Ho okumu, Kulanui a me nā kulanui
Differentials - He aha ka keia? Pehea e loaʻa i ka palena pau o ke kuleana pili i?
A me nā mea loaʻa kā lākou mau hana differentials - ia kekahi o ka walaʻauʻana manaÿo o ka palena pau calculus, i ka pauku ka papa kuhikuhi o makemakika Ka Ikepili. Like inextricably ua hoʻopili, ua piha no ia mau kenekulia hoʻomano hoʻohana 'ia ma ka hoʻoponopono Aneane pau na pilikia i ala ae la i loko o ka papa o ka iao? Iiaie oaoieei aayoaeuiinoe.
Ke muli o ka manaʻo o ka palena pau
No ka mea manawa mua i mea akaka ia ia i ka palena pau, kekahi o na Mamua (a me Isaakom Nyutonom) palena pau calculus kaulana German mathematician Gotfrid Vilgelm Leybnits. Before mea mathematicians 17th kenekulia. hoʻohana loa unclear a me vague manaʻo o kekahi infinitesimal "undivided" o kekahi ike hana, no ko ka he uuku loa ikaika cia akā,ʻaʻole like ia 'Aʻohe, ma lalo i nā loina o ka hana hiki ole e wale. Nolaila ka mea, ua wale kekahi anu u i ka hoike ana o notions o infinitesimal ńpanakahi o ke kuleana pili i kekahi manaʻo hoʻopiʻi kū'ē a me ko lakou mau ńpanakahi o ka mana e hiki ke kāheaʻia ma ka hua'ōlelo o nā mea loaʻa o ka hope. A Ua laweʻia aneane ka poʻe i kēia koe i ka luna nuiʻepekema elua.
Ma muli o ka pono e hoʻoponopono 'ana i ka manaʻo mea hiki mechanics pilikia e' alo 'epekema hoʻomaka koke ka ulu' hana a me ka enehana, Newton a me Leibniz i hana i ka pono aoao o loaa ia i ka mana o ka uku o ka loli (oi loa aku hoi me ka pili ana i ka me mechanical māmā holo o ke kino o ka mea ike ala), a alakai ai i ka hoʻokomo 'ia' ana o ia mau manaÿo, like me ka derivative papa, a me ka palena pau, a i ike i ka algorithm inverse pilikia pāʻoihana like ikeia no hoon (ee iaaanu aey) ka mämä holo kaahele aku la oia e imi i ke ala e i alakai ai i ka manaʻo o ka integral Ala.
Ma na hana o Leibniz a me Newton ka manaʻo mua ka mea iʻikeʻia e ka differentials - ka hapaʻauhau i ka xi o ke kumu o kekahi manaʻo hoʻopiʻi kū'ē Δh xi Δu oihana e hiki ke ana i nā hoʻomaulia i ka waiwai o ka mea hope. Ma na olelo e ae, ka mea i ikeia i kekahi kuleana pili i xi e ia i kekahi wahi (i loko o kona kikowaena o ka ho'ākāka 'ana) ua olelo ia ma kona derivative nā Δu = y me' (m) Δh + αΔh kahi α Δh - koena, hoʻolaha i ka 'Aʻohe like Δh → 0, he nui ma mua o ka mea maoli Δh.
E like me na Mamua o ka makemakika Ka Ikepili, na differentials - kēia mea like me ka mua manawa i loko o xi o kekahi hana. E like me ka mea maopopo intuitively he maopopo hoakaka kali manaʻo kaʻina i ka palena pau waiwai o ka derivative kahakai āu e hana ia Δh → 0 - Δu / Δh → y me '(m).
Haʻalele Newton, ka mea i kuhikuhi he physicist a me ka makemakika i hoʻomākaukauʻia manaoia me he kākoʻo koʻi no ka hoʻopaʻa 'ana i physical pilikia, Leibniz uku nui hoʻolohe i kēia kanaka Polenekia, me ke ano o ka pilikia makapō a me ka comprehensible hoʻi makemakika nā loina. Ua i ka mea nana i manaoia i ka maʻamau notation o differentials papa dy = y me '(m) dx, dx, a me ka derivative o ka papa hoʻopaʻapaʻa me kā lākou pili y me' (x) = dy / dx.
