Calculus Ukwahlukaniswa kwemisebenzi elilodwa kanye eziguquguqukayo eziningana

calculus Ukwahlukaniswa kuyinto igatsha izingabunjalo, okuyinto uhlola esuselwe, nomehluko nokusetshenziswa kwawo ekuhloleni imisebenzi.

Indaba

calculus Ukwahlukaniswa sekukhona njengenhlangano isiyalo ezimele engxenyeni kwekhulu le-17 yesibili, sibonga umsebenzi Newton kanye Leibniz, ngubani washayelwa amalungiselelo eziyisisekelo kulesi sibalo nomehluko futhi waphawula uxhumano emkhatsini wekuhlanganisa kanye kwamangqamuzana. Njengoba isiyalo athuthukiswe kanye ekubalweni integrals, kanjalo akha ngesisekelo izingabunjalo. Ukubukeka lezi calculi wavula inkathi entsha yesimanje emhlabeni zezibalo futhi wabangela ukuvela eyala entsha kwezesayensi. Futhi selule kungenzeka yokusebenzisa izibalo ngesayensi yemvelo noma kwezobunjiniyela.

imiqondo eyisisekelo

calculus Ukwahlukaniswa kusekelwe imiqondo eyisisekelo yezibalo. Ziyakwazi: wangempela inombolo, lenqubekela phambili kanye nokusebenza umkhawulo umsebenzi. Ngemva kwesikhathi esithile, bathatha a look zanamuhla, sibonga calculus ebalulekile futhi umehluko.

Inqubo yokudala

Kumiswa calculus umehluko ngesimo isicelo, bese ukusebenzisa izindlela zesayensi livelile ngaphambi ukuvela theory zefilosofi, okuyinto sidalwe Nikolay Kuzansky. Umsebenzi wakhe ubhekwa i ukuziphendukela kwemvelo isayensi lokwahlulela lasendulo. Naphezu kweqiniso lokuthi sefilosofi yena kwakungelona sezibalo, ngeqhaza lakhe ekuthuthukiseni isayensi zezibalo akunakuphikwa. WaseCusa, omunye aphume lokuqala Ukucabangela izibalo njengoba nesayensi enembile kakhulu, izibalo ngokubeka isikhathi ku umbuzo.

Ngo zezibalo lasendulo umbandela jikelele kwaba iyunithi, kuyilapho sefilosofi ezihlongozwayo njengoba isilinganiso infinity entsha ukubuyela inani eliqondile. Ngokuqondene nalokhu ukumelwa okuhlanekezelwe lokunemba kwezesayensi zezibalo. Ulwazi lwesayensi, ngokombono wakhe, ihlukaniswe enengqondo futhi ehlakaniphile. Eyesibili esinembile, ngokuvumelana usosayensi, kusukela wangaphambili unika Imiphumela eseduze kuphela.

umqondo

Umqondo oyinhloko kanye nomqondo calculus umehluko elihlobene ne-umsebenzi endaweni encane amaphuzu athile. Ukuze lokhu kuyadingeka ukuze udale ama-apharathasi zezibalo ukusebenza izifundo oziphatha endaweni encane amaphuzu efakwe eduze ukuziphatha umsebenzi komugqa noma polynomial. Kususelwa le ncazelo esuselwe futhi umehluko.

Ukuvela nomqondo esuselwe kwabangelwa inqwaba izinkinga wesayensi yemvelo nezibalo ezingamaGreki, imfundo okwaholela ukuzimisela umkhawulo amagugu yohlobo olufanayo.

Munye walemisebenti main ukuthi zinikwa njengesibonelo, uthome endala amakilasi esikoleni, kutfola ijubane motion iphuzu emgceni locondzile kanye nokwakhiwa komugqa tangent kule ijika. I umehluko ihlanganiswe kule, njengoba kungenzeka esondele umsebenzi endaweni encane iphuzu umsebenzi komugqa.

Kuqhathaniswa nomqondo esuselwe umsebenzi kwenombolo Okuyınqo, kwencazelo nomehluko idlule phezu umsebenzi imvelo jikelele, ikakhulukazi umfanekiso isikhala Euclidean kwenye.

esuselwe

Ake uhamba iphuzu bebheke-y-eksisi, ngokokuqala sithatha x, okuyinto kukalwa kusukela ekuqaleni isikhashana. Chaza ukunyakaza okunjalo kungenzeka yi y umsebenzi = (x) f, okuyinto lihlotshaniswa ngalinye isikhathi iphuzu x Ukudidiyela iphuzu displaceable. Le kholi umsebenzi e Mechanics ukuthatha umthetho motion. Isici esiyinhloko we ezinyakazayo, ikakhulukazi engalingani, kuyinto the velocity leso. Lapho iphuzu idluliswa eduze y-eksisi ngokomthetho ka Mechanics, okungahleliwe isikhathi iphuzu ke ozuza Ukudidiyela x f (x). Esikhathini iphuzu x + Δh, lapho Δh limelela anyuswe isikhathi, kuyoba kordinaty f (x + Δh). ifomula kwakhiwa Ngakho Δy = f (x + Δh) - f (x), okuthiwa i umsebenzi wokwengeza. It is a iphuzu endleleni siwela phakathi nesikhathi kusuka x ukuze x + Δh.

