Simvol Le vi Chiviti matematichnij simvol sho vikoristovuyetsya v tenzornomu analizi Nazvanij na chest italijskogo matematika Tullio Levi Chiviti Poznachayetsya e i j k displaystyle varepsilon ijk Tut navedeno simvol dlya trivimirnogo prostoru dlya inshih rozmirnostej zminyuyetsya kilkist indeksiv div nizhche Inshi nazvi Absolyutno antisimetrichnij odinichnij tenzor Povnistyu antisimetrichnij odinichnij tenzor Absolyutno kososimetrichnij ob yekt Tenzor Levi Chiviti simvol Levi Chiviti ye komponentnim zapisom cogo tenzoru Kososimetrichnij simvol Kronekera danij termin vikoristovuvavsya v pidruchniku z tenzornogo chislennya Akivisa i Goldberga Zmist 1 Oznachennya 2 Odinichnij antisimetrichnij tenzor v dovilnij sistemi koordinat 3 Oriyentaciya mnogovida ta dzerkalni vidobrazhennya 4 Tenzornij dobutok odinichnogo antisimetrichnogo tenzora na sebe 5 Kovariantna pohidna odinichnogo antisimetrichnogo tenzora 6 Tenzori metrichnoyi matroshki 7 Zastosuvannya metrichnoyi matroshki 7 1 Poslidovna zgortka dobutku odinichnogo antisimetrichnogo tenzora na sebe 7 2 Virazhennya zovnishnogo dobutku cherez tenzor metrichnoyi matroshki 8 PosilannyaOznachennyared nbsp Zobrazhennya simvolu Levi Chiviti U trivimirnomu prostori u pravomu ortonormovanomu bazisi abo vzagali u pravomu bazisi z odinichnim viznachnikom metriki simvol Levi Chiviti oznachayetsya nastupnim chinom e i j k 1 P i j k 1 1 P i j k 1 0 i j j k k i displaystyle varepsilon ijk begin cases 1 amp P i j k 1 1 amp P i j k 1 0 amp i j bigvee j k bigvee k i end cases nbsp tobto dlya parnoyi perestanovki P i j k dorivnyuye 1 dlya trijok 1 2 3 2 3 1 3 1 2 dlya neparnoyi perestanovki P i j k dorivnyuye 1 dlya trijok 3 2 1 1 3 2 2 1 3 a v inshih vipadkah dorivnyuye nulyu pri povtorenni Dlya komponent e i j k displaystyle varepsilon ijk nbsp u livomu bazisi berutsya protilezhni chisla Odinichnij antisimetrichnij tenzor v dovilnij sistemi koordinatred Perejdemo vid specialnoyi sistemi koordinat u 1 u 2 u n displaystyle hat u 1 hat u 2 dots hat u n nbsp rozglyanutoyi v poperednomu punkti do dovilnoyi u 1 u 2 u n displaystyle u 1 u 2 dots u n nbsp Metrichnij tenzor zapishetsya 8 g i j u k u i u l u j g k l u k u i u l u j d k l k 1 n u k u i u k u j A T A i j displaystyle 8 qquad g ij partial hat u k over partial u i partial hat u l over partial u j hat g kl partial hat u k over partial u i partial hat u l over partial u j delta kl sum k 1 n partial hat u k over partial u i partial hat u k over partial u j A T A ij nbsp de buvoyu A displaystyle A nbsp poznacheno matricyu perehodu iz dovilnoyi sistemi koordinat do specialnoyi 9 A i j u i u j displaystyle 9 qquad A ij partial hat u i over partial