U Vikipediyi ye statti pro inshi znachennya cogo termina Sfera znachennya Sfe ra vid grec sfaῖra kulya zamknuta poverhnya geometrichne misce tochok rivnoviddalenih vid danoyi tochki sho ye centrom sferi Sfera ye okremim vipadkom elipsoyida u yakogo vsi tri pivosi odnakovi SferaRadius r sferi Zmist 1 Vlastivosti 2 Rivnyannya 3 Zamknenij ob yem 4 Plosha poverhni 5 Ob yemi deyakih figur otrimanih vnaslidok kombinacij zi sferoyu 6 Tenzor Richchi ta skalyarna krivina sferi 7 Formuli 8 N vimirna sfera 9 Div takozh 10 Primitki 11 Dzherela 12 PosilannyaVlastivosti RedaguvatiVidrizok sho spoluchaye centr sferi z yiyi tochkoyu a takozh jogo dovzhina nazivayetsya radiusom vidrizok sho spoluchaye dvi tochki sferi hordoyu horda sho prohodit cherez centr sferi nazivayetsya yiyi diametrom Sferu mozhna rozglyadati takozh yak poverhnyu obertannya pivkola navkolo jogo diametra Chastina prostoru yaka obmezhena sferoyu i mistit yiyi centr nazivayetsya kuleyu Pereriz sferi dovilnoyu ploshinoyu ye kolo Vono nazivayetsya velikim koli ploshina prohodit cherez centr sferi vsi inshi pererizi ye malimi kolami U sferi najmensha plosha poverhni z pomizh vsih til sho zamikayut danij ob yem ta najbilshij zamknenij ob yem pri danij ploshi poverhni Z ciyeyi prichini sfera chasto zustrichayetsya u prirodi krapli vodi v nevagomosti planeti globuli i t in Ploshinu pryamu yaka maye zi sferoyu tilki odnu spilnu tochku nazivayut dotichnoyu ploshinoyu pryamoyu do sferi Yaksho dvi sferi mayut tilki odnu spilnu tochku govoryat sho voni dotikayutsya v cij tochci Rivnyannya RedaguvatiU analitichnij geometriyi sfera u dekartovij sistemi koordinat z koordinatami centru O x 0 y 0 z 0 displaystyle O x 0 y 0 z 0 i radiusom r r ye geometrichnim miscem usih tochok x y z displaystyle x y z sho opisuyetsya rivnyannyam x x 0 2 y y 0 2 z z 0 2 r 2 displaystyle x x 0 2 y y 0 2 z z 0 2 r 2 U sferichnij sistemi koordinat bud yaku tochku sferi mozhna podati yak x x 0 r sin 8 cos f displaystyle x x 0 r sin theta cos varphi y y 0 r sin 8 sin f 0 f lt 2 p 0 8 p displaystyle y y 0 r sin theta sin varphi qquad 0 leqslant varphi lt 2 pi 0 leqslant theta leqslant pi z z 0 r cos 8 displaystyle z z 0 r cos theta Sfera dovilnogo radiusu z centrom u pochatku koordinat zadayetsya diferencialnim rivnyannyam x d x y d y z d z 0 displaystyle x dx y dy z dz 0 Ce rivnyannya vidobrazhaye fakt sho vektori shvidkosti ta koordinat tochki sho ruhayetsya po poverhni sferi postijno ortogonalni odin do odnogo Krivina Gausa dlya sferi postijna i viznachayetsya yak 1 r 2 displaystyle frac 1 r 2 Kolo yake lezhit na sferi tak sho centri kola ta sferi zbigayutsya nazivayetsya velikim kolom sferi Veliki kola ye geodezichnimi liniyami na sferi bud yaki dvi z nih peretinayutsya v dvoh tochkah Inshimi slovami veliki kola sferi ye analogami pryamih na ploshini Vidstan mizh tochkami na sferi viznachayetsya yak dovzhina dugi velikogo kola sho prohodit cherez zadani tochki Kutu mizh pryamimi na ploshini vidpovidaye dvogrannij kut mizh ploshinami velikih kil Bagato teorem geometriyi na ploshini mayut misce i v sferichnoyi geometriyi isnuyut analogi