Na kēia ho'ākāka 'ana
He aha ka ka palena pau ma ka hua'ōlelo o kēia makemakika? Ua Ua pili loa aku i ka manaʻo o ka ee iaaanu aey xi. Inā ka ee iaaanu aey y me lawe i ka waiwai mua o y me y me = 1, laila y me = y me 2, ua kapaia ka xi cia y me ke koena unuhi y me 2 ─ y me 1.
Inā ka waiwai Δu ākeʻakeʻa kumu papa y me = f (x) i e na poe like Δu = A Δh + α, kahi A i ole kaukaʻi ma Δh, 'ole. E. A = const no ka haawiia mai m, a me ka makahiki α ka wā Δh → 0 mālama i ia mea i wikiwiki ma mua o ka mea maoli Δh, alaila, o ka mua ( "haku") he kau hapaʻauhau Δh, a me ka mea no ka y me = f (x) palena pau, denoted dy ole df (m) (heluhelu "y me de", "ʻo eff mai X"). No ia mea, differentials - i "ka papa kuhikuhiE" laina me ka pili ana i na eiiiiiaiou o ńpanakahi Δh hana.
mechanical e hoakaka
E ke = f (Halau) - ka mamao ma ka pololei o ka laina e neʻe mea wahi mai o ka loiloi mua wahi (Halau - Hotele manawa). Xi Δs - o ke alaʻoi i loko o ka manawa wā mawaena Δt, a me ka palena pau DS =. F '(Halau) Δt - keia alanui, i wahi makemake e paa no ka mea ia manawa Δt, ina ka mea, hoopaa iho la i ka mämä holo. F' (k), hiki ma ka manawa 'ole . Ka wā i infinitesimal Δt DS ka mana ala ÿokoÿa mai ka maoli Δs infinitesimally me ke kiekie mea me ka manao i Δt. Inā i ka mämä holo i ka manawa na mea i like ke 'Aʻohe, ka He kokekau wale cia DS haawi uuku pāʻewaʻewa wahi.
geometric ano
E ka laina L o ka pakuhi o ka y me = f (x). Alaila Δ m = MQ, Δu = QM '(e nānā. Kii ma lalo nei). Tangent HI, mai hemo mai Δu oki i loko o nā māhele, QN a me HI '. Mua a me ka Δh mea hapaʻauhau QN = MQ ∙ tg (huina QMN) = Δh. F '(m),' ole. E QN mea dy palena pau.
I ka lua o ka aoao o ke koena unuhi Δu NM'daet ─ dy, ka wā Δh → 0 HI lōʻihi 'emi i wikiwiki ma mua o ka xi o ka manaʻo hoʻopiʻi kū'ē, oa ka mea, i ka mea o ka uuku luna ma mua o Δh. Ma keia hihia, ina [illegible] '(m) ≠ 0 (' ole-laua tangent bipi) Hoʻohana QMʻi QN like paha; i loko o nā hua'ōlelo HI 'emi koke (mea o ka uuku o kona kiekie) ma mua o ka huina xi Δu = QM'. He mea akaka ma ke Kii (holo mai ana Hoʻohana M'k M NM'sostavlyaet a pau uuku pakeneka QM 'Hoʻohana).
No laila, graphically palena pau ākeʻakeʻa kumu kuleana pili i mea like me ka xi o ka ordinate o ka tangent.
Derivative a me ka palena pau
A ololi i loko o ka mua kau o ka 'aoʻao o xi kuleana pili i mea like me ka waiwai o kona derivative. F' (m). Ke'ī mai,ʻo kēia hoahanau - dy =. F '(m) Δh ole df (x) =. F' (m) Δh.
Ua ua ike ia i ka xi o ke kuokoa UAOAaIOIUE mea like me kona palena pau Δh = dx. Alaila, ua hiki ke kākau: [illegible] '(m) dx = dy.
Loaa ia (kekahi manawa, i aku la ia ia i ka "olelo hooholo") differentials ua hana ia e ka ia na rula like no na nā mea loaʻa. A papa inoa o ia ua haawi mai ai ma lalo nei.