u j nbsp Yak vidno z formuli 7 viznachnik metrichnogo tenzora dorivnyuye kvadratu viznachnika matrici A displaystyle A nbsp 10 g det g i j det A T A det A 2 displaystyle 10 qquad g det g ij det A T A left det A right 2 nbsp Budemo rozglyadati tilki taki zamini sistemi koordinat yaki ne zminyuyut oriyentaciyu tobto viznachnik matrici A displaystyle A nbsp dodatnij 11 det A gt 0 det A g displaystyle 11 qquad det A gt 0 qquad det A sqrt g nbsp Teper rozglyanemo yak zminyatsya komponenti odinichnogo antisimetrichnogo tenzora pri perehodi v dovilnu sistemu koordinat 12 e 12 n u i 1 u 1 u i 1 u 1 u i 1 u 1 e i 1 i 2 i n det A g displaystyle 12 qquad varepsilon 12 dots n partial hat u i 1 over partial u 1 partial hat u i 1 over partial u 1 cdots partial hat u i 1 over partial u 1 hat varepsilon i 1 i 2 dots i n det A sqrt g nbsp Kovariantni koordinati virazhayutsya cherez simvol Levi Chivita tak 13 e i 1 i 2 i n g e i 1 i 2 i n displaystyle 13 qquad varepsilon i 1 i 2 dots i n sqrt g hat varepsilon i 1 i 2 dots i n nbsp a kontravariantni koordinati oskilki peretvorennya kontravariantnih koordinat zdijsnyuyetsya cherez obernenu do A displaystyle A nbsp matricyu tak 14 e i 1 i 2 i n 1 g e i 1 i 2 i n displaystyle 14 qquad varepsilon i 1 i 2 dots i n 1 over sqrt g hat varepsilon i 1 i 2 dots i n nbsp Oriyentaciya mnogovida ta dzerkalni vidobrazhennyared Dlya danoyi tochki mnogovida mozhna bagatma sposobami vibrati specialnu sistemu koordinat taku sho g i j d i j displaystyle hat g ij delta ij nbsp Napriklad mayuchi odnu z nih mozhna zdijsnyuvati nad neyu taki ortogonalni peretvorennya yak povoroti i dzerkalni vidobrazhennya zminu napryamku odniyeyi z kordinatnih osej Yaksho mi mayemo dvi taki dzerkalni sistemi koordinat to matricya perehodu mizh nimi bude vid yemnoyu Fomuli 13 i 14 budut spravedlivi tilki dlya odniyeyi iz cih sistem koordinat nazvemo taku sistemu koordinat pravoyu Dlya inshoyi livoyi sistemi koordinat obidvi ci formuli budut zi znakom minus Otzhe isnuye dilema yaku iz dvoh sistem koordinat vzyati za pravu a otzhe z yakim znakom zadavati odinichnij antisimetrichnij tenzor Mozhna napriklad zdijsniti dzerkalne vidobrazhennya a potim v novij sistemi koordinat yaka ranishe bula livoyu vznachiti odinichnij metrichnij tenzor formulami 13 i 14 Tobto teper formalno odinichnij metrichnij tenzor ne zminivsya Vnaslidok ciyeyi dilemi vsi tenzori ta skalyari yaki mozhna utvoriti zgortkoyu z odinichnim antisimetrichnim tenzorom divno povodyatsya pri dzerkalnih vidobrazhennyah V fizici isnuye termin aksialnogo vektora yakij utvoryuyetsya v 3 vimirnomu prostori pri vektornomu dobutku zvichajnih vektoriv w a b displaystyle boldsymbol