Rārangi me te whārite pārōnaki ōrite o te tikanga tuatahi. tauira o rongoā

whakaaro ahau e ti'a ia tatou tīmata ki te hītori o te taputapu pāngarau kororia rite whārite pārōnaki. Ka rite ki te pārōnaki katoa me tuanaki wāhi, i whakaaro enei whārite e Newton i te mutunga o te rau tau 17. whakapono ia ko reira tona kitea kia nui e ara te karere whakamuna, e i teie mahana e taea te iritihia e whai ake: "nga ture katoa o te natura whakaahuatia e whārite pārōnaki." Kia mea te reira i te faarahiraa, engari te reira pono. Tetahi ture o te ahupūngao, matū, te koiora, e taea te whakaahuatia e ēnei whārite.

He koha nui ki te whanaketanga me te hanga a te ariā o whārite pārōnaki i pāngarau o Hēura me Lagrange. Kua i roto i te rau tau 18 kitea e ratou, ka whakawhanakehia te mea kei te ako i teie nei i te akoranga whare wānanga matua.

Te hoê mahana hou i roto i te ako o ngā whārite pārōnaki timata whakawhetai ki Anri Puankare. hanga e ia he "ariā kounga o whārite pārōnaki", e, ngā ki te ariā o ngā mahi o ngā taurangi matatini wāhi tino ki te turanga o te āhuawhatunga - te pūtaiao o te wāhi, me ona āhuatanga.

He aha e whārite pārōnaki?

Ko wehi o te parau tokomaha iwi "whārite pārōnaki". Heoi, i roto i tenei tuhinga ka whakaturia e matou i roto i roto i te taipitopito te mauri o tenei taputapu pāngarau tino whai hua i te mea mau e kore rite uaua rite te mea te reira i te taitara. I roto i te tikanga ki te timata ki te kōrero e pā ana ki te whārite pārōnaki tuatahi-tikanga, me te tuatahi kia mohio koutou ki te ariā taketake e kua ere e pā ana ki tenei whakamāramatanga. A ka tīmata tatou ki te pārōnaki.

pārōnaki

e matau ana tokomaha te iwi tenei wā mai kura nui. Heoi, tonu noho ki runga ki taua mea i roto i te taipitopito. A feruri na i te kauwhata o te pānga. Ka taea e tatou ake i reira ki te whānuitanga taua e riro tetahi o ona wāhanga te rārangi tika. Ka tango te reira e rua ngā e e ore tata ki tahi i te tahi. Ko te rerekētanga i waenganui i o ratou taunga (x y ranei) he ore. A huaina ana e ia ko te pārōnaki me pūāhua tautapa dy (pārōnaki o y) me te dx (te pārōnaki o x). He mea nui ki te matau, e kore te mea te pārōnaki te uara hopea, a ko te tikanga, me te mahi matua i tenei.

A inaianei me whakaaro koutou nga mea timatanga e whai ake nei, e ka ti'a ia tatou ki te whakamārama i te ariā whārite pārōnaki. Reira - pārōnaki.

pārōnaki

Kua rongo me katoa o tatou i te kura, me tenei ariā. mea ratou e te pārōnaki - ko te tere o te tupu heke o te mahi ranei. Heoi, riro tenei whakamāramatanga atu huanane. Kia tamata tatou ki te whakamārama i te ngā pärönaki o nga Pārōnaki. Kia haere a hoki ki te mahi ore āputa ki rua ngā, e kua kei i te tawhiti iti i ia atu. Ko ara tua atu tenei mahi tawhiti, ko te wa ki te huri i ki etahi uara. A ki te whakaahua i taua tauiraa, me te haere mai ki te pārōnaki e pai te kore e tuhituhia rite te ōwehenga o te Pārōnaki ake: f (x) '= DF / dx.

