Tuanaki pārōnaki o ngā mahi o tetahi, me te maha o ngā taurangi

tuanaki pārōnaki Ko te manga o te tātaritanga pāngarau, e titiro te pārōnaki, Pārōnaki me ratou whakamahinga i roto i te ako o ngā mahi.

Ko te kōrero o

puta tuanaki pārōnaki hei ako motuhake i roto i te hawhe tuarua o te rau tau 17, te whakawhetai ki te mahi o Newton ko Leibniz, nana i waihanga nga tikanga taketake i roto i te tātai o Pārōnaki ka kite te hononga i waenganui i kōtuitui me te pārōnaki. Mai te ako whakawhanakehia e ia me te tātaitanga o pāwhaitua, reira e whakatu i te pūtake o te tātari pāngarau. Ko te ahua o enei calculi whakatuwheratia te wā hou hou i roto i te ao pāngarau a meinga te putanga o pekanga hou i roto i te pūtaiao. atu ano e te taea o te tono pāngarau i roto i nga tikanga māori me te engineering.

ariā taketake

hāngai tuanaki pārōnaki te i runga i nga ariā taketake o te pāngarau. Ko ratou: he tokomaha tūturu, tonu me te rohe o te mahi. I muri i te wa, kua tangohia e ratou i te titiro hou, whakawhetai ki te tuanaki wāhi, me te pārōnaki.

Ko te tukanga o te hanga

Hanganga o te tuanaki pārōnaki i te puka o te tono, a ka te tikanga pūtaiao puta i mua o te putanga o te ariā taumauri, i hanga nei e Nikolay Kuzansky. whakaaro tona mahi te ki he whanaketanga whanaketanga i te pūtaiao tawhito o te whakawa. Ahakoa te meka e kore ko te tangata tohe ia he mathematician, tona whai ki te whanaketanga o te pūtaiao pāngarau he whakakorekore. Cusa, tetahi o te i roto i tuatahi o te whakaaro o te tauhanga rite te pūtaiao tika tino, pāngarau e maka ana i te wa ki te pātai.

I roto i te pāngarau tawhito ko te wae paearu ao, i te tangata tohe whakaarohia rite te mutu- mehua hou hoki te tau tangohia. I roto i te hononga ki tenei kanohi Taupokia o tika i roto i te pūtaiao pāngarau. mōhiotanga pūtaiao, i roto i tona tirohanga, e tahuri iho ki kouma, me te maramarama. Ko te tuarua tika atu, rite ki te pūtaiao, mai i te mua homai hua āwhiwhiwhi anake.

whakaaro

Ko te whakaaro taketake me te ariā o te tuanaki pārōnaki e pā ana ki te mahi i roto i te tata iti o etahi ngā. Hoki tenei te mea e tika ana ki te hanga i te taputapu pāngarau ki mahi akoranga nei whanonga i roto i te tata iti o ngā tāuta tata ki te whanonga o te mahi rārangi he pūrau ranei. I runga i tenei whakamāramatanga o pārōnaki me te pārōnaki.

Ko te putanga o te ariā o te pārōnaki i meinga e te maha nui o ngā raruraru o sciences tūturu me te pāngarau, i arahina ki te faaotiraa o uara rohe o te momo kotahi.

Ko tētahi o nga mahi matua e kua hoatu hei tauira, tīmata ki te akomanga kura matamua, ko ki te whakatau i te tere o te nekehanga o te wāhi i roto i te rārangi torotika, me te hanganga o te raina pātapa ki tenei kōpiko. Ko te pārōnaki hono ki tenei, mai i te mea e taea ki te fakafuofua'i 'te mahi i roto i te tata iti o te mata o te mahi rārangi.

Ina whakaritea ki te ariā o te pärönaki o te mahi o te tāupe tūturu, te whakamāramatanga o Pārōnaki haere noa i runga i te mahi o te natura whānui, i roto i ngā te whakapakoko o te wāhi Euclidean ki tetahi.

pārōnaki

Kia te neke mata i roto i te ahunga o te tuaka-y, mo te wa tango matou x, whanganga nei i te timatanga o te kau. Te whakaahua i te kaupapa taua he taea i te pānga y = (x) f, e pā nei te ki i ia wa wāhi x taunga ira displaceable. Tenei karanga mahi i roto i rehe ki te tango i te ture o te nekehanga. Ko te āhuatanga matua o te nekehanga, ngā pāhiwihiwi, ko te tere inamata. A, no te oho te wāhi kei te taha te tuaka-y rite ki te ture o rehe, te wāhi wā matapōkere riro te reira taururuku x f (x). I roto i te wa pūwāhi x + Δh, te wahi tohu Δh te nuku o te wa, ka kordinaty reira f (x + Δh). tātai te kupu i hanga Δy = f (x + Δh) - (x) f, e kiia nei ko te mahi te nuku haere. Ko reira he wāhi o te ara traversed i roto i te wa i te x ki te x + Δh.