omega mathbf a times mathbf b nbsp sho mozhna zapisati w i e i j k a j b k displaystyle omega i varepsilon ijk a j b k nbsp Rozglyanemo dzerkalne vidobrazhennya vidnosno ploshini a b displaystyle mathbf a mathbf b nbsp Vektori a displaystyle mathbf a nbsp i b displaystyle mathbf b nbsp pri comu ne zminyatsya odinichnij metrichnij tenzor tezh formalno pokomponentno ne zminitsya A otzhe ne zminitsya i vektor w displaystyle boldsymbol omega nbsp yakij ortogonalnij do ploshini dzerkala Tenzornij dobutok odinichnogo antisimetrichnogo tenzora na sebered Vizmemo dva nabori iz n displaystyle n nbsp indeksiv i 1 i 2 i n displaystyle i 1 i 2 dots i n nbsp ta j 1 j 2 j n displaystyle j 1 j 2 dots j n nbsp i rozglyanemo funkciyu vid komponentiv metrichnogo tenzora yaka dorivnyuye nastupnomu viznachniku 15 f i 1 i 2 i n j 1 j 2 j n g i 1 j 1 g i 1 j 2 g i 1 j n g i 2 j 1 g i 2 j 2 g i 2 j n g i n j 1 g i n j 2 g i n j n displaystyle 15 qquad f i 1 i 2 dots i n j 1 j 2 dots j n begin vmatrix g i 1 j 1 amp g i 1 j 2 amp cdots amp g i 1 j n g i 2 j 1 amp g i 2 j 2 amp cdots amp g i 2 j n cdots amp cdots amp cdots amp cdots g i n j 1 amp g i n j 2 amp cdots amp g i n j n end vmatrix nbsp Velichina f i 1 i 2 i n j 1 j 2 j n displaystyle f i 1 i 2 dots i n j 1 j 2 dots j n nbsp ye tenzorom oskilki utvoryuyetsya z metrichnogo tenzora operaciyami tenzornogo dobutku i dodavannyam vidnimannyam tenzoriv Dali iz vlastivosti viznachnika po perestanovci ryadkiv i stovpciv robimo visnovok sho tenzor f displaystyle f nbsp antisimetrichnij po naboru indeksiv i 1 i 2 i n displaystyle i 1 i 2 dots i n nbsp perestanovka ryadkiv i okremo po naboru indeksiv j 1 j 2 j n displaystyle j 1 j 2 dots j n nbsp perestanovka stovpciv Takim chinom tenzor f i 1 i 2 i n j 1 j 2 j n displaystyle f i 1 i 2 dots i n j 1 j 2 dots j n nbsp mi mozhemo zapisati cherez dobutok dvoh simvoliv Levi Chivita 16 f i 1 i 2 i n j 1 j 2 j n K e i 1 i 2 i n e j 1 j 2 j n displaystyle 16 qquad f i 1 i 2 dots i n j 1 j 2 dots j n K hat varepsilon i 1 i 2 dots i n hat varepsilon j 1 j 2 dots j n nbsp Konstantu K displaystyle K nbsp znahodimo pidstavivshi chisla 1 2 n displaystyle 1 2 dots n nbsp zamist indeksiv i 1 i 2 i n displaystyle i 1 i 2 dots i n nbsp j 1 j 2 j n displaystyle j 1 j 2 dots j n nbsp 17 f 12 n 12 n g 11 g 12 g 1 n g 21 g 22 g 2 n g n 1 g n 2 g n n det g i j g displaystyle 17 qquad f 12 dots n12 dots n begin vmatrix g 11 amp g 12 amp cdots amp g 1n g 21 amp g 22 amp cdots amp g 2n cdots amp cdots amp cdots amp cdots g n1 amp g n2 amp cdots amp g nn end vmatrix det g ij g nbsp Vrahovuyuchi 13 iz formul 15 17 znahodimo sho tenzornij dobutok odinichnogo antisimetrichnogo