Differentials - chii ichi? How kuwana differential pakati mashandiro?

Pamwe nomukaka mabasa avo differentials - nayo vamwe inokosha pfungwa yacho differential calculus, chikamu chikuru pakati kwemasvomhu kuongorora. Sezvo zvakabatana chaizvo, vose emakore anoverengeka zvikuru kushandiswa kugadzirisa matambudziko anenge zvose rakamuka kosi musayenzi uye michina basa.

The kwaita pfungwa differential

Kokutanga akajekesa kuti differential akadaro, mumwe havaoni (pamwe Isaakom Nyutonom) differential calculus akakurumbira German yemasvomhu Gotfrid Vilgelm Leybnits. Tisati kuti mathematicians muzana remakore rechi17. akashandisa kujeka chaizvo uye kujeka pfungwa yevamwe "Batanidzai" kamwoyo upi anozivikanwa basa, anomirira kugara ukoshi vashomashoma asi haana kuenzana zero, pazasi iyo anokoshesa basa havagoni kuva nyore. Nokudaro paingova danho rimwe pakatanga pfungwa pamusoro kamwoyo increments ose basa nharo uye avo increments pamusoro mashandiro kuti kungaratidzwa achirifananidza nomukaka wevakabhuroka. Uye danho iri rakatorwa anenge panguva pamusoro apa masayendisiti maviri makuru.

Zvichienderana aifanira kugadzirisa okukurumidzira inoshanda hurongwa matambudziko avanosangana sayenzi nokukurumidza kutanga bhizimisi uye ano, Newton uye Leibniz akasika vanhuwo nzira yokuwana mashandiro anoita mwero kuchinja (kunyanya panyaya zvokuimba kumhanya muviri anozivikanwa trajectory), izvo zvakazoita kuti pakatanga pfungwa dzakadaro, sezvo rinobva basa uye differential, uye akawana algorithm Ivan Kuchin dambudziko mhinduro sezvo kuzivikanwa muzana se (shanduka) aimhanya zvaiyambuka kuwana nzira kuti zvakakonzera pfungwa chakakosha Ala.

In mabasa Leibniz naNewton pfungwa pakutanga akazviratidza kuti differentials - ndiyo Proportional kuti increment anokosha nharo Δh increments Δu mabasa anogona zvinobudirira kureva ngaaverenge kukosha yokupedzisira. Nemamwe mashoko, ivo vakaona kuti increment basa kunogona chero pfungwa (kwayo umambo tsanangudzo), chinoratidzwa nemashoko ayo rinobva ari Δu = y '(x) Δh αΔh + apo α Δh - Zvakasara ruchigona razero apo Δh → 0, zvikuru kupfuura chaiko Δh.

Maererano havaoni kuti akarongeka pachisara ari differentials - ndiwo izwi rokutanga increments chero mabasa chaiwo. Kunyange pasina kuva zvakajeka muganhu pfungwa sequences kunonzwisiswa intuitively kuti differential kukosha rinobva kunoita kushanda apo Δh → 0 - Δu / Δh → Y '(x).

Kusiyana Newton, uyo akanga kuchikonzerwa yefizikisi uye akarongeka midziyo vaiona webetsero chishandiso pakudzidza yokubatwa, Leibniz vainyatsochenjerera Toolkit ichi, kusanganisira kwegadziriro rinoonekwa uye inonzwisisika zviratidzo kwemasvomhu tsika. Ndiye zvinganaka mureza notation pakati differentials basa Dy = y '(x) yori, yori, uye rinobva renharo chemuTestamente kwavo ukama Y' (x) = Dy / yori.

The ano tsananguro

Ndeipi differential tichitarisa masvomhu ano? Zvinonzi chokuita pfungwa yokuti munhu shanduka increment. Kana shanduka and zvinotoda kukosha wokutanga ja ja = _1, ipapo ja = y _2, musiyano and ₂ ─ and ₁ inonzi increment ukoshi and. The increment inogona akanaka. akaipa uye razero. Shoko "increment" akasarudzwa Δ, Δu achinyora (verenga 'Delta Y') rinoreva kukosha increment and. saka Δu = y ₂ ─ and _1.

Kana mutengo Δu zvemasanga nerevane Y = d (x) zvinogona achimiririrwa Δu = A Δh + α, apo A hapana kuvimba Δh, t. E. A = const nokuda kupiwa X, uye izwi α apo Δh → 0 uderedze zviri kunyange kupfuura chaiko Δh, ipapo yokutanga ( "tenzi") izwi Proportional Δh, uye ari ja = d (x) differential, runotaurwa Dy kana df (x) (verenga "ja", "De eff kubva X"). Naizvozvo differentials - a "chikuru" Linear panyaya zvinoumba increments Δh mabasa.