Mayelana indawo okuvela kuyo amazwi Isivinini ngesikhathi esuselwe iqondiswa. I esuselwe noma yimuphi umsebenzi ngesikhathi iphuzu fixed ngokuthi umkhawulo (sicabangela ikhona). Kungaba udluliselwe izinhlamvu:

f '(x), y, y, DF / dx, dy / dx, DF (x).

Inqubo kokubala esuselwe ucingo kwamangqamuzana.

calculus Ukwahlukaniswa kwemisebenzi lokwehlukile eziningana

Le ndlela isetshenziswa lapho kubalwa umsebenzi cwaningo, eziguquguqukayo eziningana. Lapho kukhona eziguquguqukayo ezimbili x kanye y, i-esuselwe esinqunyiwe mayelana x endaweni A ubizwa ngokuthi esuselwe kule umsebenzi e x nge y isilungisiwe.

Kwangathi ishiwo izimpawu ezilandelayo:

f '(x) (x, y), u' (x), ∂u / ∂x futhi ∂f (x, y) '/ ∂x.

amakhono adingekayo

Ukuze ufunde ngempumelelo futhi bakwazi ukuxazulula diffury ladzingekako ukuhlanganiswa kanye kwamangqamuzana. Ukwenza kube lula ukuqonda zibalo umehluko, kumelwe siwaqonde sihloko esuselwe futhi nanini ebalulekile. Futhi akusho ubuhlungu ukufunda ukubheka esuselwe umsebenzi ngokuphelele. Lokhu kungenxa yokuthi enqubeni yokufunda ngokuvamile izosebenzisa integrals kanye kwamangqamuzana.

Izinhlobo umehluko zibalo

Cishe wonke umsebenzi ukulawula elihlobene ne owokuqala-oda umehluko zibalo, kukhona izinhlobo ezingu 3 of zibalo: eyohlobo, eziguquguqukayo separable, komugqa inhomogeneous.

Kukhona futhi ngaphezulu izilwane ezingavamile zibalo nge nomehluko Imininingwane, ezothando Bernoulli, futhi abanye.

Kubaluleka izixazululo

Ukuze uqale, kufanele sikhumbule kuyinto kwesibalo algebraic inkambo esikoleni. Ziqukethe eziguquguqukayo nezinombolo. Ukuze ukuxazulula kwesibalo ezivamile kufanele uthole amaningi lwezinombolo ukwanelisa isimo esithile. Ngokuvamile, lezi zibalo kufanele eyodwa izimpande, futhi sokuqinisekisa kuphela kufanele ithathe lesi silinganiso endaweni engaziwa.

I kwesibalo umehluko kufana lokhu. Ngokuvamile, i equation ye-oda kuqala yakhiwa:

• variable Independent.
• A esuselwe umsebenzi kuqala.
• Umsebenzi noma variable nasekuthengeni.

Kwezinye izimo, kungase kube kungekho muntu engaziwa, x noma y, kodwa akuyona kubaluleke njengokugwema kubalulekile ukuthi kube esuselwe lokuqala, kungekho oda nemikhiqizo ngaphezulu ukuze ikhambi futhi calculus umehluko zaziyiqiniso.

Xazulula ezothando umehluko - kusho ukuthola iqoqo yonke imisebenzi ukuthi kukhona isisho efanelekayo inikezwe. amasethi okunjalo imisebenzi evame ukubizwa ngokuthi jikelele nokulawula isixazululo.

calculus ebalulekile

calculus Integral ingenye izingxenye izingabunjalo, okuyinto uhlola nomqondo ebalulekile, izakhiwo kanye nezindlela ukubala yayo.

Ngokuvamile ukubala ebalulekile kwenzeka lapho kubalwa endaweni umumo curvilinear. Ngale ngelosi endaweni umkhawulo, maqondana okungase endaweni predetermined we elalibhalwe ipholigoni Umumo ukwenyuka kancane isandla sakhe, futhi ohlangothini idatha kwenziwe encane ukwedlula noma iliphi ngokungenasizathu lwenani elincane ngaphambili ecacisiwe.

