Differentials - dab tsi yog qhov no? Yuav ua li cas mus nrhiav tau cov differential ntawm cov nuj nqi?

Nrog rau derivatives lawv zog differentials - nws ib co ntawm cov tswv yim ntawm cov differential calculus, lub ntsiab seem ntawm zauv tsom xam. Raws li inextricably txuas, ob qho tib si ntawm lawv ob peb centuries lug siv nyob rau hauv kev daws yuav luag tag nrho cov teeb meem uas tshwm nyob rau hauv lub chav kawm ntawm scientific thiab cov kev ua si.

Lub rov tshwm sim ntawm lub tswvyim ntawm differential

Rau cov thawj lub sij hawm ua rau nws paub tseeb hais tias xws li ib tug txawv, ib tug ntawm cov founders (nrog rau Isaakom Nyutonom) differential calculus nto moo German mathematician Gotfrid Vilgelm Leybnits. Ua ntej ntawd thom nug xyoo pua 17th. siv heev tsis meej thiab vague lub tswv yim ntawm ib co infinitesimal "undivided" ntawm tej lub npe hu ua, cov sawv cev ib tug heev me me qhov nqi tab sis tsis sib npaug zos rau zero, hauv qab no uas tseem ceeb rau cov kev ua yuav tsis muaj tsuas. Li no nws tsuas yog ib kauj ruam mus rau cov kev taw qhia ntawm notions ntawm infinitesimal increments ntawm kev ua cov lus thiab lawv cov increments ntawm lub zog uas yuav qhia nyob rau hauv cov nqe lus ntawm derivatives ntawm lub yav tas. Thiab cov kauj ruam no twb npaum li cas yuav luag ib txhij cov saum toj no ob tug poj zaum.

Raws li qhov yuav tsum tau nyob rau ceev cov tswv yim mechanics teeb meem uas hais science sai tsim kev lag luam thiab technology, Newton thiab Leibniz tsim cov kev nrhiav lub zog ntawm tus nqi ntawm kev hloov (tshwj xeeb tshaj yog nrog hais txog tus neeg kho tshuab cov kev ceev ntawm lub cev ntawm cov paub trajectory), uas coj mus rau cov kev taw qhia ntawm tej tswv yim, raws li cov derivative muaj nuj nqi thiab cov differential, thiab kuj muaj nyob rau hauv lub algorithm rov qab teeb meem kev daws teeb meem raws li paub per se (nce mus nce los) speeds traversed nrhiav txoj kev uas tau coj mus rau lub tswvyim ntawm ib ALA.

Nyob rau hauv tej hauj lwm ntawm Leibniz thiab Newton lub tswv yim ua ntej nws los tshwm hais tias tus differentials - yog xwm yeem mus rau increment ntawm qhov yooj yim cov lus sib cav Δh increments Δu zog uas yuav ntse ua ntawv thov kom xam tus nqi ntawm cov caij nyoog kawg no. Nyob rau hauv lwm yam lus, lawv tau nrhiav tau hais tias ib tug increment muaj nuj nqi tej zaum yuav tom tej taw tes (tsis pub dhau nws sau ntawm txhais) yog qhia los ntawm nws cov derivative ob Δu = y '(x) Δh + αΔh qhov twg α Δh - tas, tsis dim mus pes tsawg raws li Δh → 0, sai npaum li cas tshaj lub sij Δh.

Raws li cov founders ntawm zauv tsom xam, cov differentials - qhov no yog raws nraim cov thawj lub sij hawm nyob rau hauv increments ntawm tej kev khiav dej num. Txawm tias yuav tsis muaj ib tug kom meej meej tseg txwv tswvyim sequences yog to taub intuitively hais tias cov differential nqi ntawm cov derivative nyhav ua thaum Δh → 0 - Δu / Δh → y '(x).

Tsis zoo li Newton, uas yog feem ntau ib tug physicist thiab xaam apparatus tau ua ib qho koom haum pab lub cuab tam rau txoj kev tshawb no ntawm lub cev muaj teeb meem, Leibniz them ntau xim rau no vam, xws li ib tug system ntawm kev kos duab thiab nkag cim zauv tseem ceeb. Nws yog nws leej twg npaj siab rau tus txheej txheem cim ntawm differentials muaj nuj nqi dy = y '(x) dx, dx, thiab cov derivative ntawm lub cav muaj nuj nqi raws li lawv cov kev sib raug zoo y' (x) = dy / dx.

