Bambanci ilimin lissafi na ayyuka na daya da kuma da dama canji

Bambanci ilimin lissafi ne wani reshe na ilmin lissafi analysis, wanda ya yi nazarin wanda aka samu, da kuma differentials su amfani a cikin binciken na ayyuka.

A labarin na

Bambanci ilimin lissafi fito a matsayin mai zaman kanta horo a cikin biyu da rabi na 17th karni, godiya ga aikin da Newton kuma Leibniz, wanda tsara ainihin guzuri a cikin lissafi na differentials da kuma lura da connection tsakanin hadewa da bambantawa. Tun da horo ya ci gaba tare da lissafi na integrals, game da shi, da ke cikin Dalili na ilmin lissafi analysis. Bayyanar da wadannan calculi bude wani sabon zamani lokaci a cikin ilmin lissafi duniya da kuma sa fitowan da sabon tarbiyya a kimiyya. Har ila yau, ya mika yiwuwar ake ji lissafi a cikin kimiyyar yanayin halitta da aikin injiniya.

asali Concepts

Bambanci ilimin lissafi dogara ne a kan muhimman hakkokin Concepts na lissafi. Su ne: wani real lambar, a ci gaba da iyaka na aiki. Bayan wani lokaci, sun riƙi wani zamani look, godiya ga na game da bambanci ilimin lissafi.

A aiwatar da samar

Formation na bambanci ilimin lissafi a cikin nau'i na aikace-aikace, sa'an nan da kimiyya Hanyar faru kafin fitowan da falsafa ka'idar, wanda aka halitta da Nikolay Kuzansky. Aikinsa ne dauke su wani juyin ci gaba daga zamanin d kimiyya na shari'a. Duk da cewa da Falsafa da kansa ya ba da wani lissafi, taimakonsa da ci gaban ilmin lissafi kimiyya ne undeniable. Cusa, daya daga cikin na farko daga cikin shawara na ilmin lissafi kamar yadda mafi m kimiyya, ilimin lissafi sa da lokaci a cikin tambaya.

A zamanin d lissafi duniya rarrabẽwa ta zama ɗaya, yayin da Falsafa samarwa a matsayin wani sabon ma'auni rashin iyaka koma ainihin lambar. A dangane da wannan inverted misali daidaito a ilmin lissafi kimiyya. Kimiyya da ilmi ba, a ra'ayinsa, ne zuwa kashi m kuma na fasaha. Na biyu shi ne mafi m, bisa ga masanin kimiyya, tun da tsohon ba kawai m sakamakon.

ra'ayin

Ainihin ra'ayin da kuma manufar bambanci ilimin lissafi hade da aiki a kananan unguwa da wasu muhimmai. Domin wannan shi ne zama dole don ƙirƙirar wani ilmin lissafi na'ura zuwa aiki nazarin wanda hali a wani karamin unguwa na da maki shigar kusa da hali na wani mikakke aiki ko wani polynomial. Bisa ga wannan definition of wanda aka samu da bambanci.

Ana zargin manufar da wanda aka samu aka sa ta a manyan yawan matsaloli na kimiyyar yanayin halitta da kuma lissafi, wanda ya kai ga tabbatar da dalilin da iyaka dabi'u na wannan irin.

Daya daga cikin manyan ayyuka da cewa an bai wa wani misali, lokacin da na fara da mafi tsufa azuzuwan makarantar, shi ne domin sanin gudun motsi na da wata ma'ana a cikin wani madaidaiciya line kuma gina tangent line ga wannan kwana. A bambanci nasaba da wannan, tun da shi ne yiwu ga m da aiki a kananan unguwa na gab da wani mikakke aiki.