tenzora na sebe dorivnyuye viznachniku 18 e i 1 i 2 i n e j 1 j 2 j n g i 1 j 1 g i 1 j 2 g i 1 j n g i 2 j 1 g i 2 j 2 g i 2 j n g i n j 1 g i n j 2 g i n j n displaystyle 18 qquad varepsilon i 1 i 2 dots i n varepsilon j 1 j 2 dots j n begin vmatrix g i 1 j 1 amp g i 1 j 2 amp cdots amp g i 1 j n g i 2 j 1 amp g i 2 j 2 amp cdots amp g i 2 j n cdots amp cdots amp cdots amp cdots g i n j 1 amp g i n j 2 amp cdots amp g i n j n end vmatrix nbsp Kovariantna pohidna odinichnogo antisimetrichnogo tenzorared Zgidno z oznachennyam kovariantnoyi pohidnoyi divitsya stattyu Diferencialna geometriya mayemo 19 k e i 1 i 2 i n k e i 1 i 2 i n G k i 1 s e s i 2 i n G k i 2 s e i 1 s i n G k i n s e i 1 i 2 s displaystyle 19 qquad nabla k varepsilon i 1 i 2 dots i n partial k varepsilon i 1 i 2 dots i n Gamma ki 1 s varepsilon si 2 dots i n Gamma ki 2 s varepsilon i 1 s dots i n dots Gamma ki n s varepsilon i 1 i 2 dots s nbsp Pidstavimo syudi virazi komponent tenzora za formuloyu 13 Chastinna pohidna dorivnyuye 20 k e i 1 i 2 i n k g e i 1 i 2 i n displaystyle 20 qquad partial k varepsilon i 1 i 2 dots i n partial k sqrt g hat varepsilon i 1 i 2 dots i n nbsp Rozglyanemo formulu 19 u dvoh vipadkah Pershij vipadok koli sered indeksiv i 1 i 2 i n displaystyle i 1 i 2 dots i n nbsp ye hocha b dva odnakovih napriklad i 1 i 2 i displaystyle i 1 i 2 i nbsp Todi chastinna pohidna za formuloyu 20 dorivnyuye nulyu a iz vidyemnikiv formuli 19 tilki dva pershih mozhut buti nenulovi tomu mayemo 21 k e i i i n G k i s e s i i n G k i s e i s i n G k i s e s i i n e i s i n 0 displaystyle 21 qquad nabla k varepsilon ii dots i n Gamma ki s varepsilon si dots i n Gamma ki s varepsilon is dots i n Gamma ki s varepsilon si dots i n varepsilon is dots i n 0 nbsp oskilki tenzor e i 1 i 2 i n displaystyle varepsilon i 1 i 2 dots i n nbsp antisimetrichnij po pershih dvoh indeksah Teper rozglyanemo drugij vipadok koli vsi indeksi i 1 i 2 i n displaystyle i 1 i 2 dots i n nbsp rizni U kozhnomu vid yemniku formuli 19 vidbuvayetsya dodavannya za indeksom s displaystyle s nbsp ale v comu dodavanni vidminnim vid nulya ye lishe odin v yakomu indeks s displaystyle s nbsp dorivnyuye nedostayuchomu indeksu z naboru i 1 i 2 i n displaystyle i 1 i 2 dots i n nbsp pri tenzori e displaystyle varepsilon nbsp 21 k e i 1 i 2 i n k e i 1 i 2 i n G k i 1 i 1 e i 1 i 2 i n G k i 2 i 2 e i 1 i 2 i n G k i n i n e i 1 i 2 i n displaystyle 21 qquad nabla k varepsilon i 1 i 2 dots i n partial k varepsilon i 1 i 2 dots i n Gamma ki 1 i 1 varepsilon i 1 i 2 dots i n Gamma ki 2 i 2 varepsilon i 1 i 2 dots i n dots Gamma ki n i n varepsilon i 1 i 2 dots i n nbsp k g G k i i g e i 1 i 