Cov niaj hnub txhais

Yuav ua li cas yog lub differential nyob rau hauv cov nqe lus ntawm niaj hnub kawm txog zauv? Nws yog zoo txog rau lub tswvyim ntawm ib tug nce mus nce los increment. Yog hais tias lub nce mus nce los y yuav siv sij hawm ib tug thawj nqi ntawm y y = _1, ces y = y _2, qhov txawv y ₂ ─ y ₁ yog hu ua tus increment nqi y. Lub increment yuav ua tau zoo. tsis zoo thiab pes tsawg. Lo lus "increment" yog xaiv Δ, Δu cov ntaubntawv povthawj siv (nyeem 'delta y') qhia tus nqi ntawm cov increment y. yog li Δu = y ₂ ─ y _1.

Yog hais tias tus nqi Δu arbitrary muaj nuj nqi y = f (x) tej zaum yuav muaj tuaj raws li Δu = A Δh + α, qhov twg A yog tsis muaj dependence rau Δh, t. E. A = const rau cov muab x, thiab lub sij hawm α thaum Δh → 0 nyhav nws yog sai tshaj cov sij Δh, ces tus thawj ( "master") ib lub sij hawm xwm yeem Δh, thiab yog rau cov y = f (x) txawv, denoted dy los yog df (x) (nyeem "y de", "de eff los ntawm X"). Yog li ntawd differentials - ib tug "ntsiab" linear nrog hwm rau lub Cheebtsam ntawm increments Δh zog.

txhua yam piav

Cia s = f (t) - qhov kev ncua deb nyob rau hauv ib tug ncaj kab tsiv khoom point los ntawm cov thawj txoj hauj lwm (t - mus txawv tebchaws lub sij hawm). Increment Δs - yog txoj kev taw tes thaum lub sij hawm ib lub sij hawm luv Δt, thiab cov differential ds = f '(t) Δt - txoj kev no, uas taw tes yuav tuav rau tib lub sij hawm Δt, yog hais tias nws muab khaws cia tseg rau hauv lub ceev f' (t), mus txog thaum lub sij hawm t . Thaum ib tug infinitesimal Δt ds xav txog tej yam kev txawv los ntawm lub sij Δs infinitesimally muaj ib tug ntau dua kev txiav txim nrog rau kev hwm mus Δt. Yog hais tias cov kev ceev ntawm lub sij hawm t yog tsis sib npaug zos rau zero, kwv yees tus nqi ds muab me me kev tsis ncaj ncees point.

geometric txhais

Cia cov kab L yog lub teeb ntawm y = f (x). Ces Δ x = MQ, Δu = QM '(saib. Daim duab hauv qab no). Tangent MN lov Δu txiav mus rau hauv ob qhov chaw, QN thiab NM '. Thawj thiab Δh yog proportional QN = MQ ∙ TG (lub kaum sab xis QMN) = Δh f '(x), t. E QN yog dy differential.

Qhov thib ob yog ib feem ntawm qhov sib txawv Δu NM'daet ─ dy, thaum Δh → 0 NM ntev 'txo sai tshaj cov increment ntawm lub cav, piv txwv li nws muaj qhov kev txiav txim ntawm smallness siab tshaj Δh. Nyob rau hauv cov ntaub ntawv no, Yog hais tias f '(x) ≠ 0 (non-thaum uas tig mus tangent twm) feem QM'i QN sib npaug; nyob rau hauv lwm yam lus NM 'txo sai (kev txiav txim ntawm smallness ntawm nws ntau dua) tshaj tag nrho cov increment Δu = QM'. Qhov no yog tshwm nyob rau hauv Daim duab (nce ya M'k M NM'sostavlyaet tag nrho cov me feem pua QM 'ya).

Yog li ntawd, graphically txawv arbitrary muaj nuj nqi yog sib npaug zos rau cov increment ntawm lub ordinate ntawm lub tangent.

Derivative thiab differential

Ib tug zoo tshaj nyob rau hauv thawj lub sij hawm ntawm kev qhia increment muaj nuj nqi yog sib npaug zos rau cov nqi ntawm nws cov derivative f '(x). Yog li, lub nram qab no piv - dy = f '(x) Δh los yog df (x) = f (x) Δh.

Nws yog lub npe hu hais tias lub increment ntawm tus neeg sib cav yog sib npaug zos rau nws differential Δh = dx. Raws li, peb yuav sau: f '(x) dx = dy.

Nrhiav (tej zaum hais tias yuav lub "kev txiav txim siab") differentials yog ua los ntawm cov cai raws li rau qhov derivatives. Ib daim ntawv ntawm lawv yog muab hauv qab no.

