Differentials - ke eng see? Mokhoa oa ho fumana differential tsa mosebetsi joang?

Hammoho le derivatives mesebetsi ea bona differentials - e ba bang ba dikgopolo tse go motheo tsa differential calculus, karolong e ka sehloohong ea tshekatsheka lipalo. Ka inextricably amahanngoa, bobeli ba bona ba lilemong tse makholo tse 'maloa ho pharaletseng sebediswa ho rarolla hoo e ka bang mathata ohle a ileng a tsoha ka tsela ea tšebetso ea saense le botekgeniki.

Ho hlaha ha khopolo ea differential

Ka lekhetlo la pele o ile a hlakisa hore ho na le differential joalo, e mong oa bathehi ba (hammoho le Isaakom Nyutonom) differential calculus tummeng Jeremane setsebi sa lipalo Gotfrid Vilgelm Leybnits. Pele hore thuto ea lipalo tsa lekholong la lilemo la bo17 la lilemo. sebelisoa se se sa haholo le sa hlakang maikutlo a ba bang infinitesimal "undivided" tsa mosebetsi ofe kapa ofe tsebahalang, e emelang e nyenyane haholo boleng kamehla empa ha a lekane le lefela, tlase tse ananela mosebetsi o ke ke oa ba mpa feela. Kahoo e ne e le mohato o le mong feela ho qaloa ho maikutlo a ea increments infinitesimal la mabaka a mosebetsi le increments tsona oa mesebetsi e le hore a ka ho bolela hore ho latela derivatives ea bobeli. Le mohato ona o ile a isoa hoo e ka bang ka nako e ka hodimo bo-rasaense tse peli tse khōlō.

E thehiloeng ho hlokahale ho rarolla e potlakileng e sebetsang nyenyenyane mathata hore tobana saense e ka potlako ho ntshetsa pele indasteri le theknoloji, Newton le Leibniz bōptjoa litsela le tloaelehileng la ho fumana mesebetsi ya lebelo la phetoho (haholo-holo mabapi le lebelo ho phetha molao feela oa 'mele oa tsela tsebahalang), e ile ea etsa hore selelekeleng likhopolo tse joalo, e le mosebetsi sehlahisoa le differential, 'me a boela a fumana ba-tharabololo pitikololo ditharollo bothata e tsebahalang ka banayee (polygonal) lebelo ho tsamauoa ho fumana tsela e se entse hore ho khopolo ea bohlokoa Ala.

Ka mesebetsi ea maikutlo Leibniz le Newton ba pele ho ile ha bonahala eka ho differentials - ke tekanyo e lekanang ya ho increment la mabaka a motheo Δh increments Δu mesebetsi e ka ka katleho kōpo ea ho lekanya boleng ba bobeli. Ka mantsoe a mang, ba ile ba ba fumane hore ho increment mosebetsi e ka ba nako efe kapa efe (ka domain name lona ya tlhaloso), a bontša ka ho ya ka sehlahisoa lona ka Δu = featswan '(x,) Δh + αΔh moo α Δh - ba setseng atisa ho lefela ha Δh → 0, haholo ka potlako ho feta ea sebele Δh.

Ho latela bathehi ba and analysis thuto ea lipalo, le differentials - sena ke sona lentsoe pele ka increments mesebetsi efe kapa efe. Esita le ntle le ho ba le ka ho hlaka hlalosoa tatellanong moedi kgopolo ya go di utlwisiswa intuitively hore differential boleng ba sehlahisoa sa atisa ho sebetsa ha Δh → 0 - Δu / Δh → featswan '(x e).

Ho fapana le Newton, eo e neng e haholo-holo e le setsebi sa fisiks le sethusathuto thuto ea lipalo nkoa e le sesebelisoa thusang tsa ho ithuta mathata 'meleng, Leibniz ela hloko ka ho feta ho kobotlo ya didirisiwa ena, ho kenyelletswa le tsamaiso ea matshwaonyana bonwang le utloisisa makgabane lipalo. E ne e le eena ea ileng a sisintsweng tekanyetso notation tsa differentials mosebetsi Dy = featswan '(x,) dx, dx, le sehlahisoa sa khang mosebetsi e le kamano ea bona featswan' (x,) = Dy / dx.