Idan aka kwatanta da manufar wanda aka samu da wani aiki na wani real m, da definition differentials kawai ya wuce a kan aiki na janar yanayi, musamman siffar wani Euclidean sarari zuwa wani.

wanda aka samu

Bari batu motsa a cikin shugabanci na y-axis, domin lokacin da muka dauka x, wanda aka auna daga farkon lokacin. Bayyana irin wannan yunkuri shi ne zai yiwu ta aiki y = f (x), wanda ake dangantawa ga kowane lokaci batu x tsara displaceable batu. Wannan aiki kira a makanikai dauki doka na motsi. Babban halayyar da motsi, musamman m, ne da instantaneous gudu. Lokacin da aya da aka koma tare da y-axis bisa ga dokar da makanikai, da bazuwar lokaci ma'ana da ta samu daidaita x f (x). A lokacin aya x + Δh, inda Δh wakiltar increment na lokaci, shi zai kordinaty f (x + Δh). Kamar wancan kafa dabara Δy = f (x + Δh) - f (x), wanda aka kira wani increment aiki. Yana da wani batu na hanyar traversed a lokacin da lokaci daga x zuwa x + Δh.

A dangane da abin da ya faru na gudu a lokacin da wanda aka samu yana gudanar. The wanda aka samu da wani aiki a wani tsayayyen batu da ake kira iyaka (dauka ya wanzu). Yana za a iya kira a wasu haruffa:

f '(x), y', y, DF / DX, DY / DX, DF (x).

A tsari na kirga da wanda aka samu daga kira bambantawa.

Bambanci ilimin lissafi na ayyuka da dama masu canji

Wannan hanya aka yi amfani da lokacin da kirga aiki nazari, da dama canji. Lokacin da akwai guda biyu masu canji x da kuma y, da m wanda aka samu game da x a aya A ake kira da wanda aka samu da wannan aiki a cikin x da tsayayyen y.

Iya nuna wadannan alamomin:

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

ake bukata basira

Domin samu nasarar koyi da kuma iya warware diffury bukata basira a hadewa da bambantawa. Don yin shi sauki gane bambanci lissafai, dole ne a fahimci topic wanda aka samu da kuma m na game. Har ila yau, ba ciwo su koyi su duba ga wanda aka samu da a fakaice aiki. Wannan shi ne saboda da cewa a cikin aiwatar da koyo zai sau da yawa amfani integrals da bambantawa.

Iri bambanci lissafai

Kusan duk da iko aikin dangantaka da na farko-domin bambanci lissafai, akwai 3 iri lissafai: kama, tare da gaßoßi canji, mikakke inhomogeneous.

Akwai kuma mafi rare jinsuna lissafai da total differentials, Bernoulli ta lissafi, da sauransu.

muhimmai mafita

Don fara, ya kamata mu tuna da shi ne algebraic Jadawalin a makaranta hanya. Sun ƙunshi canji da lambobi. Domin warware al'ada lissafi kamata sami yalwa da lambobin cewa gamsar da kayyade yanayin. Yawanci, wadannan lissafai da daya tushen, kuma domin Ingancin kamata kawai musanya wannan darajar cikin wuri ba a sani ba.

A bambanci lissafi ne kama da wannan. A general, wani lissafi na farko domin qunshi:

• Independent m.
• A wanda aka samu daga na farko aiki.
• Aiki ko ka dogara da m.

A wasu lokuta, ƙila a sami wata daya ba a sani ba, x ko y, amma shi ne ba da muhimmanci a matsayin wajibi ne a yi na farko wanda aka samu, tare da ba fi domin Kalam zuwa cikin bayani da bambanci ilimin lissafi sun gaskiya.

Warware bambanci lissafi - yana nufin a sami sa na dukkan ayyuka da cewa su dace ba magana. Irin wannan sets na ayyuka ne sau da yawa ake kira general bayani iko.

na game ilimin lissafi

Na game ilimin lissafi ne daya daga cikin sassan da ilmin lissafi analysis, wanda ya yi nazarin ra'ayi na na game, Properties da hanyoyin da ta lissafi.

Sau da yawa da lissafi na na game faruwa a lokacin kirga fannin wani curvilinear siffar. By wannan yana nufin wani iyaka yanki, zuwa ga wanda qaddara yankin na rubũtacce polygon siffar da hankali karuwa a hannunsa, da kuma data gefen iya sanya kasa da wani baya kayyade sabani kananan darajar.

