Differential calculus mitambo mumwe uye variables akawanda

Differential calculus iri bazi kwemasvomhu pachisara iyo anoongorora rinobva, differentials uye kushandiswa kwawo kudzidza mabasa.

Nyaya

Differential calculus yakabuda sezvo yakazvimirira chirango yechipiri yezana remakore rechi17, anoonga basa Newton uye Leibniz, uyo akagadzira dzinokosha zvokudya dzemaitirwe pakati differentials uye akacherechedza kubatana pakati kusangana uye Kusiyanisa. Sezvo chirango akava pamwe kukarukureta pakati integrals, nokudaro vanoumba hwaro kwemasvomhu kuongorora. Chimiro calculi izvi akazarura mutsva nguva ano munyika kwemasvomhu nyika uye akaita kwaita anoranga musayenzi itsva. Uyewo akawedzera mukana kwokushandisa masvomhu mune zvesayenzi zvepanyama uye ouinjiniya.

zvinhu pfungwa

Differential calculus inobva chinokosha pfungwa wemasvomhu. Ivo: chaiyo nhamba, akaramba uye panogumira basa. Papera nguva, vakatora ano Kutarira, nemhaka chinokosha uye differential calculus.

Muitiro pakusika

Paumbwe differential calculus muchimiro kushanda, uye ipapo nzira yesayenzi chakaitika pamberi chidzoke kwouzivi dzidziso, hwakasikwa maererano Nikolay Kuzansky. basa rake riri kuonekwa-shanduka kukura kubva kare sayenzi mutongo. Pasinei chokwadi kuti muzivi pachake haana yemasvomhu, mupiro wake kuti pave kwemasvomhu sayenzi hazvirambiki. WokuCusa, mumwe wokutanga kubva kufunga svomhu sezvo yakarurama zvikuru sainzi, masvomhu vachiisa nguva kupinda mubvunzo.

In mathematicians yekare yezvisikwa chokuratidza aiva chairema, apo muzivi wakatangwa idzva nechiyero kusaverengeka kumudzorera nhamba chaiyo. Maererano kumiririrwa ichi inverted zvakarurama mune kwemasvomhu nesayenzi. ruzivo Scientific, ari maonero ake, rakakamurwa ouchenjeri uye vakangwara. Wechipiri zvakawanda yakarurama, maererano musayendisiti, sezvo vaimbove anopa anofungidzirwa nemigumisiro chete.

zano

The chikuru Pfungwa uye pfungwa differential calculus chokuita basa duku pedyo dzimwe pfungwa. Nokuti ichi zvakafanira kusika munhu akarongeka midziyo kushanda zvidzidzo vane maitiro duku pedyo pfungwa akagadzawo pedyo maitiro ane Linear nerevane kana polynomial. Kwakavakirwa tsanangudzo iyi rinobva uye differential.

The kwaita pfungwa yacho rinobva kwakakonzerwa yokuwanda zvinetso sayenzi dzinongoitika uye masvomhu, izvo zvakazoita kuti kutsunga muganhu tsika imwe mhando.

Mumwe mukuru mabasa dzinopiwa somuenzaniso, kutanga vakuru makirasi chikoro, ndiko kuona kumhanya inofamba ane pfungwa nenzira yakarurama mutsetse uye kuvakwa tangent mutsetse kuti munyonga ichi. The differential chokuita ichi, sezvo zvichiita kuti paiva basa duku pedyo pfungwa imwe Linear basa.

Tichienzanisa pfungwa rinobva ane basa rimwe shanduka chaiyo, tsanangudzo differentials anongoti anoudzawo mashandiro vakawanda hunhu, zvikurukuru mufananidzo Euclidean nzvimbo mumwe.

rinobva

Regai pfungwa inofamba nenzira iri and-akabatana, nokuti nguva tinotora X, riri akayera kubva pakutanga kwechinguvana. Tsanangura sangano kwakadaro kunobvira nokuda mashandiro and = d (x), izvo Rinobatanidzwa kusvikira nguva imwe neimwe pfungwa X batanidza displaceable pfungwa. Izvi basa kufona mune hurongwa kutora mutemo inofamba. The chikuru tive inofamba, kunyanya gogonera, ndiye ari nguva imwe velocity. Kana pfungwa inodedera achitevedza and-akabatana maererano nomurayiro hurongwa, asi chero nguva pfungwa nayo anowana batanidza X add (x). Panguva pfungwa X + Δh, apo Δh anomiririra increment kwenguva, izvozvo zvichaita kordinaty add (x + Δh). Saka kuumbwa fomura Δy = add (x + Δh) - F (x), unonzi ane increment chaitika. Zviri kurehwa munzira zvaiyambuka panguva kubva X kusvikira X + Δh.