2 i n displaystyle qquad left partial k sqrt g Gamma ki i sqrt g right hat varepsilon i 1 i 2 dots i n nbsp Viraz v duzhkah ostannogo virazu dorivnyuye nulyu divitsya zgortku simvoliv Kristofelya v statti Prosti obchislennya diferencialnoyi geometriyi Otzhe v usih vipadkah kovariantna pohidna odinichnogo antisimetrichnogo tenzora dorivnyuye nulyu 22 k e i 1 i 2 i n 0 displaystyle 22 qquad nabla k varepsilon i 1 i 2 dots i n 0 nbsp Tenzori metrichnoyi matroshkired U formuli 18 figuruye dosit cikava konstrukciya z metrichnogo tenzora u viglyadi viznachnika matrici n displaystyle n nbsp go poryadku Dlya doslidzhennya vlastivostej ciyeyi konstrukciyi docilno rozglyanuti taku neskinchennu seriyu tenzoriv zi shoraz bilshoyu kilkistyu indeksiv 23 g i j g i j g i 1 i 2 j 1 j 2 g i 1 j 1 g i 1 j 2 g i 2 j 1 g i 2 j 2 g i 1 j 1 g i 2 j 2 g i 1 j 2 g i 2 j 1 g i 1 i 2 i 3 j 1 j 2 j 3 g i 1 j 1 g i 1 j 2 g i 1 j 3 g i 2 j 1 g i 2 j 2 g i 2 j 3 g i 3 j 1 g i 3 j 2 g i 3 j 3 displaystyle 23 qquad begin matrix g ij begin vmatrix g ij end vmatrix g i 1 i 2 j 1 j 2 begin vmatrix g i 1 j 1 amp g i 1 j 2 g i 2 j 1 amp g i 2 j 2 end vmatrix g i 1 j 1 g i 2 j 2 g i 1 j 2 g i 2 j 1 g i 1 i 2 i 3 j 1 j 2 j 3 begin vmatrix g i 1 j 1 amp g i 1 j 2 amp g i 1 j 3 g i 2 j 1 amp g i 2 j 2 amp g i 2 j 3 g i 3 j 1 amp g i 3 j 2 amp g i 3 j 3 end vmatrix cdots end matrix nbsp Cyu seriyu tenzoriv i nazvemo metrichnoyu matroshkoyu Kozhen z cih tenzoriv maye dvi grupi indeksiv prichomu tenzor antisimetrichnij pri perestanovci indeksiv v mezhah odnoyi grupi oskilki viznachnik zminyuye znak pri perestanovci ryadkiv chi stovciv matrici i tenzor simetrichnij stosovno perestanovki cih dvoh grup indeksiv mizh soboyu 24 g i 1 i 2 i m j 1 j 2 j m g j 1 j 2 j m i 1 i 2 i m displaystyle 24 qquad g i 1 i 2 dots i m j 1 j 2 dots j m g j 1 j 2 dots j m i 1 i 2 dots i m nbsp oskilki viznachnik matrici ne zminyuyetsya pri transponuvanni Ochevidno sho tilki pershi n displaystyle n nbsp n displaystyle n nbsp rozmirnist mnogovida z ciyeyi seriyi tenoriv vidminni vid nulya Yaksho m gt n displaystyle m gt n nbsp to 25 g i 1 i 2 i m j 1 j 2 j m 0 m gt n displaystyle 25 qquad g i 1 i 2 dots i m j 1 j 2 dots j m 0 qquad m gt n nbsp oskilki sered indeksiv i 1 i 2 i m displaystyle i 1 i 2 dots i m nbsp obov yazkovo znajdetsya dva odnakovih Cikavo yak zminyatsya formuli 23 yaksho mi pidnimemo indeksi odniyeyi z grup Pochnemo z pidnimannya odnogo indeksa pershogo 26 g i 2 i m j 1 j m i 1 g i 1 s g s i 2 i m j 1 j m s g i 1 s g s j 1 g s j 2 g s j m g i 2 j 1 g i 2 j 2 g i 2 j m g i m j 1 g i m j 2 g i m j m displaystyle 26 qquad g i 2 dots i m j 1 dots j m i 1 g i 1 s g si 2 dots i m