Yuav ua li cas yog ntau universal: lub increment ntawm lub cav los yog nws cov differential

Ntawm no nws yog tsim nyog los ua ib co clarifications. Sawv cev tus nqi f '(x) differential Δh tau thaum twg yog xav x li ib tug sib cav. Tab sis cov nuj nqi yuav ua tau ib tug complex, nyob rau hauv uas x yuav ua tau ib tug muaj nuj nqi ntawm lub cav t. Ces tus sawv cev ntawm lub differential qhia ntawm f '(x) Δh, raws li ib tug txoj cai, nws yog tsis yooj yim sua; tsuas yog nyob rau hauv cov ntaub ntawv ntawm linear dependence x = ntawm + b.

Raws li rau cov mis f '(x) dx = dy, ces nyob rau hauv cov ntaub ntawv ntawm ywj siab sib cav x (ces dx = Δh) nyob rau hauv cov ntaub ntawv ntawm lub parametric dependence ntawm x t, nws yog differential.

Piv txwv li, cov kev qhia 2 x Δh yog rau y = x ² nws differential thaum x yog ib tug sib cav. Peb tam sim no x = t ² thiab xav tias t sib cav. Ces y = x ² = t ^4.

Qhov no yog ua raws li los ntawm (t + Δt) ² = t ² + 2tΔt + Δt ^2. Li no Δh = 2tΔt + Δt ^2. Li no: 2xΔh = 2t ² (2tΔt + Δt ^2).

Qhov no qhia yog tsis xwm yeem mus rau Δt, thiab yog li ntawd tam sim no yog 2xΔh yog tsis txawv. Nws muaj peev xwm yuav nrhiav tau los ntawm cov kab zauv y = x ² = t ^4. Nws yog sib npaug zos dy = 4t ³ Δt.

Yog hais tias peb coj tus qhia 2xdx, nws yog ib lub differential y = x ² rau kev sib cav t. Tseeb, thaum x = t ² tau dx = 2tΔt.

Yog li ntawd 2xdx = 2t ² 2tΔt = 4t ³ .DELTA.t, t. E. Cov kev qhia differentials sau cia los ntawm ob tug sib txawv ntau yam coincide.

Hloov increments differentials

Yog hais tias f '(x) ≠ 0, ces Δu thiab dy sib npaug (thaum Δh → 0); yog f '(x) = 0 (lub ntsiab lus thiab dy = 0), lawv yuav tsis sib npaug.

Piv txwv li, yog tias y = x ^2, ces Δu = (x + Δh) ² ─ x ² = 2xΔh + Δh ² thiab dy = 2xΔh. Yog hais tias x = 3, ces peb muaj Δu = 6Δh + Δh ² thiab dy = 6Δh uas yog sib npaug vim Δh ² → 0, thaum x = 0 nqi Δu = Δh ² thiab dy = 0 yog tsis sib npaug.

Qhov tseeb, ua ke nrog tej yam yooj yim qauv ntawm cov differential (m. E. Linearity nrog hwm rau Δh), yog feem ntau siv nyob rau hauv approximate xam, nyob rau hauv assumption uas Δu ≈ dy rau me me Δh. Nrhiav cov differential muaj nuj nqi yog feem ntau yooj yim dua rau xam lub caij nyoog muaj nuj nqis ntawm cov increment.

Piv txwv li, peb muaj nws yog xim hlau lub voos xwmfab nrog ntug x = 10.00 cm. Nyob rau cua sov rau cov ntug lengthened rau Δh = 0,001 cm. Yuav ua li cas nce volume lub voos xwmfab V? Peb muaj V = x ^2, yog li ntawd DV = 3x ² = Δh 3 ∙ ∙ Lub ob hlis ntuj ¹⁰ 0/01 = 3 (cm ^3). Nce ΔV sib npaug differential DV, yog li ntawd ΔV = 3 cm ^3. Tag nrho cov muab xam yuav muab ³ ΔV = 10,01 ─ Lub peb hlis ntuj ¹⁰ = 3.003001. Tab sis qhov tshwm sim ntawm tag nrho cov lej tom qab tsuas yog cov thawj unreliable; yog li ntawd, nws yog tseem tsim nyog los nyob ib puag ncig mus txog 3 cm ^3.

Obviously, qhov no mus kom ze yog pab tau yog hais tias nws yog tau los laij rau tus nqi imparted nrog kev ua yuam kev.

Txawv muaj nuj nqi: piv txwv

Wb sim mus nrhiav cov differential ntawm cov nuj nqi y = x ^3, nrhiav cov derivative. Cia peb muab lub cav increment Δu thiab txhais tau.