Chokuita panowanikwa velocity panguva rinobva chinotarisirwa. The rinobva chero mashandiro panguva yakatarwa nguva inonzi muganhu (kuzvitutumadza riripo). Zvinogona kunzi vamwe kutaurwa:

d '(x), y', y, df / yori, Dy / yori, Df (x).

Muitiro pakuverenga kuti rinobva kudanwa Kusiyanisa.

Differential calculus kuti mashandiro anoita variables akawanda

Nzira iri rinoshandiswa kana pakuverenga basa kudzidza, vanoverengeka variables. Kana pane mbiri variables X uye Y, ari kukwana rinobva noruremekedzo kuti X panguva A rinonzi rinobva kuti basa iri X aine wakaiswa Y.

Kungaratidzwa zvinotevera nemifananidzo:

d '(x) (x, Y), u' (x), ∂u / ∂x uye ∂f (x, y) '/ ∂x.

zvaida unyanzvi

Kuti vabudirire kudzidza uye kukwanisa kugadzirisa diffury zvaida unyanzvi vari kusangana uye Kusiyanisa. Kuti zvive nyore kunzwisisa differential equations, anofanira kunzwisiswa wenyaya rinobva uye nokusingagumi chinokosha. Uyewo hazvirevi kurwadzisa kudzidza kutsvaka rinobva pamusoro zvakazara basa. Izvi imhaka yokuti vari mubishi kudzidza kazhinji vanoshandisa integrals uye Kusiyanisa.

Mhando differential equations

Zvinenge zvose kudzorwa basa chokuita yomuzana murayiro differential equations, pane 3 mhando equations: homogeneous, pamwe kupatsanurwa variables, Linear inhomogeneous.

Kunewo mamwe mashoma mhando equations pamwe zvachose differentials, Bernoulli kuti mukana wokuti vanhu vanga, uye vamwe.

kunyatsogona zvingaitwa

Chokutanga, tinofanira kuyeuka ndechokuti algebraic mukana wokuti vanhu vanga Chokwadika chikoro. Ane variables uye nhamba. Kuti vagadzirise kwakajairika mukana wokuti vanhu vanga vanofanira kuwana zvakawanda nhamba kuti kugutsa chaiyo ezvinhu. Kazhinji, equations ava mudzi mumwe, uye kuti chivimbo chakasimba kunofanira chete kuisa ukoshi ichi munzvimbo USINGAZIKAMWI.

The differential mukana wokuti vanhu vanga zvakafanana ichi. Kazhinji, munhu mukana wokuti vanhu vanga wokutanga kuti unotora:

• Independent shanduka.
• A rinobva wokutanga basa.
• Basa kana muyanzvi shanduka.

Mune zvimwe zviitiko, pangava pasina isingazivikanwi, X kana Y, asi hazvisi chinokosha sezvo zvakafanira kuti rinobva wokutanga, pasina yepamusoro kuitira nomukaka kuti mhinduro uye differential calculus zvaiva zvechokwadi.

Kugadzirisa differential mukana wokuti vanhu vanga - zvinoreva kuti kuwana seti mitambo zvose zvakakodzera kupiwa mashoko. nemhando kwakadaro mitambo unowanzonzi zvavo mhinduro kuzvidzora.

chinokosha calculus

Chinokosha calculus mumwe zvikamu kwemasvomhu pachisara iyo inoongorora pfungwa chinokosha, zvivako uye nzira kwayo kukarukureta.

Kazhinji dzemaitirwe yacho musimboti kunoitika apo pakuverenga nzvimbo imwe curvilinear chimiro. By ichi zvinoreva paunogumira nharaunda, vakananga iyo akafanorongerwa kunharaunda rakanyorwa polygon chimiro ne zvishoma nezvishoma kuwedzera rwake, uye mashoko divi anogona kuitwa zvishoma kupfuura kare wainyatsotaura zvemasanga duku ukoshi.