Babban ra'ayin a cikin lissafi na yanki na wani lissafi da siffar da aka kirga fannin wani murabba'i mai dari, sa'an nan akwai shaida cewa ta yanki ne daidai da samfurin na tsawon da nisa. Idan ya zo ga lissafi, to, duk da constructions ake yi amfani da wani shugaba da kamfas, sa'an nan da rabo daga tsawon to nisa ne m darajar. Lokacin da kirga fannin wata dama alwatika za a iya ƙaddara cewa idan ka sa a gaba alwatika, wani murabba'i mai dari aka kafa. A cikin yankin na parallelogram aka lasafta a cikin irin wannan amma dan kadan mafi rikitarwa hanya, a cikin wani murabba'i mai dari da alwatika. A fannin wani polygon yana dauke da triangles kunshe a shi.

A kayyade rahamar sabani, wannan hanya ba ya shige da kwana. Idan muka karya shi a cikin mutum murabba'ai, shi zai zama unfilled wuraren. A wannan yanayin, kokarin yin amfani da biyu dasu, tare da rectangles sama da kasa, a sakamakon wadanda sun hada da jadawali na aikin da bai hada. Muhimmanci a nan shi ne hanyar warware wadannan rectangles. Har ila yau, idan muka dauki hutun more kuma mafi rage, yankin na saman da kasa kamata converge a kan wani daraja.

Ya kamata komawa zuwa wani hanya domin raba cikin rectangles. Akwai biyu rare hanyoyin.

Riemann aka wa dai da definition na integral, halitta ta Leibniz da Newton, a matsayin yanki na subgraph. A wannan yanayin, muna dauke da wani adadi ya kunshi wani yawan tsaye rectangles samu ta hanyar rarraba da tazara. Lokacin da keta wani karu akwai iyaka ga abin da rage yanki na irin wannan adadi, ana kiran wannan kayyadewa da Riemann na game da wani aiki a wani ajali tazara.

A na biyu Hanyar ne gina Lebesgue na game, kunsha a gaskiyar cewa a cikin wuri na rabuwa sanya yankin a kan wani ɓangare na integrand da Tattara bayanan sa'an nan da na game Naira Miliyan Xari da dabi'u samu a cikin wadannan sassa, a kan jinkiri raba ta kewayon dabi'u, sa'an nan summed da daidai matakan kishiya images of wadannan integrals.

zamani AIDS

Daya daga cikin manyan abũbuwan amfãni a gare nazarin bambanci da na game ilimin lissafi Fikhtengol'ts rubuta - "na bambanci da na game ilimin lissafi." Ya littafi ne mai asali ga kayan aiki da nazarin ilmin lissafi bincike, wanda ya msa da yawa bugu da fassara shi zuwa wasu harsuna. Created ga dalibai da kuma na dogon lokaci a yi amfani da wani iri-iri na makarantun a matsayin daya daga cikin babban amfanin da binciken. Yana bada msar tambayar bayanai da kuma m basira. Farko da aka buga a 1948.

Shikenan lissafin bincike aiki

Don bincika hanyoyin da bambanci ilimin lissafi aiki, kana bukatar ka bi an riga an ba algorithm:

1. Nemo yankin na aiki.
2. Nemo da tushen da aka lissafi.
3. Lissafi da tsakaitãwa. Don yin wannan, mun lissafta da wanda aka samu da kuma batu inda shi ne ya daidaita da sifili.
4. Muka musanya darajar samu a Eq.

Irin bambanci lissafai

Control na farko domin (in ba haka ba, bambanci ilimin lissafi na daya m) da kuma su iri:

• Tare da gaßoßi canji lissafi: f (y) DY = g (x) DX.
• A sauki lissafi ko bambanci ilimin lissafi aiki na daya m, da ciwon da dabara: y '= f (x).
• A mikakke farko-domin nonuniform iko: y '+ P (x) y = Q (x).
• Bernoulli bambanci lissafi: y '+ P (x) y = Q (x) y wata.
• Jadawalin total differentials tare da: P (x, y) DX + Q (x, y) DY = 0.

A bambanci lissafai na biyu domin kuma su iri:

• Kama mikakke biyu domin bambanci lissafi tare da m ^{coefficients: y n + py '+ qy} = 0 p, q nasa R.
• Inhomogeneous mikakke biyu domin bambanci lissafi tare da m coefficients ^{darajar: y n + py '+ qy} = f (x).
• Kama mikakke bambanci ^{lissafi: y n + p (x)} y '+ q (x) y = 0, kuma inhomogeneous na biyu domin ^{lissafi: y n + p (x)} y' + q (x) y = f (x).

