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 red Vidrizok 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 red U 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 nbsp i radiusom r displaystyle r nbsp ye geometrichnim miscem usih tochok x y z displaystyle x y z nbsp 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 nbsp 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 nbsp 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 nbsp z z 0 r cos 8 displaystyle z z 0 r cos theta nbsp 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 nbsp 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 nbsp 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 nbsp 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 nbsp ta 90 displaystyle 90 circ nbsp Zamknenij ob yem red nbsp 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 nbsp de r displaystyle r nbsp 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 displaystyle x nbsp inkrementnij ob yem d V displaystyle delta V nbsp dorivnyuye dobutku ploshi poperechnogo pererizu kruga pri x displaystyle x nbsp i jogo tovshini d x displaystyle delta x nbsp d V p y 2 d x displaystyle delta V approx pi y 2 cdot delta x nbsp 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 nbsp 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 nbsp Dlya bud yakogo x displaystyle x nbsp pryamokutnij trikutnik poyednuye x displaystyle x nbsp y displaystyle y nbsp ta r displaystyle r nbsp iz pochatkom koordinat znachit yaksho zastosuvati teoremu Pifagora otrimayemo y 2 r 2 x 2 displaystyle y 2 r 2 x 2 nbsp Vikoristayemo pidstanovku V r r p r 2 x 2 d x displaystyle V int r r pi r 2 x 2 dx nbsp 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 nbsp 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 nbsp 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 nbsp Dlya bilshosti praktichnih cilej ob yem vseredini sferi vpisanoyi v kub mozhe buti nablizhene do 52 4 displaystyle 52 4 nbsp ob yemu kuba cherez te sho V p 6 d 3 displaystyle V tfrac pi 6 d 3 nbsp de d displaystyle d nbsp ye diametrom sferi i v toj zhe chas dovzhinoyu storoni kuba i p 6 0 5236 displaystyle tfrac pi 6 approx 0 5236 nbsp Napriklad sfera diametru 1 displaystyle 1 nbsp metr maye 52 4 displaystyle 52 4 nbsp ob yemu kuba z dovzhinoyu rebra 1 displaystyle 1 nbsp metr abo blizko 0 524 displaystyle 0 524 nbsp m3 Plosha poverhni red Plosha poverhni sferi radiusu r displaystyle r nbsp S 4 p r 2 displaystyle S 4 pi r 2 nbsp 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 displaystyle r nbsp bo ob yem vseredini sferi radiusu r displaystyle r nbsp mozhe rozglyadatis yak suma neskinchennoyi kilkosti sferichnih obolonok neskinchenno maloyi tovshini de kozhna nastupna vpritul obgortaye poperednyu vid nulovogo radiusu do radiusu r displaystyle r nbsp 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 displaystyle r nbsp prosto ye dobutkom ploshi poverhni na radiusi r displaystyle r nbsp ta neskinchenno maloyu tovshinoyu Dlya bud yakogo danogo radiusu r displaystyle r nbsp 5 pririst ob yemu d V displaystyle delta V nbsp dorivnyuye dobutku ploshi poverhni na radiusi r displaystyle r nbsp S r displaystyle S r nbsp ta tovshinoyu obgortki d r displaystyle delta r nbsp d V S r d r displaystyle delta V approx S r cdot delta r nbsp Povnij ob yem sferi dorivnyuye sumi vsih ob yemiv obolonok V S r d r displaystyle V approx sum S r cdot delta r nbsp Dlya granici funkciyi koli d r displaystyle delta r nbsp nablizhayetsya do nulya 2 ce rivnyannya nabuvaye viglyadu V 0 r S r d r displaystyle V int 0 r S r dr nbsp Pidstavimo V displaystyle V nbsp 4 3 p r 3 0 r S r d r displaystyle frac 4 3 pi r 3 int 0 r S r dr nbsp Yaksho vzyati pohidnu po r displaystyle r nbsp z oboh bokiv rivnyannya to otrimayemo S displaystyle S nbsp yak funkciyu vid r displaystyle r nbsp 4 p r 2 S r displaystyle 4 pi r 2 S r nbsp Zvichajno ce zapisuyetsya yak S 4 p r 2 displaystyle S 4 pi r 2 nbsp de r displaystyle r nbsp 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 nbsp 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 nbsp 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 nbsp nbsp 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 nbsp de r displaystyle rho nbsp ce shilnist vidnoshennya masi do ob yemu Ob yemi deyakih figur otrimanih vnaslidok kombinacij zi sferoyu red Nehaj mayemo sferu z radiusom R displaystyle R nbsp 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 nbsp 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 nbsp ob yem konusa vpisanogo v sferu za umovi sho osovij pereriz konusa rivnobedrenij trikutnik z kutom 2 a displaystyle 2 alpha nbsp 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 nbsp 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 nbsp 