(eaeoeey 1)
Ii?aaeaeaiea ooieoeee ianeieueeeo ia?aiaiiuo.
Ia?aiaiiay u iacuaaaony f(x,y,z,..,t), anee aeey ethaie niaieoiiinoe
cia/aiee (x,y,z,..,t) noaaeony a niioaaonoaea aiieia ii?aaeaeaiiia
cia/aiea ia?aiaiiie u.
Iiiaeanoai niaieoiiinoae cia/aiea ia?aiaiiie iacuaatho iaeanoueth
ii?aaeaeaiey o-oeee.
G – niaieoiiinoue (x,y,z,..,t) – iaeanoue ii?aaeaeaiey .
Ooieoeee 2-o ia?aiaiiuo.
Ia?aiaiiay z iacuaaaony ooieoeeae 2o ia?aiaiiuo f(x,y), anee aeey ethaie
ia?u cia/aiee (x,y) ( G noaaeony a niioaaonoaea ii?aaeaeaiiia cia/aiea
ia?aiaiiie z.
I?aaeae ooieoeee 2-o ia?aiaiiuo.
Ionoue caaeaia ooieoeey z=f(x,y), ?(o,o)-oaeouay oi/ea, ?0(o0,o0)-
?anniao?eaaaiay oi/ea.
Ii?. Ie?anoiinoueth oi/ee ?0 iacuaaaony e?oa n oeaio?ii a oi/ea ?0 e
?aaeeonii (. ( = ((o-o0)2+(o-o0)2(
*enei A iacuaaaony i?aaeaeii ooieoeee |a oi/ea ?0, anee aeey ethaiai
Lim f(x,y)
p(p0
neieue oaiaeii iaeiai /enea ( iiaeii oeacaoue oaeia /enei ( (()>0, /oi
i?e anao cia/aieyo o e o, aeey eioi?uo ?annoiyiea io o. ? aei ?0 iaiueoa
( auiieiyaony ia?aaainoai: (f(x,y) ( A(((, o.a. aeey anao oi/ae ?,
iiiaaeathueo a ie?anoiinoue oi/ee ?0, n ?aaeeonii (, cia/aiea ooieoeee
ioee/aaony io A iaiueoa /ai ia ( ii aaniethoiie aaee/eia. A yoi cia/eo,
/oi eiaaea oi/ea ? i?eaeeceony e oi/ea ?0 ii ethaiio iooe, cia/aiea
ooieoeee iaia?aie/aiii i?eaeeaeaaony e /eneo A.
Iai?a?uaiinoue ooieoeee.
Ionoue caaeaia ooieoeey z=f(x,y), ?(o,o)-oaeouay oi/ea, ?0(o0,o0)-
?anniao?eaaaiay oi/ea.
Ii?. Ooieoeey z=f(x,y) iacuaaaony iai?a?uaiie a o. ?0, anee auiieiythony
3 oneiaey:
1)ooieoeey ii?aaeaeaia a yoie oi/ea. f(?0) = f(x,y);
2)o-y eiaao i?aaeae a yoie oi/ea.
Lim f(?) = (
p(p0
3)I?aaeae ?aaai cia/aieth ooieoeee a yoie oi/ea: ( = f(x0,y0);
Lim f(x,y) = f(x0,y0);
p(p0
Anee oioy au 1 ec oneiaee iai?a?uaiinoe ia?ooaaony, oi oi/ea ?
iacuaaaony oi/eie ?ac?uaa. Aeey ooieoeee 2o ia?aiaiiuo iiaoo
nouanoaiaaoue ioaeaeueiua oi/ee ?ac?uaa e oeaeua eeiee ?ac?uaa.
Iiiyoea i?aaeaea e iai?a?uaiinoe aeey ooieoeee aieueoaai /enea
ia?aiaiiuo ii?aaeaeyaony aiaeiae/ii.