zvokuimba tsananguro

Regai S = d (T) - nechekure ari yakarurama mutsetse chinofamba zvinhu pfungwa kubva pakutanga nzvimbo (T - nguva kufamba). Increment Δs - ndiyo nzira yacho panguva penguva Δt, uye differential DS = d '(T) Δt - nzira iyi, iyo pfungwa raizoitwa kuti imwe nguva Δt, kana rakaramba nokukurumidza add' (T), yakasvika panguva T . Kana ane Δt DS zvekufungidzira nzira kamwoyo yakasiyana kubva Δs chaiko Zviduku kuva yepamusoro kuti panyaya Δt. Kana pakukurumidza panguva T haana kuenzana zero, kuti fungidzira ukoshi DS anopa diki kutsvetera pfungwa.

geometric dudziro

Regai mutsetse L ndiye girafu kuti ja = d (x). Zvadaro Δ x = MQ, Δu = QM '(ona. Figure pazasi). Tangent MN Anoparadza Δu akatema migove miviri, QN uye NM '. First uye Δh ndiye Proportional QN = MQ ∙ nkm (angled QMN) = Δh F '(x), t. E QN ndiye Dy differential.

Chikamu musiyano Δu NM'daet ─ Dy, apo Δh → 0 NM urefu 'zvinoderedza kunyange kupfuura increment yacho nharo, wechipiri kureva ine murayiro smallness kupfuura Δh. Muchiitiko chino, kana e '(x) ≠ 0 (non-yakafanana tangent mombe) zvikamu QM'i QN yakaenzana; mune mamwe manzwi NM 'zvinoderedza nokukurumidza (sezvakarayirwa smallness yaro yepamusoro) kupfuura yose increment Δu = QM'. Izvi zvinooneka Figure (kusvika chidimbu M'k M NM'sostavlyaet zvose maduku kubva muzana QM 'chidimbu).

Saka, zvakajeka differential zvemasanga nerevane akaenzana increment ari ordinate yacho tangent.

Rinobva uye differential

A chinhu chokutanga izwi rwekutaura increment basa akaenzana kukosha kwayo rinobva add '(x). Saka, zvinotevera ukama - Dy = d '(x) Δh kana df (x) = F' (x) Δh.

Zvinozivikanwa kuti increment ari yakazvimirira nharo akaenzana ayo differential Δh = yori. Saka, tinogona kunyora: f '(x) yori = Dy.

Kuwana (dzimwe nguva akati kuva "Chisarudzo") differentials iri kuteererwa chete mitemo chete nomukaka. A pamazita vavo anopiwa pazasi.

Chii chinonyanya inoshanda: ari increment renharo kana ayo differential

Pano zvakafanira kuti zvimwe zvijeke. Representation ukoshi add '(x) differential Δh zvichibvira kana tichifunga X ive nharo. Asi mashandiro kunogona kuva kwakaoma, umo X kunogona kushanda nharo t. Ipapo cherudo differential kutaura e '(x) Δh, sezvinoita mutemo, hazvibviri; asi kune Linear nedoro x = panguva + b.

Sezvo kune nzira add '(x) yori = Dy, zvino zvakaitika kuzvimirira nharo X (zvino yori = Δh) mu zvakaitika parametric nedoro kuti X t, ndizvo differential.

Somuenzaniso, mashoko okuti 2 X Δh nokuda ja = X ² differential wayo kana X chiri nharo. zvino x = t ² uye taifunga t nharo. Zvadaro, y = X ² = t ^4.

Izvi zvinoteverwa (T + Δt) ² = t ² + 2tΔt + Δt ^2. Nokudaro Δh = 2tΔt + Δt ^2. Nokudaro: 2xΔh = 2t ² (2tΔt + Δt ^2).

Mashoko aya haasi Proportional kuna Δt, uye naizvozvo zvino 2xΔh haasi differential. Rinogona kuwanikwa kubva mukana wokuti vanhu vanga ja = X ² = t ^4. Hazvina wakaenzana Dy = 4t ³ Δt.

Kana isu kutora mashoko 2xdx, ndiyo differential ja = X ² upi nharo t. Zvechokwadi, kana x = t ² kuwana yori = 2tΔt.

Saka 2xdx = 2t ² 2tΔt = 4t ³ .DELTA.t, t. E. The okuti differentials hwakanyorwa variables zviviri zvakasiyana rienderane.

Pavanotsiva increments differentials

Kana d '(x) ≠ 0, ipapo Δu uye Dy yakaenzana (apo Δh → 0); kana d '(x) = 0 (zvinoreva neDy = 0), havana noenderana.

Somuenzaniso, kana ja = X ^2, ipapo Δu = (x + Δh) ² ─ X ² = 2xΔh + Δh ² uye Dy = 2xΔh. Kana x = 3, zvino tine Δu = 6Δh + Δh ² uye Dy = 6Δh kuti vari yakaenzana yakafanira Δh ² → 0, kana x = 0 ukoshi Δu = Δh ² neDy = 0 havasi noenderana.

chokwadi ichi, pamwe chete nyore nemamiriro differential (m. E. Linearity noruremekedzo kuna Δh), rinowanzoshandiswa anofungidzirwa kukarukureta, pamusoro kubvuma kuti Δu ≈ Dy nokuda Δh duku. Tsvaga differential basa Zvinowanzova nyore pane kuverenga chaiyo kukosha increment.