teoremi sinusiv teoremi kosinusiv dlya sferichnih trikutnikiv U toj zhe chas isnuye chimalo vidminnostej napriklad v sferichnomu trikutniku suma kutiv zavzhdi bilshe 180 displaystyle 180 gradusiv do troh oznak rivnosti trikutnikiv dodayetsya chetverta yih rivnist po troh kutah u sferichnogo trikutnika mozhe buti dva i navit tri pryamih kuta napriklad u sferichnogo trikutnika utvorenogo ekvatorom i dvoma meridianami 0 displaystyle 0 circ ta 90 displaystyle 90 circ Zamknenij ob yem Redaguvati Cilindr opisanij navkolo sferiV trivimirnomu prostori ob yem vseredini sferi yakij ye ob yemom kuli ye V 4 3 p r 3 displaystyle V frac 4 3 pi r 3 de r r ce radius sferi Arhimed vpershe viviv cyu formulu koli pokazav sho ob yem vseredini sferi v dva razi bilshij za riznicyu v ob yemah vseredini sferi ta vseredini opisanogo cilindra u yakogo visota ta diametr dorivnyuyut diametru sferi 1 Ce tverdzhennya mozhna otrimati z principu Kaval yeri Cya formula takozh mozhe buti otrimana za dopomogoyu integralnogo obchislennya Pri kozhnomu zadanomu x x inkrementnij ob yem d V displaystyle delta V dorivnyuye dobutku ploshi poperechnogo pererizu kruga pri x x i jogo tovshini d x displaystyle delta x d V p y 2 d x displaystyle delta V approx pi y 2 cdot delta x Povnij ob yem dorivnyuye sumi vsih inkrementnih ob yemiv V p y 2 d x displaystyle V approx sum pi y 2 cdot delta x Dlya granici funkciyi koli dx nablizhayetsya do nulya 2 ce rivnyannya nabuvaye viglyadu V r r p y 2 d x displaystyle V int r r pi y 2 dx Dlya bud yakogo x x pryamokutnij trikutnik poyednuye x x y y ta r r iz pochatkom koordinat znachit yaksho zastosuvati teoremu Pifagora otrimayemo y 2 r 2 x 2 displaystyle y 2 r 2 x 2 Vikoristayemo pidstanovku V r r p r 2 x 2 d x displaystyle V int r r pi r 2 x 2 dx Ce mozhe buti obchislene shob otrimati nastupnij rezultat V p r 2 x x 3 3 r r p r 3 r 3 3 p r 3 r 3 3 4 3 p r 3 displaystyle V pi left r 2 x frac x 3 3 right r r pi left r 3 frac r 3 3 right pi left r 3 frac r 3 3 right frac 4 3 pi r 3 Alternativno cya formula mozhe buti znajdena z vikoristannyam sferichnih koordinat z elementom ob yemu en d V r 2 sin 8 d r d 8 d f displaystyle dV r 2 sin theta dr d theta d varphi i takim chinom V 0 2 p 0 p 0 r r 2 sin 8 d r d 8 d f 4 3 p r 3 displaystyle V int 0 2 pi int 0 pi int 0 r r 2 sin theta dr d theta d varphi frac 4 3 pi r 3 Dlya bilshosti praktichnih cilej ob yem vseredini sferi vpisanoyi v kub mozhe buti nablizhene do 52 4 displaystyle 52 4 ob yemu kuba cherez te sho V p 6 d 3 displaystyle V tfrac pi 6 d 3 de d d ye diametrom sferi i v toj zhe chas dovzhinoyu storoni kuba i p 6 0 5236 displaystyle tfrac pi 6 approx 0 5236 Napriklad sfera diametru 1 displaystyle 1 metr maye 52 4 displaystyle 52 4 ob yemu kuba z dovzhinoyu rebra 1 displaystyle 1 metr abo blizko 0 524 displaystyle 0 524 m3 Plosha poverhni RedaguvatiPlosha poverhni sferi radiusu r r S 4 p r 2 displaystyle S 4 pi r 2 Arhimed vpershe otrimav cyu formulu 3 z togo faktu sho proyekciya na bichnu poverhnyu opisanogo cilindra zberigaye ploshu 4 Inshij sposib otrimati cyu formulu ce vzyati pohidnu vid formuli ob yemu