He aha ka oi paʻa: ka xi o ka manaʻo hoʻopiʻi kū'ē 'ole kona palena pau
Eia ka mea, ua pono ia e kekahi clarifications. E hoohalike ai waiwai. F '(m) palena pau Δh hiki i ka wa e noonoo pono m me ka loulou. Akā, i ke kuleana pili i hiki ia i ka luna ', i i m hiki ia i ka hana o ka manaʻo hoʻopiʻi kū'ē' ole. A laila, i ka hoohalike o ka palena pau hōʻike o [illegible] '(m) Δh, e like me ka rula, he mea hiki ole; koe i loko o ka hihia o ka laina kaukaʻi m = i + b.
Like i ka haʻilula [illegible] '(m) dx = dy, laila, i loko o ka hihia o ke kuokoa i kekahi manaʻo hoʻopiʻi m (laila dx = Δh) i loko o ka hihia o ka parametric kaukaʻi o ka m t, ia mea palena pau.
No ka laʻana, ka 'aoʻao o 2 m Δh mea no ka y me = m 2 kona palena pau ia m o ka loulou. Mākou e m = na 2 a me ka lawe 'ole loulou. Alaila, y me = m 2 = 'ole 4.
Kēia Ua ukaliʻia (Halau + Δt) 2 = 'ole 2 + 2tΔt + Δt 2. Nolaila Δh = 2tΔt + Δt 2. Nolaila: 2xΔh = 2t 2 (2tΔt + Δt 2).
Kēia 'aoʻao o ka ole hapaʻauhau iā Δt, a no ia mea, i kēia manawa 2xΔh ua ole palena pau. Ua hiki ke loaʻa mai i ka helu kaulike y me = m 2 = 'ole 4. He like dy = 4t 3 Δt.
Inā mākou e lawe i ka 'aoʻao o 2xdx, ia mea m ka palena pau y me = 2 no kekahi manaʻo hoʻopiʻi kū'ē' ole. Ua, ka wā m = 'ole 2 loaa dx = 2tΔt.
No laila, 2xdx = 2t 2 2tΔt = 4t 3 .DELTA.t, 'ole. E. Ka' aoʻao o differentials hoʻopaʻaʻia e nā okoa aiaiiuo coincide.
Waiho hou ńpanakahi differentials
Inā. F '(m) ≠ 0, laila Δu a me dy like paha (wā Δh → 0); ina. f '(x) = 0 (ʻano a me ka dy = 0), ka mea, i ole like paha.
No ka laʻana, ina y me = m 2, laila Δu = (m + Δh) 2 ─ m 2 = 2xΔh + Δh 2 a me dy = 2xΔh. Inā m = 3, laila mākou i Δu = 6Δh + Δh 2 a me dy = 6Δh i ka mea like paha ma muli o Δh 2 → 0, ka wā m = 0 waiwai Δu = Δh 2 a me ka dy = 0 mea e like paha.
Keia mea, hui pū me ka poʻe noʻonoʻo 'ole o ka palena pau (m. E. Linearity me ka mahalo i Δh), ua pinepine ia i loko o He kokekau wale i ka ho omaulia, ma ka mea mahuʻi i Δu ≈ dy no ka uuku Δh. E huli i ka papa palena pau ka IeAUPIIe, IAa IO hiki pono ma mua, e hoʻomaulia i ka kiko'ī cia o ke xi.
No ka laʻana, ua i mea hao kupa me ka maka m = 10,00 knm, Ma paio hahana maoli nō i ke kihi hooloihiia ma Δh = 0,001 knm. Pehea mahuahua ka leo kupa V? Mākou i V = m 2, no laila, ua dV = 3x 2 = Δh 3 ∙ ∙ February 10 0/01 = 3 (knm 3). Hoomahuahua ae la lakou ΔV like palena pau dV, no laila, ua ΔV = 3 knm 3. Full i ka ho omaulia e haawi ΔV = 10,01 ─ 3 o 10 3 = 3.003001. Akā, i ka hopena o nā huahelu, koe ka unreliable mua; nolaila, he mea nō e pono e hoopoepoe mai i 3 knm 3.
ʻae nō hoʻi, ua kokoke loa i kahi mea pono keia mau mea wale nō, inā paha he mea hiki i ka manaʻo i ka waiwai haawiia'ku me ka hewa.
Palena pau ho'āʻo: examples
E ka ho'āʻo e loaʻa i ka palena pau o ka papa y me = m 3, loaa ia makou ka derivative. E iā mākou e hāʻawi aku i ka manaʻo hoʻopiʻi kū'ē xi Δu, a hoakaka.