Nepfungwa huru iri kuverenga kwenzvimbo upi geometric chimiro ari pakuverenga nzvimbo imwe rectangle, saka pane uchapupu hwokuti nzvimbo yayo akaenzana zvibereko urefu pedyo upamhi. Kana totaura geometry, ipapo zvose zvakagadzirwa dzinoitwa kushandisa mubati kambasi, uye ipapo nhamba urefu kuti upamhi chinhu ndaisagona ukoshi. Kana pakuverenga nzvimbo netriangle zvakanaka inogona tsunga kuti kana waisa rinotevera, triangle, mumwe rectangle aumbwa. Munzvimbo yacho parallelogram vari dzaona mune nzira yakafanana asi zvishoma zvikuru kwakatooma, mukati rectangle uye netriangle. Munzvimbo imwe polygon runoonekwa zvokuridza zvaisanganisira mariri.

Pakusarudza ngoni zvemasanga, nzira iyi hairevi munyonga. Kana isu kuiputsa kupinda munhu dzinoungana, izvozvo vacharamba unfilled nzvimbo. Muchiitiko chino, edza kushandisa shati mbiri, pamwechete rectangles kumusoro uye pasi, somugumisiro avo zvinosanganisira girafu pakati mashandiro uye haibatanidzi. Kukosha pano ndiyo nzira kuvhuna rectangles izvi. Uyewo, kana isu kutora rotonhorera zvakanyanya kuderedzwa, nzvimbo kumusoro uye pasi vanofanira wekuita rimwe ukoshi.

Zvinofanira kudzokera nzira kuti kupatsanura kupinda rectangles. Pane vaviri dzakakurumbira nzira.

Riemann akanga formalized tsanangudzo chinokosha, wakasikwa Leibniz uye Newton, sezvo nzvimbo subgraph. Muchiitiko chino, isu vaiona munhu zvinosanganisira imwe nhamba kutwasuka rectangles yaunganidzwa vachigoverana penguva. Kana nokuputsa vasanyanye pane muganhu kuti izvo kwakaderedza nzvimbo munhu akadaro, muganhu uyu anonzi Riemann chinokosha ane mashandiro panguva yakati penguva.

A Nzira yechipiri kuti vavake Lebesgue chinokosha, wakaumbwa chokuti munzvimbo kuparadzana yakasarudzwa nzvimbo iri chikamu integrand uye rokunyora ipapo chinokosha Verengai tsika vakawana muzvikamu izvi, pane zvikamu akaparadzanisa kwayo siyana tsika, uye ipapo muchidimbu nemashoko runoenzanirana matanho Ivan Kuchin mifananidzo integrals izvi.

anobatsira ano

Mumwe mukuru zvinobatsira pakudzidza differential uye chinokosha calculus Fikhtengol'ts akanyora - "pamusoro differential uye chinokosha calculus." Bhuku rake chinhu chinokosha chokushandisa pakudzidza kwemasvomhu pachisara izvo hwevaPeresia shanduro dzakawanda uye shanduro kupinda mimwe mitauro. Created vadzidzi uye kwenguva refu rakashandiswa dzakasiyana-siyana nemasangano dzidzo somumwe chikuru kubatsira pakudzidza nguva. Rinopa yokungotaura ruzivo uye unyanzvi hunobatsira. First rakabudiswa muna 1948.

Algorithm tsvakurudzo basa

Kuongorora nzira differential calculus basa, unofanira kutevera atopiwa kupiwa algorithm:

1. Find umambo mashandiro.
2. Find Midzi kupiwa mukana wokuti vanhu vanga.
3. Ngaaverenge kunyanyisa. Kuti aite izvi, isu ngaaverenge rinobva uye pfungwa apo akaenzana razero.
4. We kuisa ukoshi kuwanikwa muna Eq.

Mhando dzakasiyana-siyana differential equations

Kudzora rokutanga kurongeka (zvikasadaro, differential calculus mumwe shanduka) uye avo mhando:

• With kupatsanurwa variables mukana wokuti vanhu vanga: f (Y) Dy = d (x) yori.
• The nyore mukana wokuti vanhu vanga kana differential calculus kushanda rimwe shanduka, ane nzira: y '= d (x).
• The Linear vomuzana hurongwa nonuniform kuzvidzora: y '+ P (x) Y = Q (x).
• Bernoulli differential mukana wokuti vanhu vanga: y '+ P (x) Y = Q (x) and a.
• Mukana wokuti vanhu vanga zvachose differentials pamwe: P (x, y) yori + Q (x, y) Dy = 0.

