLuecTByeT.
flnn 0yHKAMM AByx nepeMeHHbix cMTyauMB He Tax npocTa, noTOMy HTO Mbi MO>KeM no3BonnTb (x,
y)
npn6nn>KaTbcn K (a, b) c 6ecKOHeHHoro nucna HanpaaneHHM nioObiM cnoco6oM (CM.
pwc. 3) AO Tex nop. noKa (x, y) ocTaeTcn B oOnacTM f. M