Umcondvo lomcoka endzabeni kulesi sibalo endaweni yanoma iluphi Jomethri is kuhlaziywa indawo isikwele ke kunobufakazi bokuthi endaweni yayo ilingana umkhiqizo ubude by ububanzi. Uma kuziwa geometry, bese zonke izakhiwo zenziwa ukusebenzisa irula lapho kusetshenzwa bese ikhampasi, khona-ke isilinganiso obuphelele ebubanzini liyinani okunengqondo. Lapho kubalwa kwendawo unxantathu wesokudla kungaba azimisele ukuthi uma ufaka unxantathu esilandelayo, bakha calandze is. Endaweni ye-parallelogram kubalwa e indlela efanayo kodwa kancane more xaka, ngaphakathi isikwele nonxantathu. Endaweni ethile ipholigoni ibhekwa onxantathu azifaka kuwo.

Ekunqumeni isihe kabi amandla akhe, le ndlela angahambisani ijika. Uma sihlukanisa ke tikwele ngamunye, izohlala izindawo gcwalisiwe. Kulesi simo, zama ukusebenzisa namabhantshi amabili, nge calandze nangaphezulu, ngenxa labo zihlanganisa igrafu umsebenzi futhi ayihlanganisi. Okubalulekile lapha kuyindlela ukuyiyeka le calandze. Futhi, uma sithatha ikhefu encishisiwe kakhulu, endaweni phezulu futhi ngezansi kufanele bukhomba inani elithile.

Kufanele ubuyele indlela ekwehlukaniseni ku calandze. Zimbili izindlela athandwayo.

Riemann waba semthethweni kwencazelo ebalulekile, adalwe Leibniz futhi Newton, njengoba indawo subgraph. Kulokhu, sixoxe sibalo ehlanganisa kwenani elithile calandze mpo etholwe sokuhlukanisa isikhawu. Lapho breaking ukwehla kukhona umkhawulo lapho endaweni ukuncipha abe semqoka kangaka, lo mkhawulo ubizwa ngokuthi Riemann ebalulekile umsebenzi ngesikhathi isikhawu esibekiwe.

Indlela yesibili ukwakha Lebesgue ebalulekile, owakhiwa yokuthi endaweni ngokuhlukana esiqokiwe endaweni ku ingxenye integrand futhi kwakhiwa ke isamba ebalulekile amagugu etholwe kulezi zingxenye, ngezikhawu ihlukaniswe uhla yayo amagugu, bese wafingqa ne izinyathelo okuhambisanayo izithombe ephambene lezi integrals.

nezinsiza yesimanje

Enye yezinzuzo esiyinhloko ekutadisheni umehluko kanye calculus ebalulekile Fikhtengol'ts wabhala - "we umehluko kanye calculus ebalulekile." ebhukwinitifundvo wakhe iyithuluzi okuyisidingo ekutadisheni izingabunjalo, okuyinto samelana edishini eziningi ezinguqulweni zezinye izilimi. Wadala for abafundi futhi isikhathi eside isetshenziswa ezihlukahlukene izikhungo zemfundo njengoba omunye izinzuzo main yocwaningo. Inikeza ulwazi theory kanye amakhono awusizo. Okokuqala eshicilelwe ngo-1948.

Algorithm ucwaningo umsebenzi

Ukuze ukuhlola izindlela umehluko calculus umsebenzi, kudingeka ulandele isivele inikezwe algorithm:

1. Thola esizindeni umsebenzi.
2. Thola izimpande kwesibalo inikezwe.
3. Bala ukweqisa. Ukuze wenze lokhu, thina ukubala esuselwe kanye iphuzu lapho ilingane no-zero.
4. Thina bamelele ukubaluleka etholwe EQ.

Izinhlobonhlobo umehluko zibalo

Ukulawula oda kuqala (uma kungenjalo, umehluko calculus elilodwa variable) nezinhlobo zabo:

• Eziguquguqukayo separable kwesibalo: f (y) dy = g (x) dx.
• Indlela elula kwesibalo noma umehluko calculus umsebenzi owodwa variable, kokuba ifomula: y '= f (x).
• I komugqa wokuqala-oda control nonuniform: y '+ P (x) y = Q (x).
• Bernoulli umehluko kwesibalo: y '+ P (x) y = Q (x) y.
• Equation nomehluko Imininingwane nge: P (x, y) dx + Q (x, y) dy = 0.