Ooieoeeth o?ao ia?aiaiiuo iaaiciiaeii ecia?aceoue a?aoe/anee, a ioee/ea
io ooieoeee 2o ia?aiaiiuo.
Aeey ooieoeee 3o ia?aiaiiuo iiaoo nouanoaiaaoue oi/ee ?ac?uaa, eeiee e
iiaa?oiinoe ?ac?uaa.
*anoiia i?iecaiaeiie.
?annii?ei ooieoeeth z=f(x,y), ?(o,o)- ?anniao?eaaaiay oi/ea.
Aeaaeei a?aoiaioo o i?e?auaiea (o; o+(o, iieo/ei oi/eo ?1(o+(o,o),
au/eneei ?aciinoue cia/aiee ooieoeee a oi/ea ?:
(oz = f(p1)-f(p) = f(x+(x,y) – f(x,y) ( /anoiia i?e?auaiea ooieoeee
niioaaonoaothuaa i?e?auaieth a?aoiaioa o.
Ii?. *anoiia i?iecaiaeiie ooieoeee z=f(x,y) ii ia?aiaiiie o iacuaaaony
i?aaeae ioiioaiey /anoiiai i?e?auaiey yoie ooieoeee ii ia?aiaiiie o e
yoiio i?e?auaieth, eiaaea iineaaeiaa no?aieony e ioeth.
(z = Lim (xz
(x (x(0 (x
( (z = Lim f(x+(x,y) – f(x,y)
(x (x(0 (x
Aiaeiae/ii ii?aaeaeyai /anoiia i?iecaiaeiie ii ia?aiaiiie o.
Iaoiaeaeaiea /anoiuo i?iecaiaeiuo.
I?e ii?aaeaeaiee /anoiuo i?iecaiaeiuo eaaeaeue ?ac eciaiyaony oieueei
iaeia ia?aiaiiay, inoaeueiua ia?aiaiiua ?anniao?eaathony eae iinoiyiiua.
A ?acoeueoaoa eaaeaeue ?ac iu ?anniao?eaaai ooieoeeth oieueei iaeiie
ia?aiaiiie e /anoiay i?iecaiaeiie niaiaaeaao n iau/iie i?iecaiaeiie yoie
ooieoeee iaeiie ia?aiaiiie. Ionthaea i?aaeei iaoiaeaeaiey /anoiuo
i?iecaiaeiuo: /anoiiy i?iecaiaeiay ii ?anniao?eaaaiie ia?aiaiiie euaony
eae iau/iay i?iecaiaeiayooieoeee iaeiie yoie ia?aiaiiie, inoaeueiua
ia?aiaiiua ?annoao?eaathony eae iinoiyiiua aaee/eiu. I?e yoii
ieacuaathony ni?aaaaeeeauie ana oi?ioeu aeeooa?aioee?iaaiey ooieoeee
iaeiie ia?aiaiiie (i?iecaiaeiiy noiiu, i?iecaaaeaiey, /anoiiai).
(Eaeoeey ? 2)
Iieiue aeeooa?aioeeae o-oeee 2-o ia?aiaiiuo.
z=f(x,y) a iaeanoe D.
p(x,y) ( D – ?anniao?eaaaiay oi/ea. Aeaaeei o i?e?auaiea (o, o – (o.
Iieo/ei ?1(o+(o, o+(o). Au/eeei cia/aiea ooieoeee. Iieiui i?e?auaiea
ooieoeee iacuaaaony ?aciinoue:
(z = f(p1)-f(p)
(z = f(x+(x,y+(y) ( f(x,y)
Ii?. Iieiui aeeooa?aioeeaeii ooieoeee z=f(x,y) iacuaaaony aeaaiay
eeiaeiay /anoue i?e?auaiey yoie ooieoeee, anee i?e?auaiea iiaeii
i?aia?aciaaoue e aeaeo:
(z = A(x + B(y + (
A, A – ia caaenyo io (o, (o;
( – caaeneo io (o e (o e i?e yoii
Lim ( = 0
((0 (
( – ?annoiyiea iaaeaeo oi/eaie ? e ?1
S = ??1 = ((o2 +(o2(
( yaeyaony aaneiia/ii iaeie, aieaa aunieiai ii?yaeea, /ai (
I?e oiaiooaiee (o e (o ((0 auno?aa, /ai (. Ec ii?aaeaeaiey neaaeoao, /oi
iieiue aeeooa?aioeeae ooieoeee ?aaai
z = A(x + B(y
I?e iaeuo (o e (o eiaao ianoi ?aaainoai (z ( dz.