Bambanci lissafai na mafi girma umarni da su iri:

• A bambanci lissafi, kyale rage daga cikin ^{tsari: F (x, y (k} ), y (k + ^{1), .., y (n)} = 0.
• A mikakke Jadawalin mafi girma domin ^{_{kama: y (n) + f (}} n- ₁₎ y ^(n-1) + ... + f ₁ y '+ f ₀ y = 0, kuma ^{_{inhomogeneous: y (n) + f (}} n _-1) y ^(n-1) + ... + f ₁ y '+ f ₀ y = f (x).

Matakai na warware matsalar da bambanci lissafi

Da taimakon ramut an warware ba kawai lissafi ko ta jiki da matsaloli, amma kuma matsaloli daban-daban na nazarin halittun, tattalin arziki, ilimin halayyar zaman jama'a da sauransu. Duk da fadi da dama batutuwa, kamata ya bi guda dabaru jerin domin warware wadannan matsaloli:

1. Jawo sama iko. Daya daga cikin mafi wuya, saukarwa, wanda na bukatar iyakar daidaito, saboda wani kuskure zai kai ga gaba daya ba daidai sakamakon. Wajibi ne a yi la'akari da dukkan abubuwan da shafi aiwatar da kuma sanin farko yanayi. Ya kamata kuma a bisa gaskiya da ma'ana karshe.
2. Domin warware lissafai. Wannan tsari ne mai sauki a batu na farko, tun da shi na bukatar kawai m aiwatar da ilmin lissafi lissafin.
3. Analysis da kimantawa da sakamakon. Samu bayani kamata a kiyasta ga shigarwa na m da msar tambayar darajar da sakamakon.

Wani misali na amfani da bambanci lissafai a magani

Amfani da ramut a fagen magani da aka samu a cikin shiri na ANNOBA ilmin lissafi model. Kada mu manta cewa wadannan lissafai an kuma samu a ilmin halitta da kuma sunadarai, wanda suke kusa da magani, saboda shi taka muhimmiyar rawa nazarin daban-daban nazarin halittu alƙarya da kuma sunadarai matakai a cikin jikin mutum.

A cikin wannan misali, da annoba baza kamuwa da cuta za a iya bi a wani zama ruwan dare al'umma. The mazaunan kasu kashi uku iri:

• Cutar, da yawan x (t), wanda ya kunshi mutane, dauke da kwayar cutar dako, wanda kowannensu yana da cututtuka (shiryawa zamani ne takaice).
• Nau'i na biyu ya hada da saukin kamuwa mutane y (t), za a iya kamuwa da lamba tare da kamuwa da.
• The uku irin hada da tsaurin mutane z (t), wanda suke rigakafi ko rasa saboda rashin lafiya.

Yawan mutane da kullum, kiyaye haihuwa, na halitta mutuwa, da kuma hijirarsa ba a dauke. A core zai zama biyu shiriritar.

Kashi cuta a wasu lokaci nufi shi ne daidai x (t) y (t) (tushen zato a kan ka'idar cewa yawan lokuta a gwargwado ga yawan intersections tsakanin marasa lafiya da kuma m mambobi, wanda a farko kimantawa ne na gwargwado ga x (t) y (t)), a saboda haka yawan lokuta na karuwa ne, da kuma yawan saukin kamuwa tayi a wani kudi wanda aka lasafta ta da dabara gatari (t) y (t) (a> 0).

Yawan wadanda ba karɓãwa, dabbobi da cewa ya mutu ko samu rigakafi, ya karu a wani kudi wanda shi ne na gwargwado ga yawan lokuta, bx (t) (b> 0).

A sakamakon haka, za ka iya kafa wani tsarin na lissafai tare da dukan uku Manuniya a kan tushen da ta karshe.

