Laina a me homogeneous palena pau helu kaulike o ka mua o ka papa. examples o nā pāʻoihana

I manaʻo mākou e hoʻomaka me ka mōʻaukala o ka nani makemakika paahana like palena pau nā helu kaulike. Like a pau i ka palena pau a me ka integral calculus, i invented mau nā helu kaulike ma Newton ma ka hopena o 17th kenekulia. He manaoio ka mea, ua i kona loaʻa nui no laila, i ke ahiahi, i ka hoʻopā'āluaʻia olelo, i keia la hiki ke unuhi 'ia penei: ". A pau i nā kānāwai o ke ano i hoakakaia ma ka palena pau nā helu kaulike" A kohu mea, i ka exaggeration, akā, heʻoiaʻiʻo. Kekahi kānāwai o ke kālaikūlohea, kemika, kālaimeaola, hiki ke ho'ākāka 'ia ma kēia mau nā helu kaulike.

An, alahula na makana i ke ana a me ka hanaia'na o ka pähola 'o ka palena pau nā helu kaulike i makemakika o Euler a me Lagrange. Ua i loko o ka 18th kenekulia ka mea, kaunana a hoʻomohala mea ua manawa këia i ka papa kiʻekiʻe kulanui papa.

A hou milestone i loko o ke ao o ka palena pau nā helu kaulike hoomaka aku la e Anri Puankare. He hana mai i kekahi "qualitative pähola 'o ka palena pau nā helu kaulike", a, hui me ke kumumanaʻo o ka mana o ka luna' aiaiiuo ka pilina me ka nui loa i ke kumu o ka topology - ka 'epekema o ka makahiki a me kona waiwai.

He aha nā palena pau nā helu kaulike?

He nui nā poʻe e makaʻu o ka hopuna'ōlelo "palena pau helu kaulike". Naʻe, i loko o kēia 'atikala, e hoonoho aku makou i mai i ka au mamuli o ka ae pā o kēia loa pono keia mau mea makemakika hoʻopololei i mea nae i like huikau like ia mea, mai ka inoa. I mea e hoʻomaka e kamailio e pili ana i ka mua-mea palena pau helu kaulike, e pono mua loaaʻi ka ike lea i ke kumu o manaÿo i i inherently pili i kēia ho'ākāka 'ana. A mākou e hoʻomaka me ka palena pau.

palena pau

He nui nā kanaka i keia manawa, mai ke kula kiʻekiʻe. Eia naʻe, nō e noho ma luna o ka mea i loko o au mamuli. Hoi i noonoo ai i ka pakuhi o ke kuleana pili i. Mākou ke hoomahuahua ia i ka ia he nui i kekahi o kona Hoʻohana i pololei ka laina. Ka mea, e lawe i elua mauʻai i nā infinitely kokoke i kekahi i kekahi. Ke koena unuhi ma waena o ko lākou mau wahi (m a me y me) mea infinitesimal. A ka mea, ua kapaʻia palena pau a me nā huapalapala i hoʻokohu 'ia dy (palena pau o ka y me) a me ka dx (i ka palena pau o ka m). He nui no ka maopopo i ka palena pau mea i ka hope loa cia, a me keia mea i keʻano, a me ka papa kuhikuhiE kuleana pili i.

Ano oe e noonoo i ke kēia mau hehee wale, a mākou e pono e wehewehe i ka palena pau helu kaulike manaʻo. It - derivative.

derivative

pono i lohe a pau o ko mākou ma ke kula a me kēia manaʻo. I mai la lakou i ka derivative - o ka uku o ka ulu a me ka emi ana o ke kuleana pili i. Naʻe, i kēia ho'ākāka 'ana i oi i hookahuli ai. E mākou ho'āʻo e wehewehe i ka derivative hua'ōlelo o nā differentials. E ka hele hoʻi i ka infinitesimal wā mawaena hana a me nā heluʻai, a i Aia ma ka palena iki mamao mai kekahi i kekahi. Akā, a hiki ma kela aoao o keia papa mamao o ka manawa e hoʻololi i kekahi nui. A me ka wehewehe i ka hoololi, a hele mai me ka derivative i makemake ole ia e palapala like me ka lākiō o ka differentials: f (x) '= df / dx.