Ii?. Anee ooieoeey z=f(x,y) eiaao iieiue aeeooa?aioeeae a oi/ea ?, oi
iia iacuaaaony aeeooa?aioee?oaiie a yoie oi/ea.
Oai?aia. Iaiaoiaeeiua oneiaey aeeooa?aioee?oaiinoe ooieoeee.
Anee ooieoeey z=f(x,y) aeeooa?aioee?oaia a oi/ea ?, oi iia eiaao
/anoiua i?iecaiaeiua a yoie oi/ea e i?e yoii au?aaeaiea iiiiai
aeeooa?aioeeaea A = (z/(x B = (z/(y, o.a. iieiue aeeooa?aioeeae iiaeao
auoue caienae a aeaea:
dz = (z/(x (x + (z/(y (y
Aeie-ai: Ii ii?aaeaeaieth aeeooa?aioee?oaiinoe i?e?auaiea ooieoeee
iiaeao auoue caienaii a aeaea:
(z = A(x+B(y +( i?e ethaii (o e (o.
?anniio?ei 2 /anoiuo neo/ay
1)(o(0 (o = 0
I?e yoii (z=A(x+( /(x e ia?aeaeai e i?aaeaeo. Iieiia i?e?auaiea
ooieoeee i?aa?auaaony a /anoiia i?e?auaiea.
Lim (xz/(x = Lim A+(/(x
(x(0 (x(0
(z/(x= A+Lim((x(0)(/(x =0 o.e. (=(o
A ?acoeueoaoa iieo/aai A=(z/(x
2)(x=0 (y(0
I?e yoii aiaeiae/iui ia?acii iieo/ei, /oi A=(z/(y
Oai?aia aeieacaia. Eae neaaenoaea ( iieiue aeeooa?aioeeae
aeeooa?aioee?oaiie ooieoeee ii?aaeaeyaony ii oi?ioea:
dz=(z/(x((x+(z/(y((y, anee i?e yoii o/anoue, noi i?e?auaiea iacaaeneiuo
ia?aiaiiuo o e o ?aaiu eo aeeooa?aioeeaeai (x=dx, (y=dy, oi
ieii/aoaeueii iieo/ei:
dz=(z/(x(dx+(z/(y(dy
Oai?aia 2. Aeinoaoi/iia oneiae aeeooa?aioee?oaiinoe ooieoeee.
Anee z=f(x,y) eiaao a oi/ea ?(o,o) iai?a?uaiua /anoiua i?iecaiaeiua, oi
iia aeeooa?aioee?oaia a yoie oi/ea, o.a. iia eiaao iieiue
aeeooa?aioeeae.
Iieiue aeeooa?aioeeae aeey ooieoeee ianeieueeeo ia?aiaiiuo.
Aeey ooieoeee iiiaeo ia?aiaiiue iieiue aeeooa?aioeeae ii?aaeaeyaony
aiaeiae/ii, i?e yoii:
u=f(x,y,z,…,t)
du=(u/(x(dx+(u/(y(dy+(u/(z(dz+…+(u/(t(dt
I?eiaiaiea iieiiai aeeooa?aioeeaea aeey i?eaeeaeaiiuo au/eneaiee.
Ionoue caaeaia ooieoeey z=f(x,y) ?anniio?ei aa iieiia i?e?auaiea.