The tlhaloso ea kajeno

ke differential ya ka dipehelo tsa thuto ea lipalo ea kajeno ke efe? E haufi-ufi amanang le khopolo ea ho increment polygonal. Ha polygonal featswan nka boleng pele la featswan featswan = _1, moo featswan = featswan _2, phapang featswan ₂ ─ y el ₁ o bitsoa increment boleng featswan. The increment e ka ba khothatsang. mpe le lefela. Lentsoe "increment" le khethoa Δ, Δu rekota (bala 'makgobokgobo featswan') le bolela bohlokoa ba increment featswan. kahoo Δu = featswan ₂ ─ y el _1.

Ha boleng Δu mosebetsi hatellang featswan = o f (x,) ka hlahisoa e le Δu = A Δh + α, moo A na ho itšetleha ka Δh, 't. E. A = const bakeng sa: x fanoeng,' me lentsoe α ha Δh → 0 atisa ho ke esita le ka lebelo le fetang la sebele Δh, joale e ntan'o ba pele ( "mong'a") e leng lentsoe le tekanyo e lekanang ya Δh, 'me ke ea featswan = o f (x,) differential, denoted Dy kapa DF (x,) (bala "y el de", "de Khao eff tloha X"). Ka hona differentials - e le "se seholo" guttate mabapi le metsoako ea increments mesebetsi Δh.

tlhaloso phetha molao

A re ke 's = o f (sa t) - the hole ka mola ka kotloloho ho fallela ntlha lintho tse bonahalang ho tswa ho boemo ba ka lekhetlo la pele (sa t - nako leeto). Increment Δs - ke tsela ntlha nakong karohano Δt, le differential bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb = f e '(sa t) Δt - tseleng ena, eo ntlha ne ho tla tšoaroa ka nako e tšoanang Δt, haeba ho bolokoa lebelo f e' (sa t), e ile ea fihla ka nako-t . Ha e infinitesimal Δt bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb inahanelang tseleng e fapana ho tloha Δs sebele infinitesimally ho ba le taelo e phahameng mabapi le Δt. Ha lebelo ka nako e sa t ke ha a lekane le lefela, ho hakanyetsoa boleng bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb fana nyane leeme ntlha.

thutatekanyo tlhaloso

A re ke re moleng L ke kerafo ya featswan = o f (x e). Ebe Δ: x = MQ, Δu = bener bgt '(bona. Utloisisa ka tlase). Tangent MN roba Δu khaola likoto tse peli, QN le NM '. Pele le Δh ke tekanyo e lekanang ya QN = MQ ∙ tg (angle QMN) = Δh f e '(x,),' t. E QN ke Dy differential.

E leng karolo ea bobeli ea phapang Δu NM'daet ─ Dy, ha Δh → 0 NM bolelele 'decreases esita le ka lebelo le fetang la increment ba ngangisana, ke hore, ho na le taelo ea bonyenyane phahame ho feta Δh. Tabeng ena, ha o f '(x,) ≠ 0 (bao e seng e tšoanang le tangent pholo) likarolo QM'i QN lekanang; ka mantsoe a mang NM 'decreases potlako (taelo ea bonyenyane ba phahame lona) ho feta kakaretso ea increment Δu = bener bgt'. Sena se bonahala ka ho Figure (atamela karolo M'k M NM'sostavlyaet tsohle nyenyane Peresente bener bgt 'karolo).

Ho joalo, ka ho hlaka differential mosebetsi hatellang e lekanang le increment tsa golaganya tsa tangent ena.

Sehlahisoa le differential

Ntho e 'A ka lentsoe pele ba ho hlahisa maikutlo increment mosebetsi e lekanang le boleng ba f e lona sehlahisoa' (x e). Kahoo, e latelang mabapi - Dy = f e '(x,) Δh kapa DF (x,) = f e' (x,) Δh.

E o tsejoa hore increment la khang e ipusang o lekana le differential lona Δh = dx. Ka lebaka leo, re ka ngola: f e '(x,) dx = Dy.