po r r bo ob yem vseredini sferi radiusu r r mozhe rozglyadatis yak suma neskinchennoyi kilkosti sferichnih obolonok neskinchenno maloyi tovshini de kozhna nastupna vpritul obgortaye poperednyu vid nulovogo radiusu do radiusu r r Dlya neskinchenno maloyi tovshini riznicya mizh vnutrishnoyu ta zovnishnoyu ploshami poverhon dlya bud yakoyi obolonki ye neskinchenno maloyu a element ob yemu en na radiusi r r prosto ye dobutkom ploshi poverhni na radiusi r r ta neskinchenno maloyu tovshinoyu Dlya bud yakogo danogo radiusu r r 5 pririst ob yemu d V displaystyle delta V dorivnyuye dobutku ploshi poverhni na radiusi r r S r displaystyle S r ta tovshinoyu obgortki d r displaystyle delta r d V S r d r displaystyle delta V approx S r cdot delta r Povnij ob yem sferi dorivnyuye sumi vsih ob yemiv obolonok V S r d r displaystyle V approx sum S r cdot delta r Dlya granici funkciyi koli d r displaystyle delta r nablizhayetsya do nulya 2 ce rivnyannya nabuvaye viglyadu V 0 r S r d r displaystyle V int 0 r S r dr Pidstavimo V V 4 3 p r 3 0 r S r d r displaystyle frac 4 3 pi r 3 int 0 r S r dr Yaksho vzyati pohidnu po r r z oboh bokiv rivnyannya to otrimayemo S S yak funkciyu vid r r 4 p r 2 S r displaystyle 4 pi r 2 S r Zvichajno ce zapisuyetsya yak S 4 p r 2 displaystyle S 4 pi r 2 de r r rozglyadayetsya yak fiksovanij radius sferi Alternativno element ploshi en na sferi zadanij v sferichnih koordinatah yak d S r 2 sin 8 d 8 d f displaystyle dS r 2 sin theta d theta d varphi V pryamokutnij sistemi koordinat element ploshi viglyadaye yak d S r r 2 i k x i 2 i k d x i k displaystyle dS frac r sqrt r 2 displaystyle sum i neq k x i 2 prod i neq k dx i forall k Dlya bilsh detalnogo rozglyadu vidvidajte element ploshi en Povna plosha takim chinom mozhe buti otrimana za dopomogoyu integruvannya S 0 2 p 0 p r 2 sin 8 d 8 d f 4 p r 2 displaystyle S int 0 2 pi int 0 pi r 2 sin theta d theta d varphi 4 pi r 2 Kartinka odniyeyi z najbilsh tochnih sfer zroblenih lyudinoyu yaka zalomlyuye foto Ejnshtejna na foni Sfera virobnictva NASA z plavlenogo kvarcu dlya vikoristannya v giroskopi Gravity Probe B Ce odna z najtochnishih sfer koli nebud stvorenih lyudinoyu sho vidriznyayutsya za formoyu vid idealnoyi sferi ne bilshe nizh na 40 atomiv tovshini mensh nizh 10 nanometriv Vvazhayetsya sho tilki nejtronni zirki ye gladkishimi 1 lipnya 2008 roku bulo ogolosheno sho avstralijski vcheni stvorili she bilshe idealnih sfer z tochnistyu do 0 3 nanometriv yak chastina mizhnarodnogo poshuku novogo globalnogo standartu kilogram 6 Sfera maye najmenshu ploshu poverhni z usih poverhon sho mistyat pevnij ob yem i mistit najbilshij ob yem sered usih zakritih poverhon iz zadanoyu plosheyu poverhni Tomu sfera z yavlyayetsya v prirodi napriklad bulbashki ta neveliki krapli vodi ye priblizno sferichnimi oskilki poverhnevij natyag lokalno minimizuye ploshu poverhni Plosha poverhni vidnosno masi kulki nazivayetsya pitoma poverhnya i mozhe buti virazhena z vishenavedenih rivnyan yak S S A A V r 3 r r displaystyle mathrm SSA frac A V rho frac 3 r rho de r rho ce shilnist vidnoshennya masi do ob yemu Ob yemi