Nazariyai sathho

Nazariyai sathho yak fasli geometriyai differensiali ast, ki khosiyati sathhoro meomuzad. Yake az vazifahoi asosii Nazariyai sathho chen namuda­ni buzurgihoi gunogun dar sathho meboshad. Majmui dalelhoe, ki dar natijai in chenkunino ba dast meoyad, geometriyai dokhilii sathhoro tashkil medihad. Mafhumhoi darozii khat, kunji bayni du samt, masohati soha, inchunin khathoi geodezi, kajii geodezii khat va gayra mafhuhnoi geometriyai dokhiliand. Geometriyai dokhiliro shakli kvadratii asosii yakumi sath

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ds² = Edu² +  2Fdudv + Gdv² (1)

E = g   , F = r _u r _v, G= r , r= r{u, r)

— radius-vektori nuqtai tagyiryobandai sath, i, v koordinatahoi kaqkhattai in nuqta muayyan mekunad, ki kvadrati differeisialii kamoni khat dar sath meboshad.

Sokhti fazoii atrofi nuqta dar sath tavassuti shakli (formai) kvadratii asosii duyum

2 h = Ldu³ + 2Mdudv + Ndv²;           (2)

ki dar on L=r_uun, M =g _uvn, N=

r_uun,

vektori vooidii normal ba sato meboshand, tadqiq, karda meshavad. Agar buzurgihoi beintiho khurdi tartibashon nisbat ba tartibi du, dv balandtar ba nazar girifta nashavad, buzurgii h dar formulai (2) masofa az nuqtai M^r(u+ du_b v+dv) to hamvorii rasanda u dar nuqtai M(i, i) meboshad; in masofa vobasta ba on ki nuqtai M dar kadom tarafi hamvorii u mekhobad, bo alomati + yo — qabul karda meshavad. Agar shakli (2) alomati muayyan doshta boshad, sath dar atrofi kofi khurdi nuqtai M dar yak tarafi hamvorii u mekhobad va nuqtai M nuqtai e l l i p s i  nomida meshavad. Agar shakli (2) alomathoi tagyiryobanda doshta boshad, sath dar atrofi nuqtai M dar har du ta­rafi hamvorii u joy megirad va nuqtai M nuqtai giperboli nomida meshavad. Agar alomati shakli (2) muayyan boshad, vale baze azoho qimathoi sifriro qabul kunand (hangomi yakbora ba sifr barobar nabudani du va dv) nuqtai M nuqtai paraboli nomida me­shavad ( yake az misolhoi sokhti sath dar atrofi nuqtai paraboli nishon doda shudaast).

Baroi sahehtar tadqiq namudani shakli fazoii sath mafhumi kajii normalistifoda meshavad. Dar mavridi bo hamvorii normal buridani sath burishi normal hosil meshavad. Kajii burishi normal kajii normali sath nom dorad. Agar samti hamvorii buranda ba samti muvofiq  oyakajii normal (k) az rui formulai

hisob karda meshavad. Qimathoi ekstremalii kajii normal k₁, k₂ k a j i h o i asosi, samthoi muvofiq dar sath samthoi asosi nom dorand. Hosili zarbi kajihoi asosi K = k • k₂-ro kajii p u r r a  yo kajii Gauss mvnomand va onro az rui formulai Gauss

(3)

hisob mekunand. Kajii purra darajai kajshavii sathhoro dar atrofi nuqtai dodashuda nishon medihad. Masalan, agar K > 0 boshad — nuqtai ellissi K < 0 boshad — nuqtai giperboli va K = 0 boshad — nuqtai pa­raboli hosil meshavad.

Agar bayni nuqtahoi du sath chunin muvofiqiti yakkimmata mavjud boshad, ki darozii khathoi muvofiq dar in sathho barobar boshand, in sathhoro sathhoi izometri menomand. Geometriyai dokhilii sathhoi izometri yak khel boshad ham, va­le sokhti fazoii onho az ham farq mekunand.

Yake az mafhumhoi muhimi Nazariyai sathho kajii miyona ba hisob meravad, ki ba nisfi hosili jami kajihoi aso­si  barobar meboshad. Agar kajii miyonai sath dar har nuqtaash ba sifr barobar boshad, sathro mi- nimali menomand, ki on dar fizi­ka va digar sohahoi ilm ahamiyati kalon dorad.

Digar masalai muhami Nazariyai sathho prob­lemai qatshavii sathho meboshad. Dar in bobat tadqiqoti olimi soveti N. V. Efimov jolibi diqqat ast.

Ad.: P o g o r e l o v A. V., Differen­tsialnaya geometriya, v izd.. M., 1974.