I roto i te hononga ki te takanga o te tere i te wa pārōnaki whakahaere te. Ko te pärönaki o tetahi mahi i te wāhi i whakaritea ka karanga te rohe (mana'o te vai te reira). Ka taea te tuku te reira ki etahi pūāhua:

f '(x), y', ÿ, DF / dx, dy / dx, DF (x).

Ko te tukanga o te tātai i te pārōnaki o karanga pārōnaki.

tuanaki pārōnaki o ngā mahi o te maha taurangi

tono i tēnei tikanga te ka tātai ako mahi, e rave rahi taurangi. A, no te reira e rua taurangi x me te y, te pārōnaki wāhanga ki te faatura ki x i te pūwāhi A i huaina ko te pärönaki o tenei mahi i roto i te x ki te y whakaritea.

Kia tohua e te tohu e whai ake nei:

f '(x) (x, y), u' (x), ∂u / ∂x me ∂f (x, y) '/ ∂x.

pūkenga e hiahiatia ana

I roto i te tikanga ki te ako angitu, ka e taea ki te whakaoti diffury pūkenga e hiahiatia ana i roto i te kōtuitui me te pārōnaki. Hei hanga i te reira māmā ki te mahino te whārite pārōnaki, me i matau kaupapa pärönaki, me te tuatau wāhi. Hoki e kore e kino ia ki te ako ki te titiro mo te pärönaki o te mahi paruhi. Ko e tika ana ki te meka e maha ka whakamahi pāwhaitua me te pārōnaki i roto i te tukanga o te ako tenei.

Ngā momo o whārite naki

Tata te mahi mana katoa e pā ana ki te whārite pārōnaki tuatahi-tikanga, i reira e 3 ngā momo o whārite: ōrite, ki taurangi te wehewehe, rārangi inhomogeneous.

He hoki nui atu te utu momo whārite ki Pārōnaki katoa, whārite o Bernoulli, me ētahi atu.

Fundamentals rongoā

Hei timata, kia tatou mahara he whārite taurangi o te akoranga kura. Kei roto ratou i te taurangi me tau. I roto i te tikanga ki te whakaoti i te whārite tikanga kia kitea nui o tau e makona i te huru whakaritea. Tikanga, enei whārite i tetahi pakiaka, a mo te whakamanatanga e te whakauru anake tenei uara ki wahi unknown.

Ko te rite ki tenei te whārite pārōnaki. I roto i te whānui, he whārite o te tikanga tuatahi ngā:

• taurangi motuhake.
• He pärönaki o te mahi tuatahi.
• Taumahi te taurangi whakawhirinaki ranei.

I roto i ētahi wā, i reira kia kia kahore tetahi unknown, x y ranei, engari e kore e he te reira rite nui rite he reira e tika ana ki te whai i te pārōnaki tuatahi, me kahore he pärönaki tikanga teitei ki te otinga me te tuanaki pārōnaki pono i.

Whakaoti i te whārite pārōnaki - te tikanga reira ki te kitea te huinga o nga mahi katoa e te hunga e tika ana faaiteraa i homai. Kei te maha i huaina taua huinga o ngā mahi te mana otinga whānui.

tuanaki tōpu

tuanaki wāhi ko tetahi o nga wāhanga o tātari pāngarau, e titiro te ariā o wāhi, āhuatanga me ngā tikanga o tona tātaitanga.

E mea pinepine te tātaitanga o te wāhi puta, ka tātai i te rohe o te āhua-pewa. Na roto i tenei tikanga te wāhi rohe, ki e kia hanga ai he wāhi tomu'a o te āhua tapamaha tuhituhi ki te hua āta i tona ringa, me te taha raraunga iti iho i tetahi uara noho tohua i mua iti.