deyakih figur otrimanih vnaslidok kombinacij zi sferoyu RedaguvatiNehaj mayemo sferu z radiusom R R Todi ob yem kuba vpisanogo v sferu dorivnyuye V c u b e 8 3 R 3 9 displaystyle V cube frac 8 sqrt 3 R 3 9 ob yem cilindra vpisanogo v sferu za umovi sho osovij pereriz cilindra kvadrat dorivnyuye V c y l i n d e r 2 p R 3 2 displaystyle V cylinder frac sqrt 2 pi R 3 2 ob yem konusa vpisanogo v sferu za umovi sho osovij pereriz konusa rivnobedrenij trikutnik z kutom 2 a displaystyle 2 alpha pri vershini dorivnyuye V c o n e 2 p R 3 sin 2 2 a cos 2 a 3 displaystyle V cone frac 2 pi R 3 sin 2 2 alpha cos 2 alpha 3 ob yem konusa vpisanogo v sferu za umovi sho osovij pereriz konusa rivnobedrenij pryamokutnij trikutnik dorivnyuye V c o n e p R 3 3 displaystyle V cone frac pi R 3 3 ob yem konusa vpisanogo v sferu za umovi sho osovij pereriz konusa rivnostoronnij trikutnik dorivnyuye V c o n e 3 p R 3 8 displaystyle V cone frac 3 pi R 3 8 ob yem kuba opisanogo navkolo sferi dorivnyuye V c u b e 8 R 3 displaystyle V cube 8R 3 ob yem cilindra opisanogo navkolo sferi dorivnyuye V c y l i n d e r 2 p R 3 displaystyle V cylinder 2 pi R 3 ob yem konusa opisanogo navkolo sferi za umovi sho osovij pereriz konusa rivnobedrenij trikutnik z kutom 2 a displaystyle 2 alpha pri vershini dorivnyuye V c o n e 8 p R 3 cos a 1 sin a 3 3 sin a displaystyle V cone frac 8 pi R 3 cos alpha 1 sin alpha 3 3 sin alpha ob yem konusa opisanogo navkolo sferi za umovi sho osovij pereriz konusa rivnobedrenij pryamokutnij trikutnik dorivnyuye V c o n e p R 3 2 2 3 3 displaystyle V cone frac pi R 3 2 sqrt 2 3 3 ob yem konusa opisanogo navkolo sferi za umovi sho osovij pereriz konusa rivnobedrenij trikutnik z kutom 2 a displaystyle 2 alpha pri vershini dorivnyuye V c o n e 9 3 p R 3 displaystyle V cone 9 sqrt 3 pi R 3 Tenzor Richchi ta skalyarna krivina sferi RedaguvatiGeometriyu sferi mozhna prosto opisati predstavivshi yiyi vkladenoyu v fiktivnij chotirivimirnij prostir x 1 2 x 2 2 x 3 2 x 4 2 R 2 d l 2 d x 1 2 d x 2 2 d x 3 2 d x 4 2 1 displaystyle x 1 2 x 2 2 x 3 2 x 4 2 R 2 quad dl 2 dx 1 2 dx 2 2 dx 3 2 dx 4 2 qquad 1 Vvedennyam koordinatx 1 R c o s ps x 2 R c o s f s i n ps s i n 8 x 3 R s i n f s i n ps s i n 8 x 4 R c o s 8 s i n ps displaystyle x 1 Rcos psi quad x 2 Rcos varphi sin psi sin theta quad x 3 Rsin varphi sin psi sin theta x 4 Rcos theta sin psi mozhna zadovolniti 1 displaystyle 1 a elementi dovzhin na poverhni matimut viglyad elementarno pereviryayetsya pidstanovkoyu d l 2 R 2 d ps 2 s i n 2 ps d 8 2 s i n 2 8 d f 2 2 displaystyle dl 2 R 2 d psi 2 sin 2 psi d theta 2 sin 2 theta d varphi 2 qquad 2 Yak vidno metrichnij tenzor maye specifichnu strukturu ye diagonalnim pershij diagonalnij element riven odinici drugij zalezhit vid pershoyi zminnoyi tretij vid pershoyi i drugoyi a vid tretoyi zminnoyi zalezhnosti nemaye sho pevnoyu miroyu vidpovidaye izotropiyi prostoru Vihodyachi iz cogo mozhna viznachiti virazi dlya simvoliv Kristoffelya mayuchi zagalnij virazG j l k 1 2 g k m l g m j j g m l m g j l displaystyle Gamma jl k frac 1 2 g km partial l g mj partial j g ml partial m