Na ko reira e tika ana ki te whakaaro i te āhuatanga taketake o te pārōnaki. E toru anake:

1. Ka taea te māngai moni pärönaki te rerekētanga ranei rite te moni ranei rerekētanga o te pärönaki: (he + b) '= he' + b ', me (AB)' = a'-b '.
2. hono te taonga tuarua ko te ki whakarea. mahi pārōnaki - ko te moni o nga mahi o tetahi mahi ki tetahi pārōnaki: (he * b) '= he' * b + he * b '.
3. Ka taea te tuhituhi te pärönaki o te rerekētanga rite te whārite e whai ake nei: (i te / b) '= (he' * ba * b ') / b ^2.

haere mai enei āhuatanga katoa i roto i te pepa mo te te kimi rongoā ki te pārōnaki whārite o te tikanga tuatahi.

Ano, i reira he pärönaki kanohi. Tera pea to tatou he mahi o te z, e tei runga i te taurangi x me te y. Hei tātai i te pārōnaki wāhanga o tenei mahi, hei tauira, i roto i te x, Me matou ki te tango i te y tāupe mō te tamau, me te ngāwari ki te rerekëtanga.

wāhi

Ko tētahi atu ariā nui - wāhi. I roto i te meka ko reira te ritenga o te pārōnaki. Pāwhaitua He maha ngā momo, ko te otinga māmā o whārite pārōnaki, Me matou te tino faufaa ore pāwhaitua tuatau.

Na, he aha te mea te wāhi? Kia mea a to tatou i te tahi hononga f o x. tangohia matou i reira i te wāhi me te whiwhi i te F pānga (x) (kei te maha i te reira ki rite te tahito ra), i te mea he pärönaki o te mahi taketake. Na reira F (x) '= f (x). fakahu'unga hoki tenei e te wāhi o te pārōnaki he rite ki te mahi taketake.

I roto i te whakaoti whārite pārōnaki he mea tino nui ki te mahino te tikanga me te mahi o te wāhi, mai i te tino maha i ki te tango ia ratou ki te kitea rongoā.

He rerekē nga whārite i runga i to ratou natura rānei. I roto i te wāhanga o muri e titiro tatou ki ngā momo o whārite kia pārōnaki tuatahi, a ka ako e founga ki te whakaoti ia ratou.

Karaehe o whārite naki

"Diffury" wehea e te tikanga o pärönaki whai wāhi i roto i a ratou. Ko te kupu i reira ko te tuatahi, te tuarua, tuatoru atu ranei kia. taea hoki te wehea ratou ki te maha akomanga: mau, me te kanohi.

I roto i tenei tuhinga, ka whakaaro matou nga whārite pārōnaki noa o te tikanga tuatahi. Tauira me rongoā tatou aparau i roto i te wāhanga e whai ake nei. whakaaro tatou i te TAC anake te mea ko reira te momo tino noa o whārite. Noa wehea ki momo: ki taurangi te wehewehe, ōrite, me te heterogeneous. Next ka ako koe e founga rerekē ratou i ia atu, a ka ako e founga ki te whakaoti ia ratou.

I tua atu, e taea te ngā enei whārite, kia ai i muri i te whiwhi tatou i te pūnaha o ngā whārite pārōnaki o te tikanga tuatahi. Taua pūnaha, titiro hoki tatou ki a ako e founga ki te whakaoti.

He aha e whakaaro tatou anake te tikanga tuatahi? No te mea he mea e tika ana ki te tīmata ki te ohie, me te whakaahua e pā ana ki ngā whārite pārōnaki katoa, i roto i te tuhinga kotahi tenei e taea e.

Whārite ki taurangi te wehewehe

He pea tenei te whārite raupapa pārōnaki tuatahi tino ohie. He tauira e taea te tuhituhia rite enei: y '= f (x) * f (y). Hei whakaoti i tēnei whārite e hiahia ana tatou ki te tātai māngai o te pārōnaki rite te ōwehenga o te Pārōnaki: y '= dy / dx. Ki te reira whiwhi tatou i te whārite: dy / dx = f (x) * f (y). Na e taea tatou te tahuri ki te tikanga o te whakaoti tauira paerewa: wehe te taurangi i roto i nga wahi, arā nohopuku whakamua te y tāupe katoa i roto i te wahi te wahi i reira he dy, ka hanga hoki i te x tāupe ... whiwhi tatou i te whārite o te puka: dy / f (y) = f (x) dx, tutuki nei e tango i te pāwhaitua o nga wahi e rua. Kaua e wareware e pā ana ki te tamau e hiahia ana koe ki te hoatu i muri i kōmitimiti.