Ko te whakaaro matua i roto i te tātaitanga o te rohe o tetahi āhua āhuahanga te tātai i te rohe o te tapawhā, ka i reira he taunakitanga e tona rohe, ko te rite ki te hua o te roa i te whānui. No te haere mai te reira ki te Geometry, ka e hanga nga waihanga katoa te whakamahi i te rangatira me te awhi, a ka te ōwehenga o te roa ki te whanui ko te uara kouma. A, no te tātai i te rohe o te tapatoru matau taea te whakaritea e te mea e hoatu ki a koutou i te tapatoru i muri, i hanga te tapawhā te. I roto i te rohe o te tapawhā whakarara e tātai i roto i te tikanga rite engari paku atu uaua, i roto i te tapawhā me te tapatoru. I roto i te rohe o te taparau whakaaro e tapatoru ngā roto reira.

I roto i te whakatau i te mahi tohu o te noho, e kore e tenei tikanga pai te ānau. Ki te whati i tatou i te reira ki tapawhā takitahi, ka noho i te reira wahi, tahanga kau. I roto i tenei take, ngana ki te whakamahi i nga koti e rua, ki te tapawhā i runga, me te raro, rite ki te hua o te hunga ngā te kauwhata o te pānga, me te kore e ngā. Nui i konei ko te ara ki te wahi i enei tapawhā. Hoki, ki te tango tatou i te whati atu me te ake iti, kia kōrero te rohe o te runga, me te raro i runga i te tetahi uara.

kia hoki te reira ki te tikanga mō te wehe i roto i ngā tapawhā. E rua ngā tikanga rongonui.

i whakamanatia Rēimāna whakamāramatanga o te wāhi, i hanga e Leibniz ko Newton, rite te rohe o subgraph. I roto i tenei take, whakaaro matou he ahua tënei o te maha etahi o tapawhā poutū whiwhi e wehewehe i te wā. A, no te whawhati i te hekenga i reira ko te rohe i te rohe heke o te ahua taua, ka karanga ki nei ko tenei rohe i te wāhi Rēimāna o te mahi i te wā i whakaritea.

He tikanga tuarua ko ki hanga te Lebesgue wāhi, arā, i roto i te meka e i roto i te wahi o te wehenga tohua rohe i runga i te wahi o te integrand me te whakahiato ka te moni wāhi o te uara whiwhi i roto i enei wahi, i wehea wā tona whānuitanga o ngā uara, me ka whakarapopototia ki te mehua hāngai whakapakoko kōaro o ēnei pāwhaitua.

taonga hou

tuhituhi e taha o nga painga matua mo te ako o te pārōnaki me te tuanaki wāhi Fikhtengol'ts - "o te pārōnaki me te tuanaki wāhi." Tona pukapuka ko te taputapu taketake mō te ako o te tātari pāngarau, i turia maha putanga me whakamāoritanga ki ētahi atu reo. Hanga hoki ngā ākonga, me te mo te wa roa e whakamahia ana i roto i te whānuitanga o ngā pūtahi mātauranga rite tetahi o nga painga matua o te ako. E horo'a te reira mōhiohio ariā me ngā pūkenga mahi. Tuatahi whakaputaina i roto i te 1948.

Hātepe mahi rangahau

Ki te tūhura i te tikanga o te pārōnaki tuanaki mahi, e hiahia ana koe ki te whai i te kua hoatu hātepe:

1. Kimihia te rohe o te mahi.
2. Kimihia nga pakiaka o te whārite homai.
3. Tātaihia te tirohanga e. Ki te mahi i tenei, tātai tatou te pārōnaki me te wāhi i reira he reira rite ki te kore.
4. whakakapi tatou te uara whiwhi i roto i EQ.

Ngā momo o whārite naki

Mana o te tikanga tuatahi (te kore, pārōnaki tuanaki o tetahi tāupe) me o ratou momo:

• Ki te taurangi te wehewehe whārite: f (y) dy = g (x) dx.
• Ko te whārite pārōnaki ranei māmā tuanaki mahi o tetahi tāupe, he te tātai: y '= f (x).
• Ko te tuatahi-raupapa mana nonuniform rārangi: y '+ P (x) y = Q (x).
• whārite pārōnaki Bernoulli: y '+ P (x) y = Q (x) y he.
• Whārite Pārōnaki katoa ki: P (x, y) dx + Q (x, y) dy = 0.