g jl de metrichnij tenzor g l j displaystyle g lj maye viglyadg l j d i a g R 2 R 2 s i n 2 ps R 2 s i n 2 ps s i n 2 8 g l j d i a g R 2 R 2 s i n 2 ps R 2 s i n 2 ps s i n 2 8 3 displaystyle g lj diag R 2 R 2 sin 2 psi R 2 sin 2 psi sin 2 theta quad g lj diag R 2 R 2 sin 2 psi R 2 sin 2 psi sin 2 theta qquad 3 dlya chastinnih vipadkiv viraziv mozhna otrimatiG l l k 1 2 g k m 2 l g m l m g l l g k m l g m l 1 2 g k k k g l l 1 2 g k k k g l l 4 displaystyle Gamma ll k frac 1 2 g km 2 partial l g ml partial m g ll g km partial l g ml frac 1 2 g kk partial k g ll frac 1 2 g kk partial k g ll qquad 4 G k k k 1 2 g k m 2 k g k m m g k k 1 2 g k k k g k k 0 5 displaystyle Gamma kk k frac 1 2 g km 2 partial k g km partial m g kk frac 1 2 g kk partial k g kk 0 qquad 5 oskilki v silu strukturi metrichnih tenzoriv 0 g 00 h g h h 0 displaystyle partial 0 g 00 partial h g hh 0 G l k k 1 2 g k m l g m k k g m l m g l k 1 2 g k k l g k k 6 displaystyle Gamma lk k frac 1 2 g km partial l g mk partial k g ml partial m g lk frac 1 2 g kk partial l g kk qquad 6 G l j k 3 1 2 g k m l g m j j g m l m g l j 1 2 g k k l g k k d k j 1 2 g k k j g k k d k l 1 2 g k k k g j l d j l G l k k d j k G j k k d l k G l l k d j l 7 displaystyle Gamma lj k 3 frac 1 2 g km partial l g mj partial j g ml partial m g lj frac 1 2 g kk partial l g kk delta kj frac 1 2 g kk partial j g kk delta kl frac 1 2 g kk partial k g jl delta jl Gamma lk k delta j k Gamma jk k delta l k Gamma ll k delta j l qquad 7 Teper mozhna sprostiti yakomoga bilshe zmenshiti kilkist sum viraz dlya tenzoru Richchi mayuchi zagalne viznachennya R l j 3 k G j l k l G j k k G j l k G k s s G l s k G j k s 8 displaystyle R lj 3 partial k Gamma jl k partial l Gamma jk k Gamma jl k Gamma k sigma sigma Gamma l sigma k Gamma jk sigma qquad 8 ta virazi 4 7 displaystyle 4 7 dlya tenzora mozhna otrimati suma lishe po indeksam k s displaystyle k sigma R l j 3 j G l j j l G j l l k G l l k d j l l G j k k G j k k G l j j G l k k G j l l G k s s G l l k d j l G j k k G l k k G j l l G l j j 2 G k j l G l l k d j l G l l j G j j l 9 displaystyle R lj 3 partial j Gamma lj j partial l Gamma jl l partial k Gamma ll k delta j l partial l Gamma jk k Gamma jk k Gamma lj j Gamma lk k Gamma jl l Gamma k sigma sigma Gamma ll k delta j l Gamma jk k Gamma lk k Gamma jl l Gamma lj j 2 Gamma kj l Gamma ll k delta j l Gamma ll j Gamma jj l qquad 9 Dovedennya Dijsno vikoristovuyuchi virazi 4 7 displaystyle 4 7 dlya dodankiv 8 displaystyle 8 mozhna otrimati nastupni virazi Pershij dodanok k G j l k k G l k k d j k k G j k k d l k k G l l k d j l j G l j j l G j l l k G l l k d j l displaystyle partial k Gamma jl k partial k Gamma lk k delta j k partial k Gamma jk k delta l k partial k Gamma ll k delta j l partial j Gamma lj j partial l Gamma jl l partial k Gamma ll k delta j l Drugij dodanok zalishayetsya bez zmin Tretij dodanok G k s s G j l k G k s s G l k k d j k G k s s G j k k d l k G k s s G j j k d l j G j s s G l j j G l s s G j l l G k s s G l l k d j l displaystyle Gamma k sigma sigma Gamma jl k Gamma k sigma sigma Gamma lk k delta j k