Somuenzaniso, tine esimbi kuutio pamwe unopinza x = 10.00 cm. On muchidziisa mupendero miforo musi Δh = 0,001 cm. Sei rakawedzerwa kuutio V? Tine V = X ^2, saka kuti DV = 3x ² = Δh 3 ∙ ∙ February ¹⁰ 0/01 = 3 (masendimita ^3). Inosimudzira ΔV noenderana differential DV, saka kuti ΔV = 3 masendimita ^3. Full kukarukureta aizopa ³ ΔV = 10,01 ─ March ¹⁰ = 3.003001. Asi mugumisiro digits zvose kunze wokutanga kuvimbika; naizvozvo, zvichiri zvakakodzera kuti kumativi kusvika 3 masendimita ^3.

Zviri pachena, nzira iyi inobatsira chete kana zvichibvira kufungidzira kukosha tigovane nokukanganisa.

Differential basa: mienzaniso

Ngatiedzei kuwana differential pakati mashandiro and = x ^3, kuwana rinobva. Regai tipe nharo increment Δu uye kutsanangura.

Δu = (Δh + x) ³ ─ x ³ = 3x ² + Δh (Δh 3xΔh ² + ^3).

Pano, kuti coefficient A = 3x ² hakubvi Δh, saka kuti chokutanga izwi iri Proportional Δh, vamwe nhengo 3xΔh Δh ² + ³ apo Δh → 0 zvinoderedza kupfuura increment renharo. Somugumisiro, nhengo 3x ² Δh ndiye differential kuti ja = x ^3:

Dy = 3x ² Δh = 3x ² yori kana D (x ³⁾ = 3x ² yori.

Pakazoti D (x ³⁾ / yori = 3x ^2.

Dy ikozvino Tinowana basa and = 1 / X pedyo rinobva. Zvadaro D (1 / x) / yori = ─1 / X ^2. Naizvozvo Dy = ─ Δh / X ^2.

Differentials zvinhu algebraic mabasa dzinopiwa pazasi.

Inotorwa Masvomhu kushandisa differential

Kuongorora mashandiro add (x), uye ayo rinobva e '(x) pa x = imwe kunowanzoomera, asi kuita zvakafanana pedyo x = munhu hakusi nyore. Zvadaro kubatsira ari anofungidzirwa okuti

d (a + Δh) ≈ add '(a) Δh + F (a).

Izvi zvinopa anofungidzirwa kukosha mashandiro ari increments diki kuburikidza kwayo differential Δh F '(a) Δh.

Naizvozvo, nzira iyi zvinopa anofungidzirwa okuti mashandiro panopera pfungwa chikamu urefu Δh sezvo chitsama kukosha kwayo pane pokutangira kuverenga chikamu (x = a) uye differential mune imwe pokutangira. Rakarurama nzira nokuda Pakusarudza tsika mashandiro pazasi inoratidza mufananidzo.

Asi kuzivikanwa uye chaiyo okuti kukosha mashandiro x = munhu + Δh wakapiwa payakavakirwa finite increments (kana, neimwe nzira, Lagrange kuti fomura)

d (a + Δh) ≈ F '(ξ) Δh + F (a),

apo pfungwa x = munhu + ξ iri penguva kubva x = mumwe kune x = munhu + Δh, kunyange zvazvo nzvimbo yacho chaiyo haizivikanwi. Chaiwo payakavakirwa unobvumidza kuti vaongorore kukanganisa anofungidzirwa payakavakirwa. Kana tikaisa muna Lagrange payakavakirwa ξ = Δh / 2, kunyange haizorambi yakarurama, asi anopa, sezvo mutemo mumwe nzira nani zvikuru kupfuura pakutanga okuti tichitarisa ari differential.

Kukoshesa Masvomhu kukanganisa nokushandisa differential

Kuyera zviridzwa , pfungwa, kururama, uye muuye nokuyerwa Data chaienzanirana kukanganisa. Ivo dzinoratidzwa nokusanyanya kukanganisa Mhedziso, kana, mune pfupi, muganhu kukanganisa - akanaka, zvakajeka kwazvo kukanganisa mu Mhedziso ukoshi (kana kupfuura nahwo). Kuderedza mwero nokukanganisa anonzi quotient yaunganidzwa kupatsanura parutivi Mhedziso kukosha kuyerwa ukoshi.

Regai chaiyo payakavakirwa Y = d (x) basa rinoshandiswa vychislyaeniya Y, asi kukosha X ndiko kuyerwa mugumisiro, uye saka unounza and kukanganisa. Zvadaro, kuwana vakaomerwa kukanganisa mhedziso │Δu│funktsii Y, uchishandisa nzira

│Δu│≈│dy│ = │ d '(x) ││Δh│,

apo │Δh│yavlyaetsya shoma kukanganisa nharo. │Δu│ uwandu anofanira vakagumisa kumusoro, sezvo kururama kukarukureta pachayo ndiyo yokutsiva iri increment pamusoro differential kukarukureta.