The differential equations chechipiri kurongeka uye avo mhando:

• Homogeneous Linear rechipiri kuti differential mukana wokuti vanhu vanga vane kugara ^{coefficients: y N + PY '+ qy} = 0 S., Q ndezvaiye R.
• Inhomogeneous Linear rechipiri kuti differential mukana wokuti vanhu vanga vane kugara coefficients ^{ukoshi: y N + PY '+ qy} = d (x).
• Homogeneous Linear differential ^{mukana wokuti vanhu vanga: y N + S. (x)} Y '+ Q (x) Y = 0, uye inhomogeneous wechipiri kurongeka ^{mukana wokuti vanhu vanga: y N + S. (x)} Y' + Q (x) Y = F (x).

Differential equations kuvakuru zvirevo vavo mhando:

• The differential mukana wokuti vanhu vanga, vachibvumira rutapudzo pamusoro ^{murayiro: F (x, y (K} ), y (K + ^{1), .., y (n)} = 0.
• A Linear mukana wokuti vanhu vanga kuvakuru kurongeka ^{_{homogeneous: y (n) + F (}} n- ₁₎ Y ^(n-1) + ... + f ₁ Y '+ add ₀ Y = 0, uye ^{_{inhomogeneous: y (n) + F (}} N _-1) Y ^(n-1) + ... + f ₁ Y '+ add ₀ Y = d (x).

Zvokukura kugadzirisa dambudziko pamwe differential mukana wokuti vanhu vanga

Achibatsirwa kure kuzvidzora vari kugadziriswa kwete chete masvomhu kana matambudziko eutano asiwo matambudziko akasiyana-siyana zvipenyu, zvoupfumi, evanhu uye vamwe. Pasinei-siyana dzokukurukura, vanofanira kutevera imwe chete pfungwa kutevedzana kuti matambudziko aya:

1. Kubudisa zvokuita. Mumwe yakaomesesa matanho, chinoda kunonyanyisa zvakarurama, nekuti chero chikanganiso kuchaita kuti migumisiro zvakaipa zvachose. Zvakakodzera kufunga zvose zvinokonzera kukanganisa kwacho uye kuona kutanga ezvinhu. Zvinofanira dzechokwadi uye musoro mhedziso.
2. Nokuti kugadzirisa equations. Izvi zviri nyore kutanga pfungwa, sezvo zvinoda chete kuzviomesera Implementation pamusoro masvomhu.
3. Analysis uye kunyatsoongorora nemigumisiro. Yaiwana mhinduro inofanira kuongororwa kuti kugadzwa hunobatsira uye yokungotaura kukosha mugumisiro.

Muenzaniso kushandisa differential equations mukurapa

Uchishandisa kure rokudzora zvokurapa inowanikwa rokuvaka epidemiological kwemasvomhu muenzaniso. Hatifaniri kukanganwa kuti equations ivava kuwanikwa zvipenyu uye kemisitiri, izvo pedyo mushonga, nokuti inoita basa rinokosha pakudzidza akasiyana zvoutachiona vanhu uye namakemikari pezvakaitwa mumuviri womunhu.

In muenzaniso uyu, denda kupararira chirwere chinogona kubatwa kwayo yoga munharaunda. Vagari vari muzvikamu zvitatu mhando:

• Neutachiona, uwandu X (T), iro raiva nevanhu, zvinotapukira vatakuri, mumwe nomumwe riri zvinotapukira (incubation nguva ipfupi).
• Chechipiri mhando zvinosanganisira muvamwe vanhu Y (T), zvinogona utachiona kuonana hweHIV.
• Wechitatu mhando zvinosanganisira refractory vanhu Z (T), izvo zvirwere kana vakarasika nokuda chirwere.

Number vanhu nguva dzose, nokuchengeta kubereka, dzinongoitika nzufu uye kutama hakunganzi. Musimboti vachava hypotheses maviri.

Muzana chirwere panguva imwe nguva nguva akaenzana X (T) Y (T) (inobva kubvuma pamusoro dzidziso kuti uwandu vanenge zvakaenzanirana nhamba panosangana pakati kuvarwere uye vanoteerera nhengo, izvo pakutanga rakada ndiyo Proportional kuti X (T) Y (T)), mu Naizvozvo nhamba nyaya iri kuwedzera, uye uwandu muvamwe kunodzikira pamwero iyo rinowanikwa ne nzira demo (t) Y (t) (a> 0).

Number yemhuka asiri responders kuti akafa kana akawana kutapata, vakawedzera pamwero riri Proportional neuwandu zviitiko bx (T) (b> 0).