I umehluko zibalo oda yesibili nezinhlobo zabo:

• Eyohlobo komugqa wesibili oda umehluko kwesibalo ne ^{okuza njalo: y n + PY '+ qy} = 0 p, q fanele R.
• Inhomogeneous komugqa ukuze yesibili umehluko kwesibalo ne okuza njalo ^{nani: y n + PY '+ qy} = f (x).
• Eyohlobo lwento umehluko ^{kwesibalo: y n + p (x)} y '+ q (x) y = 0, futhi inhomogeneous wesibili oda ^{ezothando: y n + p (x)} y' + q (x) y = f (x).

zibalo Ukwahlukaniswa oda ephakeme kanye nezinhlobo zazo:

• I kwesibalo umehluko, okuvumela nokwehlisa ^{oda: F (x, y (k} ), y (k + ^{1), .., y (n)} = 0.
• A kwesibalo komugqa lesezingeni lelisetulu ^{_{eyohlobo: y (n) + f (}} n- ₁₎ y ^(n-1) + ... + f ₁ y '+ f ₀ y = 0, futhi ^{_{inhomogeneous: y (n) + f (}} n _-1) y ^(n-1) + ... + f ₁ y '+ f ₀ y = f (x).

Izigaba yokuxazulula le nkinga nge ezothando umehluko

Ngosizo elawulwa kude isixazululekile hhayi kuphela wezibalo noma izinkinga ezingokomzimba, kodwa futhi izinkinga ezihlukahlukene biology, ezomnotho, yokuhlalisana kwabantu kanye nabanye. Naphezu ezihlukahlukene izihloko, kufanele ulandele olulodwa logic ukulandelana ukuxazulula lezi zinkinga:

1. Ukudweba ukulawula. Omunye izigaba nzima kakhulu, edinga ukunemba esiphezulu, ngoba kukhona iphutha kuyoholela Imiphumela okungalungile ngokuphelele. Kuyadingeka ukuba acabangele zonke izici ezithinta inqubo kanye ukunquma izimo kokuqala. Kufanele ezisekelwe emaqinisweni futhi eziphethweni ezinengqondo.
2. Ukuze ukuxazulula zibalo. Le nqubo kulula iphuzu lokuqala, ngoba kudinga ukuqaliswa kuphela okuqinile izibalo ezithile.
3. Ukuhlaziywa nokuhlaziya imiphumela. isixazululo Etholakale kufanele tihlolwe ukufakwa SIZUZISA futhi ezincwadini umphumela.

Isibonelo ukusetshenziswa umehluko zibalo kwezokwelapha

Ukusebenzisa i-remote control e emkhakheni wezokwelapha itholakala ukwakhiwa epidemiological imodeli zezibalo. Akufanele sibakhohlwe ukuthi lezi zibalo atholakala biology kanye ne-chemistry, okuyizinto eduze imithi, ngoba linendima ebalulekile ekutadisheni ezahlukene imiphakathi begazi yezinqubo zamakhemikhali emzimbeni womuntu.

Kulesi sibonelo, ubhubhane ukusabalala ukutheleleka ungelashwa umphakathi angawodwa. Izakhamuzi zihlukaniswe izinhlobo ezintathu:

• Elinegciwane, inani x (t), eyayihlanganisa ngabanye, abathwali ezithathelwanayo, ngamunye okuyinto ezithathelwanayo (bayashintshana ekufukameleni inkathi imfushane).
• Uhlobo lwesibili kuhlanganisa ekuthonyweni abantu y (t), angaba ngokuthintana elinegciwane.
• Uhlobo lwesithathu kuhlanganisa abantu refractory z (t), okuyizinto mzimba noma ilahleke ngenxa yokugula.

Inombolo abantu njalo, ukugcina lokuzalwa, nokufa yemvelo futhi ukuthuthela akubhekwa. Emongweni kuyoba imicabango ezimbili.

isifo Amaphesenti esikhathini esithile iphuzu ilingana x (t) y (t) (nokucabanga esekelwe imfundiso inamba yezimo ngokulingana nenani ezimpambanweni phakathi iziguli namalungu ophendulayo, okuyinto isilinganiselo sokuqala wakhona x (t) y (t)), e Ngakho-ke linani lemacala siyakhuphuka, futhi inani lehla singangena ngesilinganiso ebalwa ngu ngembazo ifomula (t) y (t) (a> 0).

Inombolo-non-abaphendulile izilwane bafa noma bathenga amasosha omzimba, kukhuphuke ngesilinganiso okuyinto wakhona eziningi, Bx (t) (b> 0).

Ngenxa yalokho, ungasetha uhlelo of zibalo nazo zonke izinkomba ezintathu ngesisekelo iziphetho yayo.