Ano ka mea, Ua pono, e noonoo i ka walaʻauʻana waiwai o ka derivative. Aia nā wale ekolu:

1. Derivative huina a me ke koena unuhi hiki e na poe like me ka huina a me koena unuhi o na nā mea loaʻa: (he + b) '= He' + b. ', A (ab)' = a'-e '.
2. Ka lua o ka waiwai ka pili me ka multiplication. Derivative hana - o ka huina o na hana o ke kuleana pili i kekahi i kekahi derivative: (he * b) '= i' * e + i ka * b. '.
3. Ke derivative o ke koena unuhi hiki ke kākau i nā kēia helu kaulike. (I / b) '= (he' * BA * b. ') / B ^2.

A pau kēia mau hiʻona e hele mai i loko o mākaukau no ka loaa pāʻoihana e palena pau nā helu kaulike o ka mua o ka papa.

Nō hoʻi, ma laila nō ka helehelena nā mea loaʻa. Ina mākou i ka hana o ka Z, i hilinaʻi nui ma luna o nā aiaiiuo xa me y. No ka ho omaulia i ka hapa derivative o keia papa, no ka laʻana, ma ka m, ua pono, e lawe i ka ee iaaanu aey y me ka ikaika a me ka oluolu, e differentiate.

integral

Kekahi mea nui manaʻo - integral. I mea ka mea, o ke ku pono ana o ka derivative. Integrals i mau hoailona ia mea, akā, i ka simplest pāʻoihana o ka palena pau nā helu kaulike, ua pono ka loa AICIeOEOI indefinite integrals.

No laila, he aha ka mea i ka integral? E ka olelo aku mākou i kekahi pilina [illegible] o ka m. lawe mākou mai ia i ka integral, a loaa i ka papa F (x) (ka mea, ua pinepine haawiia keia i like me ka primitive), a mea he derivative o ka palapala kuleana pili i. No ia mea, F (x) '= f (x). Ua hoi hoʻohuʻu i ka integral o ka derivative mea like me ka palapala kuleana pili i.

Ma ka hoʻoponopono palena pau nā helu kaulike ia mea nui e maopopo i keʻano a me ka hana ana o ka integral, no ka loa pinepine i, e lawe ia, e loaʻa pāʻoihana.

ʻO nā helu kaulike i okoa ke kaumaha ma muli o ko lakou ano. I ka lalo nei, e nana aku kakou, i ke ano o ka mua mea palena pau nā helu kaulike, a laila, aʻo pehea e hoʻoponopono i ia.

Classes o ka palena pau nā helu kaulike

"Diffury" i puunaueia e ka mea o nā mea loaʻa 'ole i loko o ia mau mea. Pela ka mea, he mua, lua, kolu a oi kauoha. Ka mea, hiki no hoi ke mahele ia i loko o kekahi mau papa: ano a me ka helehelena.

Ma keia 'atikala, mākou e noonoo i ka maʻamau palena pau nā helu kaulike o ka mua o ka papa. Ano he kumu hoʻohālike a me nā pāʻoihana mākou kamaʻilio ma ka hope o na pauku. E noonoo wale ka TAC, no ka mea, o ka loa e like me ka ano o nā helu kaulike. Ano o maheleia i subspecies: me separable DEBFULLNAME, homogeneous a me ka heterogeneous. Next oe e aʻo pehea e oko mai kekahi i kekahi, a ua aʻo pehea e hoʻoponopono i ia.

Eia hou kekahi, mau nā helu kaulike hiki ke hui, no laila, i ke kiʻi i kekahi ano o ka palena pau nā helu kaulike o ka mua o ka papa ma hope. Oia nenoai, ua ae mai no hoi i, a aʻo pehea e hoʻoponopono.

No ke aha lā mākou e wale i ka mua o ka papa e noonoo pono? No ka mea, Ua pono, e hoʻomaka ana me ka noonoo ole a me ka wehewehe a pau pili me ka palena pau nā helu kaulike, i loko o ka hoʻokahi 'atikala he mea hiki ole.

Nā helu kaulike me separable aiaiiuo

'O kēia paha ka loa noonoo ole mua mea palena pau nā helu kaulike. Mau mea examples e hiki ke palapalaia e like me: y me '= f (x) * f (y me). E hoʻoponopono i kēia helu kaulike, ua pono i ka hoohalike ai haʻilula o ka derivative like me ka lākiō o ka differentials: y me '= dy / dx. Me ka mea, ua loaa i ka helu kaulike: dy / dx = f (x) * f (y me). Ano, ua hiki ke huli i ke ano o ka hoʻoponopono maʻamau examples: hookaawale i ka aiaiiuo i loko o māhele, oa hookeai aku a pau i ka ee iaaanu aey y me i loko o ka hapa ma he mea dy, a no hoi e ka ee iaaanu aey m ... E loaa i kekahi pane helu kaulike o ka palapala: dy / f (y me) = f (x) dx, a ua loaʻa ma ka lawe ana i ka integrals o na wahi elua. Mai poina e pili ana i ka mea ikaika i kou makemake, e kau mai ma hope o hoʻohui '.

