Yuav ua li cas yog ib, thiab dab tsi yog nws lub cev lub ntsiab lus

Cov tsos yog lub tswvyim ntawm ib vim qhov yuav tsum tau nrhiav ib tug txheej thaum ub muaj nuj nqi ntawm nws cov derivative, thiab txiav txim seb tus nqi ntawm kev ua hauj lwm cheeb tsam complex nrhiav, deb mus ncig deb li cas, nrog rau cov tsis tau teev tseg curves los ntawm nonlinear sib npaug.

ntawm cov hoob kawm thiab cov physics peb paub hais tias cov ua hauj lwm yog cov khoom ntawm kev quab yuam ntau tshaj ib tug deb. Yog hais tias tag nrho cov zog yog ib qhov ceev los yog deb yog kov yeej nrog rau daim ntawv thov los ntawm cov tib quab yuam, ces txhua yam yog tseeb, koj tsuas multiply. Yuav ua li cas yog lub ib ntawm lub qhov? Qhov no yog ib tug linear muaj nuj nqi ntawm daim ntawv no y = kx + c.

Tab sis lub hwj chim rau cov lag luam yuav txawv thiab nyob rau hauv ib co chaw txawb kev sib raug zoo. Ib tug zoo xws li cov teeb meem no tshwm sim nrog muab xam rau ntawm deb mus ncig, yog tias cov kev ceev tsis yog tas li.

Yog li ntawd, nws yog to taub yog vim li cas muaj ib tug ib. Txhais nws raws li ib tug zaum ntawm cov khoom ntawm qhov tseem ceeb ntawm cov nuj nqi rau cov infinitesimal increment ntawm lub cav kiag li piav tus thawj xib fwb lub ntsiab lus ntawm lub sij hawm raws li qhov chaw ntawm lub cev ciaj los ntawm cov saum toj kawg nkaus txoj kab ntawm cov nuj nqi, thiab cov npoo - lub ntsiab txhais ntawm ib thaj tsam.

Jean Gaston Darboux, Fabkis mathematician, nyob rau hauv lub thib ob ib nrab ntawm cov XIX xyoo pua yog kom meej meej heev piav qhia hais tias qhov no tseem ceeb heev. Nws ua rau nws thiaj li paub tseeb hais tias ib tug tag nrho yuav tsis tau yooj yim rau to taub txawm tias ib tug schoolboy junior high school nyob rau hauv cov teeb meem no.

Piv txwv yog ib tug muaj nuj nqi ntawm tej complex zoo. y-axis, uas yog tso tus nqi ntawm cov kev sib cav, yog muab faib mus rau hauv me me Team sib, hom phiaj, lawv yog infinitely me me, tab sis vim hais tias lub tswvyim ntawm infinity yog heev paub daws teeb, nws yog txaus xav txog tej yam cia li daim me me, tus nqi ntawm cov uas yog feem ntau yog denoted los ntawm cov Greek tsab ntawv Δ (delta).

Cov kev ua tau "hlais" mus rau hauv me me blocks.

Txhua tus nqi ntawm lub cav raws li ib tug taw tes rau lub ordinate axis uas tso cov coj tseem ceeb ntawm cov nuj nqi. Tab sis raws li ib thaj tsam nyob rau hauv lub xaiv cheeb tsam ob, qhov tseem ceeb thiab kev khiav dej num kuj yuav ob los yog ntau tshaj thiab tsawg.

Lub sum ntawm cov khoom ntawm loj qhov tseem ceeb rau lub increment Δ hu ua Darboux loj npaum li cas, thiab yog raug xa mus rau raws li S. Yog li ntawd, me qhov tseem ceeb rau ib tug tsawg cheeb tsam, multiplied by Δ, ua ke tsim tau ib tug nqi tsawg tsawg Darboux s. Lub tsev kawm ntawv nws tus kheej tsa ib lub Trapezoid, li ntawd, raws li ib tug muaj nuj nqi ntawm lub curvature ntawm txoj kab vim ib infinitesimal increment nws yuav tsis saib tsis xyuas. Qhov yooj yim txoj kev uas yuav nrhiav tau cov cheeb tsam ntawm ib tug geometric zoo - ib tug ntau quav tso daim loj thiab me qhov tseem ceeb ntawm cov nuj nqi rau Δ-increment thiab faib los ntawm ob tug, uas yog txhais raws li lub xam phem.

Hais tias yog dab tsi ib Darboux:

s = Σf (x) Δ - ib tug me me npaum li cas;

S = Σf (x + Δ) Δ - loj npaum li cas.

Yog li ntawd, dab tsi yog qhov tseem ceeb heev? Cheeb Tsam bounded los ntawm ib tug kab muaj nuj nqi thiab lub ntsiab ntawm ib thaj tsam yuav ua ncaj rau:

∫f (x) dx = {(S + s) / 2} + c

Hais tias yog, tus zauv phem ntawm kev loj thiab me nqi Darbu.s - qhov nqi, resettable raws li qhov ntau yam zoo.

Raws li cov duab qhia txog cov tswvyim no, nws yuav luaj lub cev lub ntsiab lus ntawm lub ib. Square nrhiav, uas tau sau tseg muaj nuj nqi ntawm kev ceev, thiab cov tsawg lub sij hawm luv nyob rau hauv lub x-axis yuav qhov ntev ntawm qhov kev ncua deb mus.

L = ∫f (x) dx nyob rau hauv lub luv los ntawm t1 rau t2,

qhov twg

f (x) - ib tug muaj nuj nqi ntawm kev ceev, uas yog lub mis los ntawm cov uas nws hloov lub sij hawm;

L - ntev ntawm txoj kev;

t1 - lub sij hawm pib ntawm txoj kev;

t2 - lub sij hawm ntawm tiav txoj kev.

Raws nraim lub ntsiab cai tib yam yog txiav txim tau los ntawm tus nqi ntawm kev ua hauj lwm, tab sis yuav tsum tso rau hauv lub abscissa qhov kev ncua deb thiab lub ordinate - tus nqi ntawm kev quab yuam sib sib zog tsa rau txhua tus neeg taw tes.