Ko te otinga o tetahi "diffura" - ko te mahi o te x i te y (i roto i to tatou take), ki te mea i reira te mea he huru tau, te whakahoki ko te tau ranei. E hi'opoa tatou i te hi'oraa raima te akoranga katoa o te whakatau:

y '= 2y * hara (x)

Whakawhiti te taurangi i roto i te ahunga rerekē:

dy / y = 2 * hara (x) dx

Na tango i te pāwhaitua. Katoa o ratou e taea te kitea i roto i te tepu motuhake o pāwhaitua. A whiwhi tatou:

ln (y) = -2 * cos (x) + C

Ki te hiahiatia, ka taea e tatou e faaite i te "y" rite te mahi o "X". Na e taea tatou te mea e te kitea tatou whārite pārōnaki, ki te kahore i tohua huru. Ka taea te tohua huru, mo te tauira, y (n / 2) = e. Katahi matou ka whakauru noa te uara o enei taurangi i roto i te whakatau, me te kitea i te uara o te tamau. I roto i to tatou hi'oraa, he reira 1.

whārite pārōnaki kia tuatahi ōrite

Na i runga i ki nga wahi matatini ake. Ka taea te tuhituhia ōrite whārite raupapa pārōnaki tuatahi i roto i te puka whānui rite: y '= z (x, y). Me kï reira e te mahi tika o te rua taurangi he kākahu, a kore e taea te wehea ai ki e rua i runga i te āhua: z x me te z o y. Tirohia mehemea ko te whārite ōrite kahore ranei, he tino ohie: meinga tatou te whakauru x = k * x me te y = k * y. Na tapahia matou k katoa. Ki te tuturu iho enei pukapuka, na te whārite ōrite me taea puta humarie ki tona otinga. Titiro mua, mea tatou: ko ano hoki te parau tumu o te otinga o enei tauira rawa ohie.

E ti'a ia tatou ki te hanga i te whakauru: y = t (x) * x, i reira t - he mahi e hoki, tei runga x. Na ka taea e tatou e faaite i te pārōnaki: y '= t' (x) * x + t. Whakauru tenei katoa ki to tatou whārite taketake, me te whakamāmā i te reira, to tatou te tauira o te wehenga o ngā taurangi t rite x. Whakaoti i te reira, ka whiwhi i te fakafalala o t (x). A, no te ka matou i te reira, e whakawhitia noa to tatou whakauru y = t mua (x) * x. Na ka whiwhi tatou i te fakafalala o y runga x.

Ki te hanga mārama reira, ka matau tatou i te hi'oraa: x * y '= yx * e y / x.

A, no te taki i te whakakapinga o te heke katoa. Na, ko te tino ōrite te whārite. Na hanga tetahi whakauru, korero matou e pā ana ki: y = t (x) * x me te y '= t' (x) * x + t (x). I muri te whakangāwaritanga te whārite e whai ake nei: t '(x) * x = -e t. whakatau matou ki te tiki i te tauira ki te taurangi tapu, me te ^{whiwhi tatou: e -t = ln (C *} x). Me tika tatou ki te whakakapi t e y / x (no te mea, ki te y = t * x, ka t = y / x), a ka whiwhi tatou i te ^{whakahoki: e -y / x = ln (} x * C).

whārite pārōnaki rārangi o te tikanga tuatahi

Ko te wā ki te whai whakaaro ki tetahi kaupapa whanui. Ka titiro tatou heterogeneous whārite pārōnaki tuatahi-kia. Me pēhea e rerekē ratou i te rua o mua? Tatou i te reira. Ka taea te tuhituhia kupu rārangi whārite pārōnaki kia tuatahi i roto i te puka whānui o te whārite: y '+ g (x) * y = z (x). kia fakamahino'i reira e kia z (x) me te g (x) hei uara tamau.