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 nbsp ob yem kuba opisanogo navkolo sferi dorivnyuye V c u b e 8 R 3 displaystyle V cube 8R 3 nbsp 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 nbsp ob yem konusa opisanogo navkolo sferi za umovi sho osovij pereriz konusa rivnobedrenij trikutnik z kutom 2 a displaystyle 2 alpha nbsp 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 nbsp 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 nbsp ob yem konusa opisanogo navkolo sferi za umovi sho osovij pereriz konusa rivnobedrenij trikutnik z kutom 2 a displaystyle 2 alpha nbsp pri vershini dorivnyuye V c o n e 9 3 p R 3 displaystyle V cone 9 sqrt 3 pi R 3 nbsp Tenzor Richchi ta skalyarna krivina sferi red Geometriyu 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 nbsp 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 nbsp mozhna zadovolniti 1 displaystyle 1 nbsp 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 nbsp 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 nbsp de metrichnij tenzor g l j displaystyle g lj nbsp 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 nbsp 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 nbsp 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 nbsp oskilki v silu strukturi metrichnih tenzoriv 0 g 00 h g h h 0 displaystyle partial 0 g 00 partial h g hh 0 nbsp 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 nbsp 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 nbsp 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 nbsp ta virazi 4 7 displaystyle 4 7 nbsp dlya tenzora mozhna otrimati suma lishe po indeksam k s displaystyle k sigma nbsp 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 nbsp Dovedennya Dijsno vikoristovuyuchi virazi 4 7 displaystyle 4 7 nbsp dlya dodankiv 8 displaystyle 8 nbsp 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 nbsp 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 nbsp 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 nbsp 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 nbsp 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 nbsp 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 nbsp Dodavshi virazi dlya vsih dodankiv ta zaminivshi nimij indeks s displaystyle sigma nbsp na k displaystyle k nbsp mozhna otrimati 9 displaystyle 9 nbsp Teper mozhna zastosuvati sproshenij viglyad dlya tenzoru Richchi do metriki sferichnogo prostoru Treba obchisliti komponenti R i j displaystyle R ij nbsp Spochatku dovedetsya otrimati koristuyuchis 4 7 displaystyle 4 7 nbsp 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 nbsp 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 nbsp 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 nbsp 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 nbsp 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 nbsp 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 nbsp Todi napriklad komponenta 11 tenzora iz urahuvannyam cih viraziv ta 9 displaystyle 9 nbsp 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 nbsp 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 nbsp Otzhe dlya sferiR i j 2 R 2 g i j 10 displaystyle R ij frac 2 R 2 g ij qquad 10 nbsp 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 nbsp Otzhe sferichnij prostir prostir z postijnoyu dodatnoyu skalyarnoyu krivinoyu Formuli red Plosha poverhni S O 4 p r 2 displaystyle S O 4 pi r 2 nbsp Zamknenij ob yem V 4 3 p r 3 displaystyle V frac 4 3 pi r 3 nbsp Ob yem segmenta V K S h 2 p 3 3 r h displaystyle V mathrm KS frac h 2 pi 3 3r h nbsp Moment inerciyi J 2 5 m r 2 displaystyle J frac 2 5 mr 2 nbsp N vimirna sfera red Sferu v N displaystyle N nbsp vimirnomu prostori nazivayut gipersferoyu V zagalnomu vipadku rivnyannya gipersferi v N displaystyle N nbsp 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 nbsp de x 01 x 0 N displaystyle x 01 x 0N nbsp centr gipersferi a r displaystyle r nbsp yiyi radius Peretinom dvoh N displaystyle N nbsp vimirnih sfer ye N 1 displaystyle N 1 nbsp vimirna sfera yaka lezhit na radikalnij giperploshini cih sfer V N displaystyle N nbsp vimirnomu prostori mozhe poparno dotikatis odna do odnoyi ne bilshe nizh N 1 displaystyle N 1 nbsp sfera Ob yem N displaystyle N nbsp vimirnoyi kuli ob yem sho obmezhuye N displaystyle N nbsp 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 nbsp de G x displaystyle Gamma x nbsp gamma funkciya Ploshu poverhni N displaystyle N nbsp 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 nbsp Div takozh red Sferichna geometriya Sferichna trigonometriya Sferichna sistema koordinat Sferichnij segment SferometrPrimitki red 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 red Korn 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 red Sfera 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