Ho fumana (ka linako tse ling o ile a re hore e be "Qeto") differentials o etsoa ke melao e tšoanang le bakeng sa derivatives ena. Lenane la bona e fanoeng ka tlaase mona.

Ho feta moo ba bokahohleng: ho increment ea khang kapa differential lona

Mona kamoo ho hlokahalang ho etsa ba bang ba hlakiso. Boemedi boleng f e '(x,) differential Δh khoneha ha re nahana ho x, e le khang. Empa mosebetsi e ka ba tse rarahaneng, eo ho eona: x e ka ba mosebetsi oa khang sa t. Joale e ntan'o ba setšoantšo sa differential pontšo ea f e '(x,) Δh, e le busa, ho ke ke ha khoneha; ntle tabeng ea guttate itšetleha: x = ka + b.

E le ho moralo o f '(x,) dx = Dy, joale tabeng ea ikemetseng khang: x (ebe dx = Δh) tabeng ea ho itšetleha parametric ya x-t, ke differential.

Ka mohlala, polelo e reng 2: x Δh ke featswan = x, ² differential lona ha x, ke khang. Hona joale re x e = sa t ² le inkele khang sa t. Ebe featswan = x, ² = sa t ^4.

Sena se lateloa ke (sa t + Δt) ² = sa t ² + 2tΔt + Δt ^2. Kahoo Δh = 2tΔt + Δt ^2. Kahoo: 2xΔh = 2t ² (2tΔt + Δt ^2).

Polelo ena e ha se tekanyo e lekanang ya ho Δt, ka hona ke hona joale 2xΔh e sa differential. E ka fumanoa ho tswa ho abel featswan = x, ² = sa t ^4. Ho lekanang Dy = 4t ³ Δt.

Haeba re nka ka 2xdx polelo e reng, ke differential featswan = x, ² tsa khang sa t efe kapa efe. Ka sebele, ha a x = sa t ² fumana dx = 2tΔt.

Kahoo 2xdx = 2t ² 2tΔt = 4t ³ .DELTA.t, 't. E. The differentials poleloana ee e tlalehiloeng ke divariabole a mabeli a fapaneng tsamaisane.

Ho tlosa increments differentials

Ha o f '(x,) ≠ 0, joale Δu le Dy lekanang (ha Δh → 0); ha o f '(x,) = 0 (moelelo le Dy = 0), ba ne ba ha lekana.

Ka mohlala, haeba featswan = x, ^2, ka nako eo Δu = (x, + Δh) ² ─ x, ² = 2xΔh + Δh ² le Dy = 2xΔh. Ha: x = 3, joale re na le Δu = 6Δh + Δh ² le Dy = 6Δh tse lekanang ka lebaka la Δh ² → 0, ha a x = 0 boleng Δu = Δh ² le Dy = 0 sa lekanang.

Ha e le hantle ena, hammoho le mohaho o bonolo differential le (m. E. Linearity mabapi le Δh), e atisa ho sebelisoa ka bo manolotsoeng lekantsoeng, ka khopolo ea hore Δu ≈ Dy bakeng Δh nyenyane. Fumana differential mosebetsi hangata ho le bonolo ho feta ho a bale boleng e tobileng ea increment ena.

Ka mohlala, re na le metallic sekotwaneng ka bohale: x = 10.00 cm,. On futhumatsang bohale lengthened ka Δh = 0,001 cm,. Joang eketse molumo sekotwaneng V? Re na le V = x, ^2, e le hore DV = 3x ² = Δh 3 ∙ ∙ February ¹⁰ 0/01 = 3 (cm, ^3). Eketseha ΔV lekanang differential DV, e le hore ΔV = 3 cm, ^3. Feletseng manolotsoeng ne a tla fa ³ ΔV = 10,01 ─ March ¹⁰ = 3.003001. Empa ka lebaka la dinokong tsohle haese ke keng ea tšeptjoa ea pele; Ka hona, ho ntse ho hlokahala hore ho potoloha ho fihlela ho ka 3 cm ^3.

Ho hlakile hore, mokhoa ona o na le thuso feela haeba ho khoneha ho hakanya boleng fana ka phoso.

Differential mosebetsi: mehlala

A re ke re leka ho fumana differential tsa mosebetsi featswan = x, ^3, ho fumana le sehlahisoa. A re ke re fa khang increment Δu le hlalosa.