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

Ntawm no, cov coefficient A = 3x ² tsis yog nyob ntawm seb Δh, yog li ntawd cov thawj lub sij hawm yog proportional Δh, lwm tus neeg 3xΔh Δh ² + ³ thaum Δh → 0 txo sai dua lub increment ntawm lub cav. Thiaj li, ib tug tswv cuab ntawm 3 x ² Δh yog lub differential ntawm y = x ^3:

dy = 3x ² Δh = 3x ² dx los yog d (x ³⁾ = 3x ² dx.

Wherein d (x ³⁾ / dx = 3 x ^2.

Dy Peb tam sim no nrhiav tau cov nuj nqi y = 1 / x los ntawm cov derivative. Ces d (1 / x) / dx = ─1 / x ^2. Yog li ntawd dy = ─ Δh / x ^2.

Differentials yooj yim algebraic functions yog muab hauv qab no.

Approximate suav siv differential

Yuav kom ntsuam xyuas cov nuj nqi f (x), thiab nws derivative f '(x) nyob rau hauv x = ib yog feem ntau yooj yim, tab sis ua tib yam li nyob rau hauv lub cheeb tsam ntawm x = ib tsis yog ib qho yooj yim. Ces tuaj mus rau kev pab ntawm kwv yees qhia

f (a + Δh) ≈ f '(a) Δh + f (a).

Qhov no muab ib tug approximate nqi ntawm cov nuj nqi ntawm me me increments los ntawm nws cov differential Δh f '(a) Δh.

Yog li ntawd, qhov no mis muab ib tug approximate qhia rau cov nuj nqi nyob rau thaum kawg taw tes rau ib feem ntawm ib tug ntev Δh raws li ib tug zaum ntawm nws cov nuj nqis ntawm lub starting taw tes rau ntawm lub pawg (x = a) thiab cov differential nyob rau hauv tib starting taw tes. Raug tus qauv rau kev txiav txim rau qhov tseem ceeb ntawm cov nuj nqi nram qab no qhia txog ntawm daim duab.

Txawm li cas los paub thiab lub caij nyoog qhia rau tus nqi ntawm cov nuj nqi x = ib + Δh muab los ntawm mis finite increments (los yog, hloov, Lagrange lub mis)

f (a + Δh) ≈ f '(ξ) Δh + f (a),

qhov twg tus taw tes x = ib + ξ yog nyob rau hauv lub luv los ntawm x = ib rau x = ib + Δh, txawm hais tias nws caij nyoog txoj hauj lwm yog tsis paub. Lub caij nyoog mis tso cai los soj ntsuam qhov yuam kev ntawm cov approximate mis. Yog hais tias peb muab tso nyob rau hauv lub Lagrange mis ξ = Δh / 2, txawm hais tias nws tsis tu tsis tseg yuav yog, tab sis muab, raws li ib tug txoj cai, ib tug zoo npaum li cas kom ze tshaj tus thawj qhia nyob rau hauv cov nqe lus ntawm lub differential.

Kev luj xyuas cov qauv kev ua yuam kev los ntawm thov differential

Xab seev , nyob rau hauv tus, tsis muaj tseeb, thiab coj mus rau qhov ntsuas cov ntaub ntawv sib nug xov mus rau lub kev ua yuam kev. Lawv tsiag ntawv los ntawm limiting lub meej kev ua yuam kev, los yog, nyob rau hauv luv luv, cov kev txwv yuam kev - zoo, kom meej meej tshaj qhov yuam kev nyob rau hauv tsis muaj nqis (los yog nyob rau feem ntau sib npaug zos rau nws). Tswj tus kwv tij kev ua yuam kev no yog hu ua cov quotient tau los mus faib nws los ntawm lub meej nqi ntawm cov kev ntsuas tus nqi.

Cia caij nyoog mis y = f (x) muaj nuj nqi siv rau vychislyaeniya y, tab sis qhov muaj nqis ntawm x yog qhov ntsuas tshwm sim, thiab yog li ntawd coj lub y kev ua yuam kev. Ces, mus nrhiav tus txawj tsis yuam kev │Δu│funktsii y, siv cov mis

│Δu│≈│dy│ = │ f '(x) ││Δh│,

qhov twg │Δh│yavlyaetsya marginal yuam kev sib cav. │Δu│ kom muaj nuj nqis yuav tsum tau npawv upwards, raws li tsis muaj tseeb muab xam nws tus kheej yog cov kev hloov ntawm lub increment rau lub differential muab xam.