MISALI amfani tattalin arziki

Bambanci ilimin lissafi ne sau da yawa amfani da tattalin arziki analysis. Babban aiki a cikin tattalin arziki bincike da aka dauke su a nazarin da dabi'u na tattalin arziki, wanda aka rubuta a cikin nau'i na aiki. Ana amfani da a warware matsaloli kamar canje-canje a samun kudin shiga da haraji ƙaruwa nan da nan bayan, shigarwa kudade, canje-canje a cikin kudaden shiga a lokacin da canza darajar da samfurin, a abin da rabo za a iya maye gurbinsu da ritaya ma'aikata tare da sabon kayan aiki. Don warware irin wannan matsaloli, shi ake bukata don gina wani sadarwar aiki na shigowa canji, wanda, bayan ta binciken bambanci ilimin lissafi.

A cikin yanayin tattalin arziki, yana da mahimmancin samun samfurin mafi kyau duka: iyakar yawan aiki, yawan kudin shiga, farashin mafi ƙasƙanci, da sauransu. Kowane irin alamar yana aiki ne akan ɗaya ko fiye da muhawara. Alal misali, ana iya ɗaukar samfuri a matsayin aiki na fitar da aiki da babban birnin. A wannan haɗin, ana iya rage darajar da aka dace don gano iyakar ko mafi ƙarancin aiki daga ɗaya ko fiye da masu canji.

Irin waɗannan matsaloli suna haifar da matsala mafi girma a filin tattalin arziki, wanda ma'auni na daban ya zama dole. Lokacin da alamar tattalin arziki ya kamata a rage girmanta ko ƙaddara a matsayin aiki na wani alamar, to, a matsakaicin iyakar raƙumar ƙaddamar da aikin zuwa ga muhawarar ba za ta ɓace ba idan ƙaddamar da gardamar ta nuna ba kome ba. In ba haka ba, idan irin wannan hali ya nuna wani abu mai kyau ko mummunan darajar, mahimmin bayani bai dace ba, saboda tare da ƙara ko rage ƙwararrakin yana yiwuwa ya canza darajar dogara a cikin jagoran da ake bukata. A cikin maganganun da ke tattare da ma'anar bambancin wannan yana nufin cewa yanayin da ake buƙata don matsakaicin aikin shine ƙananan siffar da aka ƙera.

A cikin tattalin arziki akwai matsalolin gano matakai na aiki tare da yawancin maɓuɓɓuka, saboda alamun tattalin arziki sun ƙunshi abubuwa da dama. Irin waɗannan tambayoyin suna nazari sosai a cikin ka'idar ayyuka da dama masu yawancin dake amfani da hanyoyi na ƙididdiga daban. Irin waɗannan ayyuka sun hada da ba kawai ƙaddamar da rage girman ayyuka ba, amma har iyakancewa. Tambayoyi kamar sun shafi tsarin ilimin lissafi, kuma an warware su tare da taimakon hanyoyin da aka haɓaka musamman, kuma bisa ga wannan ɓangaren kimiyya.

Daga cikin hanyoyin hanyoyin da aka saba amfani dashi a cikin tattalin arziki, wani ɓangaren muhimmin sashi shi ne bincike mai zurfi. A cikin yanayin tattalin arziki, wannan lokacin yana nufin hanyar da za a yi nazarin alamomi da ma'ana masu sauƙi yayin canza tsarin kundin halitta, amfani, bisa tushen nazarin iyakokin su. Ƙididdiga ƙayyade shi ne ƙayyadaddun abu ko ƙayyadaddun hanyoyi tare da wasu masu canji.

Nassin bambancin da dama masu canji yana da muhimmiyar matsala daga filin bincike na ilmin lissafi. Don nazarin cikakken bayani, ana iya amfani da kayan koyarwa daban-daban ga makarantun firamare mafi girma. Ɗaya daga cikin shahararren marubucin ya halicci Fichtenholz - "Hanyar bambanci da haɗin kai." Kamar yadda yake a fili daga lakabi, ƙwarewar aiki tare da haɗin kai yana da muhimmanci ƙwarai don magance nau'o'in bambancin. Lokacin da aka bambanta nauyin aikin aiki daya, ya zama mafi sauki. Kodayake, ya kamata a lura, yana bin dokoki guda ɗaya. Domin yin aiki da aikin a cikin mahimmancin lissafi, ya isa ya bi algorithm da aka samo a yanzu, wanda aka ba a cikin digiri na sama na makaranta kuma yana da rikicewa kaɗan lokacin da aka shigar da sabon canji.