(z=f(x+(x,y+(y) – f(x,y)
I?e iaeuo (o e (o ( (z(dz (
f(x+(x,y+(y) – f(x,y) ( (z/x(((x+(z/(y(dy(
f(x+(x,y+(y)( f(x,y)+(z/(x(dx+(z/(y(dy — oi?ioea aeey i?eaeeaeaiiuo
au/eneaiee.
Yoa oi?ioea iicaieyao au/eneyoue i?eaeeaeaiiia cia/aiea ooieoeee a oi/ea
?1 ii ecaanoiiio aa a oi/ea ? e cia/aieai aa /anoiuo i?iecaiaeiuo a
oi/ea ?. *ai iaiueoa (o e (o, oai iaiueoa iia?aoiinoue.
Aeeooa?aioee?iaaiea neiaeiuo ooieoeee.
Ii?. Ia?aiaiiay z=z(t) – iacuaaaony neiaeiie ooieoeeae ia?aiaiiie t,
anee iia ii?aaeaeyaony ?aaainoaii:
z=z(t)=f[x(t),y(t)] – neiaeiay ooieoeey io t.
Oai?aia. Anee ooieoeey z=f(x,y) aeeooa?aioee?oaia a oi/ea ?(o, o), a
ooieoeee x=x(t) e y=y(t) aeeooa?aioee?oaiu a nnioaaonoaothuae oi/ea t,
oi neiaeiay ooieoeey z=z(t) oaeaea aeeooa?aioee?oaia a oi/ea t e aa
i?iecaiaeiay ii?aaeaeyaony ?aaainoaii:
dz/dt = (z/(x(dx/dt+ (x/(y(dy/dt [**]
Aeie-ai: Aeaaeei ia?aiaiiie t i?e?auaiea (t, i?e yoii o=o(t) iieo/eo
i?e?auaiea (o, a o=o(t) ( (o, a ?acoeueoaoa ia?aiaiiay z=f(x,y) iieo/eo
i?e?auaiea (z, o.e. z(o,o) – aeeooa?aioee?oaiay ooieoeey, oi yoi
i?e?auaiea iiaeao auoue i?aaenoaaeaii a aeaea:
(z=(z/(x((x + (z/(y((y + (
?acaeaeei ia (t e ia?aeaeai e i?aaeaeo
Lim((t(0)(z/(t = (z/(x(Lim((t(0)(x/(t +
+ (z/(y(Lim((t(0)(y/(t + Lim((t(0)(/(t
dz/dt = (z/(x(dx/dt + (z/(y(dy/dt + Lim((t(0) (/(((/(t ( 0
(=((x2+(y2(
Lim((t(0)(/(=0 – ii ii?aaeaeaieth aeeooa?aioeeaea.
Lim((t(0)(/(t = Lim((t(0)(((x/(t)2+((y/(t)2(=
=((dx/dt)2+(dy/dt)2(((
Oi?ioea [**] aeieacaia.
?anniio?ei /anoiue neo/ae neiaeiie ooieoeee:
z= f[x,y(x)] = z(x)
a o-ea [**] aianoi t(o, iieo/ei
dz/dx= (z/(x(dx/dx+ (z/(y(dy/dx
dz/dx= (z/(x+ (z/(y(dy/dx [***]
Oi?ioea [**] ?ani?ino?aiyaony ia neiaeiua ooieoeee aieueoaai /enea
ia?aiaiiuo.
Ionoue z=f(x,y), aaea x=x(r,s,..t), y=y(r,s,..,t) ( z=z(r,s,..,t) –
ceiaeiay ooieoeey.
I?e yoii oi?ioea [**] i?eieiaao aeae:
(z/(r=(z/(x((x/(r+(x/(y((y/(r
(z/(s=(z/(x((x/(s+ (z/(y((y/(s [****]
Eaeoeey ?3
Aeeooa?aioee?iaaiea ooieoeee, caaeaiiuo iayaii.
Ii?. Ooieoeey z=f(x,y) iac. Caaeaiiie iayaii, anee iia ii?aaeaeaia
?aaainoaii, ia?ac?aoaiiui ioiineoaeueii z .