Somugumisiro, unogona kumisa kwegadziriro equations vose zviratidzi zvitatu pachishandiswa ayo mhedziso.

MUENZANISO kushandiswa dzezvoupfumi

Differential calculus rinowanzoshandiswa zvoupfumi kuongorora. The chikuru basa iri zvoupfumi pachisara ichaongororwa kuva kudzidza tsika hwenyika, izvo zvakanyorwa muchimiro mashandiro. Rinoshandiswa kugadzirisa matambudziko akadai kuchinja mari mutero inowedzera pakarepo pashure, kupinda mari yechikoro, kuchinja zveminda apo kuchinja kukosha chigadzirwa, mune chii maererano zvinogona kutsiviwa pamudyandigere vashandi pamwe midziyo itsva. Kugadzirisa matambudziko akadai, zvinodikanwa kuvaka kukurukurirana kushanda kuuya variables, izvo, pashure pokunge adzidza kubudikidza differential calculus.

kazhinji zvakakodzera kuwana kupfuura Zvakanyanya Kunaka Panguva Yepamuviri kuita mune zvoupfumi bhora: kunonyanyisa zvibereko, chikuru mari, muduku mutengo uye zvichingodaro. Mumwe rokuti yakadaro Zvaiitwa mumwe kana kupfuura nenharo. Somuenzaniso, kugadzirwa anogona kunzi ari kushanda basa uye guru. Nezvenyaya iyi, wokuroorana ukoshi zvingaderedzwa nokuwana kufadza kana zvishoma pamusoro Zvaiitwa mumwe kana kupfuura variables.

Zvinetso zvakadaro pave boka extremal zvinetso zvemari munda, nokuti iyo unofanira differential calculus. Kana upfumi chakarurama kunodiwa kuderedza kana maximize ave Zvaiitwa nevamwe parameters, kuti increment reshiyo kunonyanyisa pfungwa mashandiro kuti nharo vanowanzotevedzera razero kana increment renharo ruchigona razero. Zvikasadaro, kana mafungiro akadaro uderedze imwe yakanaka kana akaipa kukosha, asi yakati pfungwa haisi akakodzera, nokuti inowedzera kana kuderedza nharo inogona kuchinjwa kubatsirwa kukosha mune achida mirayiridzo. In differential calculus mazwi, izvi zvingareva kuti waida zvinhu nokuti kunonyanyisa nerevane chinhu razero kukosha kwayo rinobva.

Upfumi kunowanzoitika dambudziko kuwana Extremum ane Zvaiitwa variables anoverengeka, nekuti zviratidzi kwoupfumi dzine zvinhu zvakawanda. nyaya dzakadaro zvakanaka kunzwisiswa dzidziso mashandiro anoita variables anoverengeka, maverengerwe pakuverenga ari differential. Zvinetso zvakadaro zvinosanganisira kwete chete maximized uye kushoma basa, asiwo kuita zvimwe. Mibvunzo iyi dzinesungano kwemasvomhu zvirongwa, uye vari kugadzirisa tichibatsirwa zvikuru dzakabudirira nzira vari kwakavakirwa bazi ichi nesayenzi.

Pakati nzira differential calculus yaishandiswa upfumi, unokosha chikamu ndiye bvunzo. In upfumi bhora, rinoreva kuva nzira kutsvakurudzawo shanduka kuita uye zvinoguma kana iwe kushandura vhoriyamu kwezvisikwa, kuparadzwa, kwakavakirwa kuongorora kwavo muganhu tsika. Kuderedza chiratidzo vaiona rinobva kana kukwana nomukaka pamwe variables vanoverengeka.

Differential calculus pamusoro variables akawanda - inokosha pamusoro kwemasvomhu kuongorora. Nokuti kudzidza hwakadzama, unogona kushandisa dzakasiyana-siyana pakudzidzisa dzidzo yepamusoro masangano. Imwe yakakurumbira akasika Fikhtengol'ts - "pamusoro differential uye chinokosha calculus." How zvikuru zita kuti agadzirise zviri differential equations ose zvikuru kukosha kuti unyanzvi kushanda integrals. Kana pane differential calculus mitambo mumwe shanduka, chisarudzo kunova nyore. Kunyange zvazvo, zvinofanira kucherechedzwa kuti anotevera chete zvinokosha mitemo. In tsika, kuongorora mashandiro ari differential calculus, kungoita hunotovapo algorithm, unopirwa chesekondari, uye kunzwisisa chete zvishoma pakatanga variables matsva.