Gamma k sigma sigma Gamma jk k delta l k Gamma k sigma sigma Gamma jj k delta l j Gamma j sigma sigma Gamma lj j Gamma l sigma sigma Gamma jl l Gamma k sigma sigma Gamma ll k delta j l Chetvertij dodanok G j k s G l s k G j k s G l k k d s k G j k s G s k k d l k G j k s G l l k d s l G j k k G l k k G j l s G s l l G j k l G l l k displaystyle Gamma jk sigma Gamma l sigma k Gamma jk sigma Gamma lk k delta sigma k Gamma jk sigma Gamma sigma k k delta l k Gamma jk sigma Gamma ll k delta sigma l Gamma jk k Gamma lk k Gamma jl sigma Gamma sigma l l Gamma jk l Gamma ll k Dlya dvoh ostannih dodankiv dovedetsya povtoriti cyu zh samu proceduru G s l l G j l s G s l l G j s s d l s G s l l G l s s d j s G s l l G l l s d j l G l l l G j l l G j l l G l j j G s l l G l l s d j l 8 G j l l G l j j G s l l G l l s d j l displaystyle Gamma sigma l l Gamma jl sigma Gamma sigma l l Gamma j sigma sigma delta l sigma Gamma sigma l l Gamma l sigma sigma delta j sigma Gamma sigma l l Gamma ll sigma delta j l Gamma ll l Gamma jl l Gamma jl l Gamma lj j Gamma sigma l l Gamma ll sigma delta j l 8 Gamma jl l Gamma lj j Gamma sigma l l Gamma ll sigma delta j l G l l k G j k l G l l k G j l l d k l G l l k G k l l d j l G l l k G j j l d k j G l l l G j l l G l l k G k l l d j l G l l j G j j l 8 G l l k G k l l d j l G l l j G j j l displaystyle Gamma ll k Gamma jk l Gamma ll k Gamma jl l delta k l Gamma ll k Gamma kl l delta j l Gamma ll k Gamma jj l delta k j Gamma ll l Gamma jl l Gamma ll k Gamma kl l delta j l Gamma ll j Gamma jj l 8 Gamma ll k Gamma kl l delta j l Gamma ll j Gamma jj l Otzhe G j k s G l s k G j k k G l k k G j l l G l j j G s l l G l l s d j l G l l k G k l l d j l G l l j G j j l displaystyle Gamma jk sigma Gamma l sigma k Gamma jk k Gamma lk k Gamma jl l Gamma lj j Gamma sigma l l Gamma ll sigma delta j l Gamma ll k Gamma kl l delta j l Gamma ll j Gamma jj l Dodavshi virazi dlya vsih dodankiv ta zaminivshi nimij indeks s displaystyle sigma na k k mozhna otrimati 9 displaystyle 9 Teper mozhna zastosuvati sproshenij viglyad dlya tenzoru Richchi do metriki sferichnogo prostoru Treba obchisliti komponenti R i j displaystyle R ij Spochatku dovedetsya otrimati koristuyuchis 4 7 displaystyle 4 7 yavnij viglyad dlya simvoliv Kristoffelya G 11 1 G 22 2 G 33 3 G 23 1 G 12 3 G 21 1 G 31 1 G 32 2 G 11 2 G 11 3 0 displaystyle Gamma 11 1 Gamma 22 2 Gamma 33 3 Gamma 23 1 Gamma 12 3 Gamma 21 1 Gamma 31 1 Gamma 32 2 Gamma 11 2 Gamma 11 3 0 G 13 3 1 2 g 33 1 g 33 2 s i n ps c o s ps 2 s i n 2 ps c t g ps G 12 2 G 13 3 displaystyle Gamma 13 3 frac 1 2 g 33 partial 1 g 33 frac 2sin psi cos psi 2sin 2 psi ctg psi quad Gamma 12 2 Gamma 13 3 G 22 1 1 2 g 11 1 g 22 s i n ps c o s ps displaystyle Gamma 22 1 frac 1 2 g 11 partial 1 g 22 sin psi cos psi G 23 3 1 2 g 33 2 g 33 c t g 8 displaystyle Gamma 23 3 frac 1 2 g 33 partial 2 g 33 ctg theta G 33 1 1 2 g 11 1 g 33 s i n 2 8 s i n ps c o s ps displaystyle Gamma 33 1 frac 1 2 g 11 partial 1 g 33 sin 2 theta sin psi cos psi G 33 2 1 2 g 22 2 g 33 s i n 8 c o s 8 displaystyle Gamma 33 2 frac 1 2 g 22 partial 2 g 33 sin theta cos