Ke kākoʻo pāʻoihana o kekahi "diffura" - o ka hana o ka m ma ka y me (i loko o mākou hihia), a ina he mea he laulā ana, i ka pane mea he helu. E mai hoike i ka pōhaku puna hoʻohālike i ka holoʻokoʻa papa o ka hooholo ana.

y me '= 2y * hewa (m)

Hoolilo i ka aiaiiuo ma kekahiʻaoʻao:

dy / y me = 2 * hewa (m) dx

Ano lawe i ka integrals. A pau o ia hiki ke loaʻa i loko o ka kūikawā papa o integrals. A ua loaʻa:

Ln (y me) = -2 * cos (m) + C

Inā koi 'ia, ua hiki ka hoike i ka "y" e like me ka hana o ka "X". Ano, ua hiki ke olelo aku i ka palena pau helu kaulike ua wehe pilikiaʻana, ināʻaʻole hoakaka ia ke ano. Hiki e hoakaka ia ana, no ka laʻana, y me (n / 2) = E. A laila mākou e wale panai i ka waiwai o kēia mau aiaiiuo i loko o ka olelo hooholo, a imi i ka waiwai o ka mea ikaika. I ko makou laʻana, ia mea 1.

Homogeneous mua mea palena pau nā helu kaulike

Ano, i hiki i ka hou luna 'wahi. Homogeneous mua mea palena pau nā helu kaulike hiki ke kākau i loko o mau ano like: y me '= Z (m, y me). It E e hoʻomaopopo i ka hana pono o nā aiaiiuo kaʻaʻahu, a me ka mea hiki ole ke maheleia i mau ke kaumaha ma muli: Z m, a Z o ka y me. Hōʻoia ina ka helu kaulike mea homogeneous paha ole, Ua puni hoʻi: mākou e ka hakahaka He m = ke k * xa me y = K * y me. Ano, uaʻokiʻia nā mea a pau ia ho. Inā mau palapala i haule, a laila, i ka helu kaulike homogeneous, a hiki maluhia ke hoomau aku i kona kākoʻo pāʻoihana. Ke huli nei mua, ke i aku nei au: Ua hoi ka loina hooponopono o ka pāʻoihana o kēia mau examples loa noonoo ole.

Mākou Pono e hana i ka hakahaka: y me =, t (m) * m, kahi, t - he papa ia, ua hilinaʻi nui ma ka m. A laila, ua hiki ka hoike i ka derivative: y me '=' ole '(m) * m +' ole. Hookomoia a pau i kēia i loko o mākou palapala helu kaulike a me ka simplifying ia, ke i ke kumu hoohalike o ke kaawale ana o aiaiiuo Halau like m. Wehewehe i ka mea, a loaa i ke kaukaʻi ana o na (m). I ka hōʻeaʻana ia, wale panai mākou mua hakahaka y me = 'ole (m) * m. A laila, loaa mākou i ke kaukaʻi o ka y me ka m.

E hana ia clearer, e ike mākou i laʻana: m * y me '= yx * E y me / m.

Ka wā o kéu hoʻomaopopo i ka hope o nā mea a pau emi. No laila, i ka helu kaulike mea maoli homogeneous. Ano, e kekahi hakahaka, ua kamailio e pili ana i: y me =, t (m) * xa me y '=' ole '(m) * m +' ole (m). Ma hope o simplification ke kēia helu kaulike: Halau '(m) * m = -e' ole. Mākou hoʻoholo e kiʻi i ka hāpana me ka hookaawale aiaiiuo a me ^{ka holo} o ^{kaua: E -t = Ln (C *} m). Mākou wale Pono e puku i 'ole ma ka y me / m (no ka mea, ina y me = k * m, laila, t = y me / m), a ua loaa i ka ^{mai: E -y / m = Ln (} m * C).

Laina palena pau helu kaulike o ka mua o ka papa

Ua ka manawa i noonoo i kekahi palahalaha kumuhana. Mākou e nānā heterogeneous mua-mea palena pau nā helu kaulike. Pehea ka mea, oko, mai ka mua elua? E ka maka ia. Laina mua mea palena pau nā helu kaulike i loko o ka nui 'ano o ka helu kaulike hiki ke palapalaia penei: y me' + g. (M) * y me = Z (m). It E e möakäka hou ia Z (m) a me ka g. (M) i e ikaika nā loina.