Tenei te he tauira: y '- y * x = x ^2.

E rua huarahi ki te whakaoti, a ka tono tatou kia hi'opoa tatou e rua o ratou. Ko te tuatahi - te tikanga o te rerekētanga o taimau te noho.

Hei whakaoti i te whārite i roto i tenei tikanga, he mea e tika ana ki te whakaōrite i te taha ki matau-ringa tuatahi ki te kore, a ka whakaoti i te whārite hua nei i muri i riro te whakawhitinga o ngā wāhanga:

y '= y * x;

dy / dx = y * x;

dy / y = xdx;

ln | y | = x ^2/2 + C;

y = e ^{x2 / 2} * ^C y = C ₁ * e ^{x2 / 2.}

Na ko reira e tika ana ki te whakakapi i te tamau F ₁ i runga i te v pānga (x), e ka kitea e tatou.

y = v * e ^{x2 / 2.}

Te utu i te pārōnaki whakakapinga:

^{y '= v' * e x2} / ² -x * v * e ^{x2 / 2.}

Na whakauru enei kīanga ki te whārite taketake:

v '* e ^{x2 / 2} - x * v * e ^{x2 / 2} + x * v * e ^{x2 / 2} = x ^2.

Ka taea e kite koe i e i roto i te taha maui o te ngā e rua e iti. Ki te tahi mau tauira e kihai i tupu, na kua meatia e koe te tahi mea he. tonu matou kia:

v '* e ^{x2 / 2} = x ^2.

Na whakaoti matou i te whārite mua i roto i nei e hiahia ana koe ki te wehe i te taurangi:

AM / dx = x ^{2 /} e ^x2 / ^2;

AM = x ² * e ^{- x2 / 2} dx.

Hei tango i te wāhi, whai tatou kia tono i te kōmitimiti i ngā wāhanga konei. Heoi, e kore ko te kaupapa o tenei tuhinga i tenei. Ki te kei hiahia koe, ka taea e koe te ako i runga i ratou ake ki te kawe i roto i taua mahi. Ehara i te mea uaua, a kahore he wa e kai ana ki te nui pūkenga me te tiaki.

Ni'ai ki te tikanga tuarua te otinga o te whārite inhomogeneous: tikanga Bernoulli. He aha te huarahi Ko tere me te māmā - te reira ki runga ki a koutou.

Na, ka whakaoti tenei tikanga, e ti'a ia tatou ki te hanga i te whakauru: y = k * n. Here, k me n - etahi mahi i runga i x rānei. Na ka titiro te pārōnaki rite: y '= k' * n + k * n '. Whakakapinga e rua monoraa i roto i te whārite:

k '* n + k * n ' + x * k * n = x ^2.

Rōpū ake:

k '* n + k * ( n' + x * n) = x ^2.

Na he tika reira ki te taurite te ki kore, e ko i reu. Na, ki te whakakotahi e koe te whārite hua e rua, whiwhi tatou i te pūnaha o ngā whārite pārōnaki kia tuatahi ki te kia kitea:

n '+ x * n = 0;

* K 'n = x ^2.

Ko te ōritetanga tuatahi whakatau pehea te whārite mua. Ki te mahi i tenei, hiahia koe ki te wehe nga taurangi:

dn / dx = x * v;

dn / n = xdx.

tango tatou i te wāhi me te whiwhi tatou: ln (n) = x ^2/2. Na, ki te faaite tatou n:

n = e ^{x2 / 2.}

Na fetongi i te whārite hua ki te whārite tuarua:

k '* e ^{x2 / 2} = x ^2.

A, e whakaahua ana, whiwhi tatou i te whārite taua rite i roto i te huarahi tuatahi:

dk = x ^{2 /} e ^x2 / ^2.