Ko te whārite pārōnaki o tikanga tuarua, me o ratou momo:

• Ōrite whārite pārōnaki kia tuarua rārangi ki ^{whakarea tonu: y n + PY '+ qy} = 0 p, no q R.
• Inhomogeneous whārite pārōnaki kia tuarua rārangi ki ^uara whakarea ^{tonu: y n + PY '+ qy} = f (x).
• Ōrite ^whārite pārōnaki ^{rārangi: y n + p (x)} y '+ q (x) y = 0, me inhomogeneous ^whārite tuarua ^{kia: y n + p (x)} y' + q (x) y = f (x).

whārite pārōnaki o whakahau teitei me ratou momo:

• Te whārite pārōnaki, tuku whakaiti o te ^{tikanga: F (x, y (k} ), y (k + ^{1), .., y (n)} = 0.
• He whārite rārangi o te tikanga teitei ^{_{ōrite: y (n) + f (}} N- ₁₎ y ^(n-1) + ... + f ₁ y '+ f ₀ y = 0, me ^{_{inhomogeneous: y (n) + f (}} n _-1) y ^(n-1) + ... + f ₁ y '+ f ₀ y = f (x).

Ngā wāhanga o te whakaoti i te raruraru ki te whārite pārōnaki

Ki te e faatiti'aifarohia te awhina o mana mamao pāngarau kore anake raruraru tinana ranei, engari ano hoki nga raruraru ngā o te koiora, ohaoha, pāpori me ētahi atu. Ahakoa te whānui o ngā kaupapa momo, kia whai i te raupapa arorau kotahi mō te whakaoti enei raruraru:

1. Tuhi ake mana. Ko tētahi o nga wāhanga uaua tino, e titau tika mōrahi, no te mea ka arahi tetahi hape ki hua tino kino. He mea tika ki te tango ki roto ki pūkete nga āhuatanga katoa e pa ana i te tukanga, me te whakatau i ngā āhuatanga tuatahi. kia hoki kia hāngai te reira i runga i ngā meka me whakatau arorau.
2. Hoki te whakaoti whārite. Ko tehea te ki te wāhi tuatahi tenei tukanga, mai titau te reira anake whakatinanatanga tino o tātaitanga pāngarau.
3. Tātaritanga me te aro mātai o te hua. kia otinga ahu aromatawaia mo te tāutanga o te uara mahi, me te ariā o te hua.

He tauira o te whakamahi o te pārōnaki whārite i roto i te rongoa

Mā te whakamahi i te mana mamao i roto i te mara o te rongoā kitea i roto i te hanga o te tauira pāngarau tahumaero. kia kore tatou e wareware e kitea ano enei whārite i roto i te koiora me te matū, e te hunga tata ki te rongoā, no te mea nui reira he wāhi nui te ako o rerekē taupori koiora me ngā tukanga matū i roto i te tinana tangata.

I roto i tenei tauira, e taea te tukinotia te horapa urutā o te mate i roto i te hapori taratahi. Kua wehea nga tangata ki e toru ngā momo:

• Huaketo, (he poto wā heki) te maha o te x (t), i ngā o te takitahi, kawe hopuhopu, ia o nei te mea hopuhopu.
• Ko te momo tuarua ngā haria mau taata y (t), e taea te huaketo e whakapā ki pangia.
• toru o te momo ngā takitahi whakangutungutu z (t), e he ārai ngaro tika ki te mate ranei.

Tau o takitahi tonu, e pupuri whanau, mate māori, me te hekenga te kore whakaaro. I te matua ka kia rua nga whakapae.

He rite ki te x (t) y (t) (e hāngai ana whakaaro i runga i te ariā e te maha o ngā take i roto i te rahi ki te maha o pekanga i waenganui i tūroro me ngā mema o te aro, e i roto i te āwhiwhitanga tuatahi he hautanga ki te x (t) y (t)), i roto i te mate Ōrau i etahi wāhi wā na reira kei te piki haere te maha o te wā, me te maha o heke i haria i te tere tatau nei i te toki tātai (t) y (t) (he> 0).

Tau o te kararehe responders-kore e mate riro wātea i te mate ranei, nui haere i te tere i te mea rite ki te maha o te wā, bx (t) (b> 0).