F(x,y,z)=0
x+y+z=ez – yoi ?aaainoai caaeaai iaeioi?oth ooieoeeth z=f(x,y), eioi?oth
iaeuecy au?aceoue a iieiii aeaea.
x2+y2+z2=0 – ia caaeaao ieeaeie ooieoeee.
Oai?aia: Anee o-y F(x,y,z) ( iai?a?uaia a o. ?0(x0,y0,z0) e aa
i?iecaiaeiay ii z Fz(x,y,z)(0, oi ?aaainoai F(x,y,z)=0 iaeiicia/ii
ii?aaeaeyao a iayaiii aeaea ooieoeeth z=f(x,y), i?e yoii yoa ooieoeey
aeeooa?aioee?oaia e aa i?iecaiaeiay iaoiaeeony ii oi?ioeai:
(z/(x=( F(x(x,y,z)/F(z(x,y,z)
(z/(y=(F(z (x,y,z)/F(y(x,y,z)
Aeie-ai: Iaeaeai iieiue aeeooa?aioeeae ooieoeee
dF(x,y,z)=(F/(x*dx+(F/(y*dy+(F/(x*dz
F(x0,y0,z0)=0(dF=0(
(F/(x*dx+(F/(y*dy+(F/(x*dz=0
dz=(((F/(x)/((F/(z)*dx(((F/(y)/((F/(z)*dy (*)
N ae?oaie noi?iiu:
z=f(x,y), dz=(z/(x*dx+(z/(y*dy (**)
N?aaieaay (*) e(**) (
(z/(x=( F(x(x,y,z)/F(z(x,y,z)
(z/(y=(F(z (x,y,z)/F(y(x,y,z)
*anoiua i?iecaiaeiua aunoaai ii?yaeea.
Ionoue caaeaia ooieoeey 2o ia?aiaiiuo z=f(x,y),iaeaeai aa /anoiua
i?iecaiaeiua.
(z/(x=f(x(x,y)
(z/(y=f(y(x,y)
A iauai neo/aa, yoe i?iecaiaeiua oaeaea yaeythony ooieoeeyie 2o e
iiaeii eneaoue eo /anoiua i?iecaiaeiua. I?e yoii iieo/aai /aniua
i?iecaiaeiua 2-iai e aieaa ii?yaeeia.I?iecaiaeiua, a eioi?uo
aeeooa?aioee?iaaiea i?iecaiaeeony ii ?aciui ia?aiaiiui, iacuaathony
niaoaiiuie.
Oai?aia: I iacaaeneiinoe /aniuo i?iecaiaeiuo io ii?yaeea
(iineaaeiaaoaeueiinoe) aeeooa?aioee?iaaiey.
Aeaa niaoaiiua /anoiua ?iecaiaeiua iaeiiai ii?yaeea, ioee/athueany
oieueei ii?yaeeii aeeo-y ?aaiu.
(2z/(x(y=(2z/(y(x – a neaaenoaee yoiai, i?e iaicia/aiee niaoaiiuo
/anoiuo i?iecaiaeiuo iineaaeiaaoaeueiinoue aeeo-y ia oeacuaaaony.
(nz/(xn-2(y2
Yeno?aioiu ooieoeee 2oo ia?aiaiiuo.
?anniio?ei ooieoeeth 2o ia?aiaiiuo z=f(x,y) a iaeanoe Ae, ionoue
?0(x0,y0) – aioo?aiiyy oi/ea yoie iaeanoe.
Ii?. Oi/ea ?0 iac. Oi/eie max ooieoeee, anee a iaeioi?ie ie?aniinoe
yoie oi/ee auiieiyaony ia?aaainoai:
f(x,y)0, oi a o. ?0 nou. yeno?aioi.
I?e yoii anee r>0 ?0 (min; r
Нашли опечатку? Выделите и нажмите CTRL+Enter