Δu = (Δh + m) 3 ─ m 3 = 3x 2 + Δh (Δh 3xΔh 2 + 3).
Eia, ke kaʻi lau waiwai A = 3x 2 'aʻole e hilinaʻi ma Δh, no laila, i ka mea mua kau mea hapaʻauhau Δh,ʻo nā lālā 3xΔh Δh 2 + 3 ka wā Δh → 0 emi wikiwiki ma mua o ka xi o ka loulou. Nolaila, he lâlâ 'o 3x 2 Δh o ka palena pau o ka y me = m 3:
dy = 3x 2 Δh = 3x 2 dx ole n (m 3) = 3x 2 dx.
Ma ke aha ka b (m 3) / dx = 3x 2.
Dy mākou e loaʻa i ka papa y me = 1 / m ma ka derivative. A laila, D (1 / m) / dx = ─1 / m 2. No ia mea, dy = ─ Δh / m 2.
Differentials walaʻauʻana algebraic oihana i haawiia mai ma lalo nei.
He kokekau wale ho ka hoʻohana palena pau
E loiloi i ka papa f (x), a me kona derivative [illegible] '(m) ma m = ka mea pinepine paʻakikī, akā, e hana i ka ia i loko o ka wahi o ka m = ka mea e maʻalahi. A laila, e hele mai i ke kokua ana o ka He kokekau waleʻana
f (he + Δh) ≈. f '(i) Δh + f (i).
Kēia i hāʻawi mai ka He kokekau wale waiwai o ka hana i ka poe liilii, xi ma kona palena pau Δh. F '(i) Δh.
No laila, i kēia haʻilula haawi i He kokekau wale olelo no ka hana ma ka hopenaʻoi o kekahi hapa o ka lōʻihi Δh like me ka huina o kona waiwai ma ka ho omaka o ka hapa (m = ka) a me ka palena pau i loko o ka ia ho omaka. Pololei o ka papa hana no ka hoomaopopo ana i ka aiee o ka papa lalo wehewehe i ka unuhi.
Eia naʻe iʻikeʻia, a me ka kiko'ī olelo no ka waiwai o ke kuleana pili i m = i + Δh haawiia mai ma ka haʻilula finite xi (a,ʻOkoʻa, Lagrange ka haʻilula)
f (he + Δh) ≈ [illegible] '(ξ) Δh + f (i),
kahi o kaʻoi He m = ka + ξ mea i loko o ka wā mawaena mai m = ka i m = i + Δh, eia nae kona kiko'ī kulana mea IKE OLE IA. Ke mau haʻilula e leie aku e loiloi i ka hewa o ka He kokekau wale haʻilula. Inā mākou i loko o ka Lagrange haʻilula ξ = Δh / 2, naʻe ka mea, pohaku ia e pololei, akā, e hāʻawi mai ai, e like me ka rula, he nui maikaʻi e hoʻokokoke aku i ka palapala hōʻike i ka olelo o ka palena pau.
Loiloi nā papakuhikuhi hewa ma ka palena pau ana
Ana mea , ma ka rula, paʻewa, a lawe mai i ke anaʻikepili e like me ka hewa. Ka mea, i wehewehe aku ma ke kaupalenaʻana i ka piha hewa, a, ma ka pōkole, ke kali hewa - 'oluʻolu, maopopo iho la i oi aku i ka hewa i loko o loa cia (paha ma loa e like me ia). Ke kaupalenaʻana i ka pili hewa ua kapaia ka puu hapa i loaa mai ma ka māheleʻana i ka mea ma ka piha waiwai o ke ana waiwai.
E mau haʻilula y me = f (x) kuleana pili i hoʻohana i ka vychislyaeniya y me, akā, i ka waiwai o ka m o ka huahelu hopena, a no ia mea, lawe mai i ka y me hewa. A laila, e 'imi i ka ke kaupalenaʻana loa kikowaena │Δu│funktsii y me, ka hoʻohana' ana i ka haʻilula
│Δu│≈│dy│ = │ [illegible] '(m) ││Δh│,
kahi │Δh│yavlyaetsya marginal hewa loulou. │Δu│ mea pono e ua hoʻokōʻia luna, e like me paʻewa ai i ka ho omaulia iho o ka hope o ke xi ma ka palena pau i ka ho omaulia.
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