Eia kekahi laʻana: y me '- y me * m = m ^2.

Aia nāʻaoʻaoʻelua i ke hoʻoponopono, a māua e hoʻoponopono ai E nana pu no ia mau mea. Ka mua - ka ano o ka oko o ākeʻakeʻa kumu o ia a eia.

E hoʻoponopono i ka helu kaulike i loko o keia ano, e mea pono e equate i ka'ākau-limaʻaoʻao mua ia Aʻohe, a hoʻoponopono i ke kūpono helu kaulike a ma hope o ka hoolilo ana o māhele i:

y me '= y me * m;

dy / dx = y me * m;

dy / y me = xdx;

Ln | y me | = m ^2/2 + C;

y me = E ^{X2 / 2} * ^C y me = C ₁ * E ^{X2 / 2.}

Ano ka mea, Ua pono, e puku i ka ikaika C ₁ ma luna o ka papa V (m), a mākou e loaʻa.

y me = V * E ^{X2 / 2.}

E kaha i ka hope derivative:

^{y me '= V' * E X2} / ² -x * V * E ^{X2 / 2.}

A hookomoia mau, ka 'aoʻao i loko o ka palapala helu kaulike.

V '* E ^{X2 / 2} - m * V * E ^{X2 / 2} + m * V * E ^{X2 / 2} = m ^2.

Oe ke ike i loko o kaʻaoʻao hema o ka hua'ōlelo elua i hoemiia. Inā kekahi kumu i i ole e hiki mai, a lailaʻoe i hana i kekahi mea hewa. Mākou e hoʻomau i ka:

V '* E ^{X2 / 2} = m ^2.

Ano, ua hoʻoponopono i ka mau helu kaulike i loko o ka mea au i makemake, e hoʻokaʻawale i ka aiaiiuo:

dv / dx = m ^{2 /} E ^X2 / ^2;

dv = m ² * e ^{- X2 / 2} dx.

E wehe i ka integral, mākou i e pili i ka hoʻohui 'ma ka māhele' aneʻi. Naʻe, i kēia mea,ʻaʻole i ke kumuhana o keia Haawina. Inā 'oe makemake e hoihoi, e hiki ke aʻo ma luna o lākou iho, e lawe mai ia mau hana. Ia mea,ʻaʻole paʻakikī, a me ka nui ke akamai a me ka mālama mea i manawa e hoopau ana.

Ua haawiia i ka lua o ka iaoia ka pāʻoihana o ke inhomogeneous nā helu kaulike: Bernoulli hana. He aha kokoke ka wikiwiki a me ka hiki pono o - he mai ia oe.

No laila, i ka wa a ka hoʻoponopono i kēia hana, ua pono e hana ai i ka hakahaka: y me = K * n. Eia, ke Kalapu a N - kekahi mana ke kaumaha ma muli m. A laila, e nana i ka derivative e like: y me '= K' * N + ke k * N '. Ae pani mau substitutions ma ka helu kaulike.

ke k '* N + ke k * N ' + m * ke k * N = m ^2.

Group mai:

ke k '* N + ke k * ( N' + m * n.) = m ^2.

Ano ka mea, Ua pono, e equate i Aʻohe, i mea ma parentheses. Ano, ina e hui pu iho la ka kūpono nā helu kaulike elua, ua loaa he kahua o ka mua mea palena pau nā helu kaulike e e nā mākau ':

N '+ m * N = 0;

ke k '* N = m ^2.

I ka hoʻokolokolo pono mua hoʻoholo i ka mau helu kaulike. E hana i kēia, e pono e hoʻokaʻawale i ka aiaiiuo:

dn / dx = m * V;

dn / N = xdx.

Mākou i lawe i ka integral, a ua loaa: Ln (N) = m ^2/2. A laila, inā mākou ka hoike n:

N = E ^{X2 / 2.}

Ano, e panai i ka kūpono helu kaulike i loko o ka lua o ka helu kaulike.

ke k '* E ^{X2 / 2} = m ^2.

A me ka hoano hou ana ia, ua loaa ka ia helu kaulike like i loko o ka papa hana mua:

DK = m ^{2 /} E ^X2 / ^2.

Mākou no hoi e ole kūkākūkā ka hana hou aku. Ua ua olelo mai ia ma mua mua-mea palena pau nā helu kaulike pāʻoihana ke kumu nui pilikia. Eia naʻe, heʻoi kaiapuni i loko o ke kumuhana e hoʻomaka ana e kiʻi maikaʻi a me ka maikaʻi.