Matou e kore ano e aparau i mahi atu. Kei te mea te reira e i whārite pārōnaki tuatahi-tikanga tuatahi otinga ai nui fifi. Heoi, kei te tīmata te rumaki hohonu i roto i te kaupapa ki te tiki pai me te pai.

Kei hea he whārite pārōnaki?

whārite kaha Tino pārōnaki whakamahia i roto i te ahupūngao, rite tata kua tuhituhia nga ture taketake katoa i roto i te puka pārōnaki, me aua tātai, e kite tatou - i te otinga ki enei whārite. I roto i te matū, e whakamahia ratou mo te take kotahi: kua ahu mai te ture taketake i roto i a ratou. I roto i te koiora, e whakamahia te whārite pārōnaki ki whakatauira te whanonga o ngā pūnaha, pērā i konihi - taonga. taea hoki te whakamahi ratou ki te hanga i ngā tauira o te uri, mo te tauira, koroni o microorganisms.

Ka rite ki te āwhina whārite pārōnaki i roto i te ora?

Ko te whakautu ki tēnei pātai he māmā: kahore. Ki te kore e koe i te pūtaiao mataaro ranei, ko reira pea e ka waiho ratou whai hua. Heoi, e kore i ahatia ki mohio ki ta te whārite pārōnaki me te kitea i te reira mo te whanaketanga whānui. Na ka ki te pātai o te tama, ranei tamahine, "he aha te whārite pārōnaki?" e kore e hoatu e koe ki roto ki te mutunga mate. Well, ki te he koe i te pūtaiao mataaro ranei, ka mohio koutou ki te faufaa o tenei kaupapa i roto i tetahi pūtaiao. Ko tino nui, e inaianei ki te pātai "pehea ki te whakaoti i te whārite pārōnaki o te tikanga tuatahi?" Ka tonu e taea ki te hoatu i te whakahoki kupu koe. Whakaae, e mea nehenehe tonu, no te ite koe e te mea iwi e ara wehi ki te kitea i roto i.

Te raruraru matua i roto i te ako

Ko te raruraru matua i roto i te māramatanga o tenei kaupapa ko te peu kino o mahi tuitui, me te pārōnaki. Ki te he fiemālie koe amo pärönaki me pāwhaitua, he reira pea utu atu ki te ako, kia ako ngā tikanga rerekē o te kōmitimiti, me te pārōnaki, me anake ka puta ki te ako o te rauemi e kua whakaahuatia, kua i roto i te tuhinga.

He miharo ki te ako e taea te whakawhiti taua dx ētahi iwi, rite mua (i roto i te kura) tohe e he wehewehe te hautanga dy / dx. Na e hiahia ana koe ki te pānui i te tuhinga i runga i te pārōnaki me matau e ko te reira i te huru o rahinga ore iti, e taea te rāwekeweke i roto i te whakaoti whārite.

e kore e tokomaha te iwi ite tonu e te otinga o ngā whārite pārōnaki o te raupapa tuatahi - ko te maha tenei te mahi ranei neberuschiysya wāhi, me homai tenei pohehe ratou he rota o te he.

He aha atu e taea te ako ki te matau pai?

He pai ki te tīmata rumaki atu ki te ao o te pārōnaki tuanaki o pukapuka motuhake, hei tauira, i roto i te tātari pāngarau mō ngā ākonga o tino kore-pāngarau. ka taea e koe te neke ki te tuhinga atu motuhake.

Kei te mea te reira i e, i roto i te tua ki te pārōnaki, reira e tonu whārite wāhi, kia ka tonu i koe te tahi mea ki te tutava no te me aha ki te ako.

mutunga

Te ti'aturi nei tatou e i muri i te pānui i tēnei tuhinga ka whai koe i te whakaaro o te mea nga whārite pārōnaki me pehea ki te whakaoti tika ratou.

I roto i tetahi take, pāngarau i roto i tetahi ara whai hua ki a tatou i roto i te ora. whakawhanake te reira i arorau me te whakarongo, kahore nei nga tangata, kia rite ki ehara i te ringa.