Ka rite ki te hua, ka taea e koe whakaturia ake te pūnaha o ngā whārite ki te tohu katoa e toru i runga i te pūtake o ona whakatau.

whakamahi TAUIRA ohaoha

Kei te maha whakamahia tuanaki pārōnaki i roto i te tātari ohaoha. whakaaro te mahi matua i roto i te tātari ohaoha te ki hei te ako o nga uara o te ōhanga, i kua oti te tuhituhi i roto i te puka o te mahi. Kei te whakamahia te reira i roto i hei whakaoti rapanga pēnei i ngā huringa i roto i te piki tāke whiwhinga moni i muri tonu, utu urunga, huringa i roto i te hua, ka huri i te uara o te hua, i roto i te aha e taea te whakakapi rahi e kaimahi mutu ki taputapu hou. Hei whakaoti taua raruraru, e hiahiatia ana e ia ki te hanga i te mahi kōrero o te taurangi hou, e, i muri i te ako i te tuanaki pārōnaki.

Ko reira tika maha ki te kitea te mahi tino tino pai i roto i te ao ōhanga: hua mōrahi, te moni teitei, te iti rawa utu, me te pera i runga i. taua wähanga takitaha ko te mahi o tetahi atu ranei tohenga. Hei tauira, ka taea te whakaaro te hanga kia rite ki te mahi o te mahi, me te whakapaipai. I roto i tenei hononga, te kimi i te uara e tika ana e taea kia heke ki te kimi i te mōrahi iti rawa ranei o te mahi o tetahi atu ranei taurangi.

Taua raruraru hanga he piha o ngā raruraru extremal i te mara ōhanga, mo e hiahia ana koe tuanaki pārōnaki. A, no te hiahiatia te tohu ōhanga te ki whakaiti whakanui rite te mahi o ētahi atu tawhā ranei, te nuku ōwehenga wāhi mōrahi mahi ki nga tohenga ka ahu ki te kore, ki te whangai ana te nuku o te tautohe ki te kore. Te kore, ina e whangai ana i te huru taua ki te tetahi uara pai kino ranei, e kore te mea e tika ana te wāhi i tohua, no te mea na roto i te whakanui ake i whakaiti ranei te tautohe e taea te huri uara ti'aturi i roto i te aronga e hiahiatia ana. I roto i te kupu tuanaki pārōnaki, e te tikanga tenei e nga tikanga e hiahiatia ana mo te mahi mōrahi ko te kore uara o tona pārōnaki.

e kore he raruraru nohinohi o te kimi i te extremum o te mahi o te maha taurangi te ōhanga, no te mea kei te hanga tohu ōhanga ake o ngā āhuatanga maha. E pai te matau taua take i roto i te ariā o ngā mahi o te maha taurangi, te tikanga o te tātai i te pārōnaki. e kore ngā taua raruraru whakanuia anake, ka whakaitihia mahi, engari ano hoki whāititanga. hāngai ana ēnei pātai ki te hōtaka pāngarau, me e faatiti'aifarohia ratou ki e hāngai ana hoki te awhina o tikanga rawa whakawhanakehia i runga i tenei peka o te pūtaiao.

I roto i nga tikanga o te pārōnaki tuanaki whakamahia i roto i te ōhanga, he wāhanga nui ko te whakamātautau hopea. I roto i te ao ōhanga, e pā ana te wā ki te huinga o ngā tikanga o te rangahau o te mahi tāupe me hua ina huri koe i te rōrahi o te hanganga, kohi, e hāngai ana i runga i te tātaritanga o ratou uara rohe. Whāiti tohu whakaaro pärönaki te pārōnaki wāhanga ki te maha taurangi ranei.

tuanaki pārōnaki o te maha taurangi - he kaupapa nui o te tātari pāngarau. Hoki te ako taipitopito, ka taea e koe te whakamahi i te whānuitanga o ngā taonga whakaako hoki teitei pütahitanga mätauranga. Ko tētahi o nga Fikhtengol'ts tino rongonui hanga - "o te pārōnaki me te tuanaki wāhi." Kia pehea te nui o te ingoa mo te otinga o ngā whārite pārōnaki o ngā faufaa ki te whai i te pūkenga ki te mahi ki te pāwhaitua. A, no te reira te mea he tuanaki pārōnaki o ngā mahi o tetahi tāupe, riro māmā te whakatau. Ahakoa, kia tuhia te reira, whai reira nga ture taketake taua. I roto i te mahi, ki te tirotiro i te mahi o te tuanaki pārōnaki, tika whai te hātepe kua tīariari, homai nei i roto i te kura nui, me te iti noa uaua ki te whakataki o ngā taurangi hou.