theta Todi napriklad komponenta 11 tenzora iz urahuvannyam cih viraziv ta 9 displaystyle 9 maye virazR 11 1 G 1 k k G 1 k k G 1 k k 1 G 12 2 G 13 3 G 12 2 2 G 13 3 2 2 1 c t g ps 2 c t g 2 ps 2 s i n 2 ps 2 c t g 2 ps 2 displaystyle R 11 partial 1 Gamma 1k k Gamma 1k k Gamma 1k k partial 1 Gamma 12 2 Gamma 13 3 Gamma 12 2 2 Gamma 13 3 2 2 partial 1 ctg psi 2ctg 2 psi frac 2 sin 2 psi 2ctg 2 psi 2 Analogichni vikladki pereviryayutsya povnistyu identichno poperednij dayutR i j 2 g i j i j R i j 0 i j displaystyle R ij 2g ij i j quad R ij 0 i neq j Otzhe dlya sferiR i j 2 R 2 g i j 10 displaystyle R ij frac 2 R 2 g ij qquad 10 Zgortayuchi tenzor Richchi iz metrichnim tenzorom vidpovidno do viznachennya skalyarnoyi krivini mozhna otrimati sho dlya sferi skalyarna krivina rivnaR 1 2 R 2 g i j g i j 6 R 2 displaystyle R 1 frac 2 R 2 g ij g ij frac 6 R 2 Otzhe sferichnij prostir prostir z postijnoyu dodatnoyu skalyarnoyu krivinoyu Formuli RedaguvatiPlosha poverhni S O 4 p r 2 displaystyle S O 4 pi r 2 Zamknenij ob yem V 4 3 p r 3 displaystyle V frac 4 3 pi r 3 Ob yem segmenta V K S h 2 p 3 3 r h displaystyle V mathrm KS frac h 2 pi 3 3r h Moment inerciyi J 2 5 m r 2 displaystyle J frac 2 5 mr 2 N vimirna sfera RedaguvatiSferu v N N vimirnomu prostori nazivayut gipersferoyu V zagalnomu vipadku rivnyannya gipersferi v N N vimirnomu prostori matime viglyad i 1 N x i x 0 i 2 r 2 displaystyle sum i 1 N x i x 0i 2 r 2 de x 01 x 0 N displaystyle x 01 x 0N centr gipersferi a r r yiyi radius Peretinom dvoh N N vimirnih sfer ye N 1 displaystyle N 1 vimirna sfera yaka lezhit na radikalnij giperploshini cih sfer V N N vimirnomu prostori mozhe poparno dotikatis odna do odnoyi ne bilshe nizh N 1 displaystyle N 1 sfera Ob yem N N vimirnoyi kuli ob yem sho obmezhuye N N vimirna sfera mozhlivo rozrahuvati za formuloyu V N p N 2 G N 2 1 r N displaystyle V N frac pi N 2 Gamma frac N 2 1 r N de G x displaystyle Gamma x gamma funkciya Ploshu poverhni N N vimirnoyi sfera mozhlivo rozrahuvati za formuloyu S N N p N 2 G N 2 1 r N 1 displaystyle S N frac N pi N 2 Gamma frac N 2 1 r N 1 Div takozh RedaguvatiSferichna geometriya Sferichna trigonometriya Sferichna sistema koordinat Sferichnij segment SferometrPrimitki Redaguvati Steinhaus 1969 p 223 a b Pages 141 149 E J Borowski J M Borwein 1989 Collins Dictionary of Mathematics ISBN 0 00 434347 6 Weisstein Eric W Sphere angl na sajti Wolfram MathWorld Steinhaus 1969 p 221 r rozglyadayetsya yak zminna dlya cogo obchislennya New Scientist Technology Roundest objects in the world created Arhiv originalu za 26 kvitnya 2015 Procitovano 6 sichnya 2018 Dzherela RedaguvatiKorn G A Korn T M Spravochnik po matematike dlya nauchnyh rabotnikov i inzhenerov 4 e M Nauka 1978 277 s Geometriya 10 11 klasi Tekst probnij pidruchnik Afanasyeva O M ta in Ternopil Navchalna kniga Bogdan 2003 264 s ISBN 966 692 161 8Posilannya RedaguvatiSfera ta yiyi rivnyannya Visha matematika v prikladah i zadachah Klepko V Yu Golec V L 2 ge vidannya K Centr uchbovoyi literaturi 2009 S 155 594 s Otrimano z https uk wikipedia org w index php title Sfera amp oldid 37606057