j 1 dots j m sum s g i 1 s begin vmatrix g sj 1 amp g sj 2 amp cdots amp g sj m g i 2 j 1 amp g i 2 j 2 amp cdots amp g i 2 j m cdots amp cdots amp cdots amp cdots g i m j 1 amp g i m j 2 amp cdots amp g i m j m end vmatrix nbsp d j 1 i 1 d j 2 i 1 d j m i 1 g i 2 j 1 g i 2 j 2 g i 2 j m g i m j 1 g i m j 2 g i m j m displaystyle qquad begin vmatrix delta j 1 i 1 amp delta j 2 i 1 amp cdots amp delta j m i 1 g i 2 j 1 amp g i 2 j 2 amp cdots amp g i 2 j m cdots amp cdots amp cdots amp cdots g i m j 1 amp g i m j 2 amp cdots amp g i m j m end vmatrix nbsp Ostannyu rivnist mi zapisali oskilki vnaslidok linijnosti viznachnika po pershomu ryadku mi mozhemo znak sumi vnesti v pershij ryadok matrici Poslidovno pidijmayuchi reshtu m 1 displaystyle m 1 nbsp indeksiv pershoyi grupi prihodimo do formuli 27 g j 1 j 2 j m i 1 i 2 i m d j 1 i 1 d j 2 i 1 d j m i 1 d j 1 i 2 d j 2 i 2 d j m i 2 d j 1 i m d j 2 i m d j m i m displaystyle 27 qquad g j 1 j 2 dots j m i 1 i 2 dots i m begin vmatrix delta j 1 i 1 amp delta j 2 i 1 amp cdots amp delta j m i 1 delta j 1 i 2 amp delta j 2 i 2 amp cdots amp delta j m i 2 cdots amp cdots amp cdots amp cdots delta j 1 i m amp delta j 2 i m amp cdots amp delta j m i m end vmatrix nbsp U formuli 27 mi zapisali dvi grupi indeksiv odnu pid odnoyu ce ne viklikaye dvoznachnosti oskilki tenzor matroshki simetrichnij shodo perestanovki grup indeksiv formula 24 Rozglyanemo operaciyu zgortki tenzora 27 Zgortka za dvoma indeksami v mezhah odniyeyi grupi daye nul vnaslidok antisimetriyi Rozglyanemo zgortku za dvoma indeksami z riznih grup napriklad zgornemo tenzor 27 za pershim verhnim i pershim nizhnim indeksami 28 g s j 2 j m s i 2 i m s d s s d j 2 s d j m s d s i 2 d j 2 i 2 d j m i 2 d s i m d j 2 i m d j m i m s i 2 j 2 i m j m 1 0 0 0 d j 2 i 2 d j m i 2 0 d j 2 i m d j m i m displaystyle 28 qquad g sj 2 dots j m si 2 dots i m sum s begin vmatrix delta s s amp delta j 2 s amp cdots amp delta j m s delta s i 2 amp delta j 2 i 2 amp cdots amp delta j m i 2 cdots amp cdots amp cdots amp cdots delta s i m amp delta j 2 i m amp cdots amp delta j m i m end vmatrix sum s neq i 2 j 2 dots i m j m begin vmatrix 1 amp 0 amp cdots amp 0 0 amp delta j 2 i 2 amp cdots amp delta j m i 2 cdots amp cdots amp cdots amp cdots 0 amp delta j 2 i m amp cdots amp delta j m i m end vmatrix nbsp n m 1 g j 2 j m i 2 i m displaystyle qquad n m 1 g j 2 dots j m i 2 dots i m nbsp Zastosuvannya metrichnoyi matroshkired Poslidovna zgortka dobutku odinichnogo antisimetrichnogo tenzora na sebered Iz formul 18 i 28 oderzhuyemo zv yazani indeksi za yakimi ide zgortka poznacheni tut bukvoyu s displaystyle s nbsp z pidindeksami 29 e i 1 i 