ISIBONELO ukusetshenziswa ezomnotho

calculus Ukwahlukaniswa ngokuvamile sisetshenziswa ukuhlaziywa kwezomnotho. Umsebenzi oyinhloko e ukuhlaziywa komnotho ubhekwa ekutadisheni abakuzuzile umnotho ezilotshwe ngesimo umsebenzi. Isetshenziselwa ekuxazululeni izinkinga ezifana izinguquko kwentela engenayo inyuka ngokushesha ngemuva, ukungena izimali, izinguquko izimali lapho beshintsha ukubaluleka umkhiqizo, kulokho ingxenye ungayithola enye by abasebenzi umhlalaphansi imishini emisha. Ukuze baxazulule izinkinga ezinjalo, kuyadingeka ukwakha umsebenzi zokuxhumana eziguquguqukayo engenayo, okwathi ngemva kokuba ufundwa calculus umehluko.

ngokuvamile kuyadingeka ukuze uthole ukusebenza elilungile kakhulu ezingeni kwezomnotho: ukukhiqiza esiphezulu, imali eliphakeme, okungenani izindleko nokunye. ingxenye ngayinye esinjalo umsebenzi owodwa noma ngaphezulu agumenti. Ngokwesibonelo, ukukhiqizwa kubhekwe njengento umsebenzi zomsebenzi kanye capital. Kule ndaba, yokuthola ukubaluleka efanelekayo zingancishiswa ekutholeni esiphezulu noma ubuncane umsebenzi eyodwa noma ngaphezulu eziguquguqukayo.

Izinkinga ezinjalo ukudala ekilasini zezinkinga extremal emkhakheni kwezomnotho, okuyiwona udinga calculus umehluko. Lapho Inkomba kwezomnotho liyadingeka ukunciphisa noma ukukhulisa njengoba umsebenzi nezinye nemingcele, anyuswe isilinganiso esiphezulu iphuzu umsebenzi agumenti azothanda zero uma anyuswe lwempikiswano ivame zero. Uma kungenjalo, uma isimo sengqondo esinjalo ivame ukuba inani elithile emihle noma emibi, iphuzu echaziwe asifanele, ngoba ngokwandisa noma ukunciphisa i-agumenti kungashintshwa ukubaluleka engaphansi ohlangothini oyifunayo. Ngo umehluko calculus amagama, lokhu bekungase kusho ukuthi izimo adingekayo umsebenzi esiphezulu kuyinto zero ukubaluleka esuselwe yayo.

Umnotho akuyona inkinga into engavamile lokuthola extremum yomsebenzi eziguquguqukayo eziningana, ngoba yizinkomba zezomnotho akhiwe izinto eziningi. nezindaba ezinjalo ziqondwa imfundiso yokuziphendukela imisebenzi lokwehlukile eziningana, indlela kokubala umehluko kahle. izinkinga ezinjalo zihlanganisa hhayi kuphela kunakekelwa kanye lincishisiwe umsebenzi, kodwa futhi ukulinganiselwa. Lezi mibuzo zihlobana izinhlelo zezibalo, futhi isixazululekile ngosizo izindlela ngokukhethekile nayo isekelwe lokhu legatsha isayensi.

Phakathi izindlela umehluko calculus esetshenziswa emnothweni, kusigaba okubalulekile esizidlula zonke. Esikhathini sphere kwezomnotho, leli gama libhekisela iqoqo izindlela zocwaningo ukusebenza variable futhi imiphumela uma Ukushintsha ivolumu kokudalwa ukusetshenziswa, zisuselwa ekuhlaziyweni umkhawulo wabo amanani. Linganisa inkomba kubhekwe esuselwe noma okunye ngokwengxenye eziguquguqukayo eziningana.

calculus Ukwahlukaniswa lokwehlukile eziningana - liyindaba esemqoka sokuhlaziya zezibalo. Ukutadishwa, ungasebenzisa ezihlukahlukene izinsiza-kufundisa ukuze izikhungo zemfundo ephakeme. Omunye Fikhtengol'ts edume kunazo wadala - "we umehluko kanye calculus ebalulekile." Kungakanani igama ikhambi umehluko zibalo ka okumqoka ukuze babe namakhono ukusebenza integrals. Lapho kukhona calculus umehluko nemisebenzi eyodwa variable, isinqumo kuba lula. Nakuba, kufanele kuphawulwe ukuthi lulandela imithetho efanayo eyisisekelo. Ekusebenteni, ukuphenya umsebenzi we-calculus umehluko, vele ulandele algorithm osuvele onikelwa esikoleni esiphakeme, futhi kuphela okuncane eziyinkimbinkimbi ngesethulo lokwehlukile entsha.