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

Mona, le coefficient A = 3x ² ha ea itšetleha ka Δh, e le hore lentsoe la pele ke tekanyo e lekanang ya Δh, e mong setho 3xΔh Δh ² + ³ ha Δh → 0 decreases ka potlako ho feta ya increment ea khang ea. Ka lebaka leo, setho sa 3x ² Δh ke differential tsa featswan =: x ^3:

Dy = 3x ² Δh = 3x ² dx kapa d (x, ³⁾ = 3x ² dx.

Moo le d (x, ³⁾ / dx = 3x ^2.

Dy Hona joale re fumana mosebetsi featswan = 1 / x, ke sehlahisoa sa. Ebe le d (1 / x,) / dx = ─1 / x, ^2. Ka lebaka leo, Dy = ─ Δh / x, ^2.

Differentials mantlha mesebetsi aljebra ba fuoa tlase.

dipalelo likhakanyo sebelisa differential

Ho hlahloba mosebetsi o f (x,), 'me sehlahisoa lona f e' (x,) ka: x = ka hangata ho bang thata, empa ho etsa se tšoanang sebakeng se haufi le x, = e ke ho se bonolo. Ebe phallela ba poleloana e reng likhakanyo

f-(a + Δh) ≈ f e '(a) Δh + o f (a).

Sena se etsa hore e lekantsoeng boleng ba mosebetsi wa ka increments nyane ka differential lona Δh f e '(a) Δh.

Ka lebaka leo, moralo sena se etsa hore e hakanyetsoa ho hlahisa maikutlo bakeng sa mosebetsi qetellong ntlha ea karolo e itseng ea e-ba bolelele Δh ka chelete ea boleng ba lona ka qaloang la karolo e ka (x = a) le differential ka qaloang tšoanang. Nepahala ha mokhoa oa ho beha melao ea boitšoaro ea mosebetsi e ka tlase e bontša taka.

Leha ho le joalo tsebahalang le hantle polelo e reng bakeng sa boleng ba mosebetsi-x = sa + Δh fanoeng ke moralo increments lekanyelitsoeng (kapa, ho seng jwalo, moralo Lagrange ya)

f-(a + Δh) ≈ f e '(ξ) Δh + o f (a),

moo ntlha: x = sa + ξ ke ka karohano ho tloha: x = ho x, = sa + Δh, le hoja boemo ba eona hantle e sa tsejoeng. Tsela eo hantle dumella ho hlahloba phoso ea ka moralo o lekantsoeng. Haeba re beha ka Lagrange moralo ξ = Δh / 2, le hoja ho kgaotsa ho nepahetse, empa o fana ka, e le busa, ka tsela e molemo haholo ho feta polelo pele ya ka dipehelo tsa differential ena.

Sekasekale mekhoa phoso ka ho sebelisa differential

Lekanyang liletsa , ka molao-motheo, nepahalang, 'me le tlise ho ya data ka tekanyo e lekanang ya ho phoso. Di tšoauoa ka fokotsa ho phoso ka ho feletseng, kapa, ka mantsoe a mang, tekanyong phoso - tse ntle, ka ho hlaka feteng phoso ka boleng feletseng (kapa ho fetisisa lekanngoa le bona). Fokotsa phoso a lekanyelitsoeng o bitsoa quotient fumanoa ka ho arola e ka boleng feletseng boleng lekanngoa.

A re ke hantle moralo featswan = o f (x,) mosebetsi sebelisoa ho vychislyaeniya featswan, empa boleng ba x, ke phello ea ho lekanya, 'me ka hona ho tlisa ho featswan phoso. Ka mor'a moo, ho fumana fokotsa phoso ka ho feletseng │Δu│funktsii featswan, ho sebelisa moralo wa

│Δu│≈│dy│ = │ f e '(x,) ││Δh│,

moo │Δh│yavlyaetsya thoko ho leqephe phoso khang. │Δu│ palo tlameha ho lekaneng ho ya hodimo, joalokaha nepahalang manolotsoeng ka boeona e ke Phetolo ya increment ka differential manolotsoeng.