2 i n e j 1 j 2 j n g i 1 i 2 i n j 1 j 2 j n displaystyle 29 qquad varepsilon i 1 i 2 dots i n varepsilon j 1 j 2 dots j n g i 1 i 2 dots i n j 1 j 2 dots j n nbsp e s 1 i 2 i n e s 1 j 2 j n g i 2 i n j 2 j n 1 g i 2 i n j 2 j n displaystyle qquad varepsilon s 1 i 2 dots i n varepsilon s 1 j 2 dots j n g i 2 dots i n j 2 dots j n 1 g i 2 dots i n j 2 dots j n nbsp e s 1 s 2 i 3 i n e s 1 s 2 j 3 j n 2 g i 3 i n j 3 j n displaystyle qquad varepsilon s 1 s 2 i 3 dots i n varepsilon s 1 s 2 j 3 dots j n 2 g i 3 dots i n j 3 dots j n nbsp displaystyle qquad cdots cdots cdots nbsp e s 1 s 2 s n 1 i n e s 1 s 2 s n 1 j n n 1 g i n j n displaystyle qquad varepsilon s 1 s 2 dots s n 1 i n varepsilon s 1 s 2 dots s n 1 j n n 1 g i n j n nbsp e s 1 s 2 s n e s 1 s 2 s n n displaystyle varepsilon s 1 s 2 dots s n varepsilon s 1 s 2 dots s n n nbsp Virazhennya zovnishnogo dobutku cherez tenzor metrichnoyi matroshkired Dlya zgortki dvoh vektoriv z matroshkoyu chetvertogo rangu mayemo 30 g i j k l a k b l d i k d j k d i l d j l a k b l d i k d j l d j k d i l a k b l a i b j a j b i displaystyle 30 qquad g ij kl a k b l begin vmatrix delta i k amp delta j k delta i l amp delta j l end vmatrix a k b l delta i k delta j l delta j k delta i l a k b l a i b j a j b i nbsp Analogichno zapishemo formulu dlya dobutku troh vektoriv 31 a b c i j k g i j k p q r a p b q c r displaystyle 31 qquad mathbf a wedge mathbf b wedge mathbf c ijk g ijk pqr a p b q c r nbsp Yaksho mi mayemo dva tenzora rangiv m 1 displaystyle m 1 nbsp i m 2 displaystyle m 2 nbsp vidpovidno to yihnij zovnishnij dobutok zapisuyetsya cherez zgortku cih tenzoriv z tenzorom mentrichnoyi matroshki rangu 2 m 1 m 2 displaystyle 2 m 1 m 2 nbsp 32 s t i 1 i 2 i m 1 j 1 j 2 j m 2 g i 1 i 2 i m 1 j 1 j 2 j m 2 s 1 s 2 s m 1 p 1 p 2 p m 2 s s 1 s 2 s m 1 t p 1 p 2 p m 2 displaystyle 32 qquad boldsymbol sigma wedge boldsymbol tau i 1 i 2 dots i m 1 j 1 j 2 dots j m 2 g i 1 i 2 dots i m 1 j 1 j 2 dots j m 2 s 1 s 2 dots s m 1 p 1 p 2 dots p m 2 sigma s 1 s 2 dots s m 1 tau p 1 p 2 dots p m 2 nbsp Posilannyared Hermann R ed Ricci and Levi Civita s tensor analysis papers 1975 Math Sci Press Brookline oznachennya simvolu div str 31 Charles W Misner Kip S Thorne John Archibald Wheeler Gravitation 1970 W H Freeman New York ISBN 0 7167 0344 0 Div paragraf 3 5 dlya obzoru zastosuvannya tenzoriv u zagalnij teoriyi vidnosnosti Rosijskij pereklad Ch Mizner K Torn Dzh Uiler Gravitaciya 1977 Moskva Mir Div za vkazivnikom Levi Chivity tenzor Dimitrienko Yu I Tenzornoe ischislenie M Vysshaya shkola 2001 575 s Otrimano z https uk wikipedia org w index php title Simvol Levi Chiviti amp oldid 17336596