ʻAuhea palena pau nā helu kaulike?

Loaʻeleu palena pau nā helu kaulike i hoʻohana i ke kālaikūlohea, e like me ka aneane i kākau i nā kānāwai walaʻauʻana ma ka palena pau palapala, a me ka poe nā papakuhikuhi, i ke ike - i ka pāʻoihana i kēia mau nā helu kaulike. Ma kemika, ka mea i hoʻohana no ka mea ia kumu. I loko o nā kumu o ke kanawai ma ia. Ma kālaimeaola, e hoʻohana i ka palena pau nā helu kaulike e ÷ ike i ka hana o ka 'oihana, e like me Moana - waiwai pio. Ka mea hiki i ke hoʻohana 'ia no ka ho okumu Ke Ana Hoʻohālike o nā'ōlelo, no ka laʻana, panalāʻau o microorganisms.

Like palena pau nā helu kaulike kōkua i loko o ke ola?

Ke pane i keia ninau mea puni hoʻi: iki. Inā 'oe i ole he nä känaka' epekema a me ka lunaʻenehana, ia mea olelo lealea i ka mea e e maikaʻi kēia. Eia naʻe,ʻaʻole i hanaʻeha e ike i ka mea a ka palena pau helu kaulike a me ka mea, ua wehe pilikia no ka pono kūpono 'ana. A laila, i ka ninau o ke keikikane a ke kaikamahine, "He aha ka palena pau helu kaulike?" mai kau oe i loko o ka hopena make. Ua pono, ina oe e kekahi nä känaka 'epekema a me ka lunaʻenehana, alaila oe e ike i ka nui o keia kumuhana i loko o kekahi' epekema. Akā loa importantly, i kēia manawa i ka nīnau "Pehea e hoʻoponopono i ka palena pau helu kaulike o ka mua o ka papa?" oe, e mau e hiki ke haawi aku i ka pane. E hoolaulea, ka mea, he mau kiʻi wā nui kou 'ike i ka poʻe i hiki makau i loaʻa mai.

I ka papa kuhikuhiE pilikia i loko o ke ao

I ka papa kuhikuhiE pilikia i loko o ka naauao o keia kumuhana mea he ino maʻa o ka hoʻohui 'a me ka differentiation hana. Inā 'oe i ke koikoi kuhi nā mea loaʻa a me integrals, ia mea paha ke kumukuai nui e aʻo, e aʻo kekahi mau ano o ka hoʻohui' a me ka differentiation, a wale laila, ke hoomau aku a hiki i ka hoʻopaʻa 'ana i ka mea e, ua ho'ākāka' ia ma loko o ka pauku.

Kekahi kanaka e kāhāhā e aʻo i dx hiki ke hoolilo, e like me ka mua (ma kula) ua kūkākūkāʻia i ka mahele dy / dx mea indivisible. A laila, e pono e heluhelu i ka moʻokalaleo ma ka derivative, a ike i ka mea o ka heluia, o ka mea infinitely uuku, i hiki ke manipulated i loko o ka hoʻoponopono nā helu kaulike.

He nui nā poʻe mai i koke ike i ka pāʻoihana o ka palena pau nā helu kaulike o ka mua o ka papa - keia mea pinepine ka papa a neberuschiysya integral, a me keia delusion haawi mai ia i ka puu o ka pōpilikia.

He aha hou aʻe hiki ke aʻo i ka pono maopopo?

E aho nō e hoʻomaka hou kaiapuni i loko o ke ao nei o ka palena pau calculus o ka hana kūikawā puke, no ka laʻana, ma ka makemakika Ka Ikepili no nā haumāna o ka ole-makemakika meaʻai poʻokela pepa. Oe hiki laila, neʻe i ka oi hana kūikawā moʻokalaleo.

Ua ua olelo mai ia, ma ka hou ana i ka palena pau, eia nō integral nā helu kaulike, pela no oe e mau loaa iaʻu kekahi mea e hakakā ai a me ka mea e hoʻopaʻa haʻawina.

ka hopena

Mākou manaolana i hope heluhelu i kēia 'atikala oe e i ka manaʻo o ka mea ma ka palena pau nā helu kaulike, a pehea e hoʻoponopono i ia pono.

Ma kekahi hihia, makemakika ma kekahi ala e pono iā mākou i loko o ke ola. Ua hoʻomōhala ana i ke kūpili a me ka noonoo, me ka a kela kanaka keia kanaka, e like me ka ole lima.