Kuv xav tias peb yuav tsum pib nrog keeb kwm ntawm cov cuab yeej ua lej zoo li qhov sib npaug sib txawv. Zoo li txhua qhov sib txawv thiab cov lej suav, cov kab zauv no tau tsim los ntawm Newton thaum kawg ntawm lub xyoo pua 17th. Nws suav hais tias qhov kev tshawb pom no yog qhov tseem ceeb heev uas nws tseem tau nkag mus rau cov lus, uas niaj hnub no tuaj yeem txhais tau qee yam zoo li no: "Txhua txoj cai ntawm xwm tau piav qhia los ntawm kev sib npaug sib txawv." Qhov no tej zaum yuav zoo li exaggeration, tab sis nws muaj tseeb. Txhua txoj cai ntawm physics, chemistry, biology tuaj yeem piav qhia los ntawm cov kab zauv no.
Cov kws lej Euler thiab Lagrange tau txais txiaj ntsig zoo rau kev txhim kho thiab tsim cov kev xav ntawm kev sib npaug sib txawv. Twb tau nyob rau hauv lub xyoo pua 18th, lawv tau tshawb pom thiab tsim qhov lawv tab tom kawm hauv cov chav kawm qib siab hauv tsev kawm qib siab.
Ib qho tseem ceeb tshiab hauv kev kawm txog kev sib npaug sib txawv pib ua tsaug rau Henri Poincare. Nws tsim ib tug "zoo txoj kev xav ntawm qhov sib npaug sib npaug", uas, nyob rau hauv ua ke nrog nrog rau cov kev xav ntawm lub functions ntawm ib tug complex kuj sib txawv, ua ib tug tseem ceeb pab rau lub hauv paus ntawm topology - science ntawm qhov chaw thiab nws.khoom.
Dab tsi yog qhov sib npaug sib txawv?
Ntau tus neeg ntshai ib kab lus "kev sib npaug sib txawv". Txawm li cas los xij, hauv tsab xov xwm no peb yuav nthuav dav tag nrho cov ntsiab lus ntawm cov cuab yeej siv lej no, uas yog qhov tseeb tsis nyuaj li nws zoo li los ntawm lub npe. Txhawm rau pib tham txog thawj qhov sib npaug sib npaug, koj yuav tsum xub paub txog cov ntsiab lus tseem ceeb uas muaj feem cuam tshuam nrog cov ntsiab lus no. Thiab peb mam li pib nrog qhov sib txawv.
Differential
Ntau tus paub lub tswvyim no los ntawm tsev kawm ntawv. Txawm li cas los xij, cia peb ua tib zoo saib nws. Xav txog ib daim duab ntawm ib txoj haujlwm. Peb tuaj yeem nce nws mus rau qhov uas ib qho ntawm nws cov ntu yuav coj mus rau hauv txoj kab ncaj nraim. Ntawm nws peb coj ob lub ntsiab lus uas yog infinitely ze rau ib leeg. Qhov sib txawv ntawm lawv cov kev tswj hwm (x lossis y) yuav yog tus nqi infinitesimal. Nws yog hu ua qhov sib txawv thiab qhia los ntawm cov cim dy (qhov txawv ntawm y) thiab dx (qhov txawv ntawm x). Nws yog ib qho tseem ceeb kom nkag siab tias qhov sib txawv tsis yog tus nqi kawg, thiab qhov no yog nws lub ntsiab lus thiab lub luag haujlwm tseem ceeb.
Thiab tam sim no peb yuav tsum xav txog lub ntsiab lus tom ntej, uas yuav muaj txiaj ntsig rau peb hauv kev piav qhia lub tswv yim ntawm kev sib npaug sib txawv. Nov yog derivative.
Derivative
Peb txhua tus zaum hnov hauv tsev kawm ntawv thiab lub tswv yim no. Tus derivative tau hais tias yog tus nqi ntawm kev loj hlob lossis txo qis ntawm kev ua haujlwm. Txawm li cas los xij, los ntawm cov ntsiab lus nontau dhau los ua tsis meej. Cia peb sim piav qhia qhov sib txawv ntawm qhov sib txawv. Cia peb rov qab mus rau ib ntu infinitesimal ntawm kev ua haujlwm nrog ob lub ntsiab lus uas nyob ntawm qhov tsawg kawg nkaus ntawm ib leeg. Tab sis txawm rau qhov kev ncua deb no, lub luag haujlwm tswj kev hloov pauv los ntawm qee tus nqi. Thiab los piav txog qhov kev hloov pauv no, lawv tuaj yeem muaj qhov sib txawv, uas tuaj yeem sau ua qhov sib txawv ntawm qhov sib txawv: f(x)'=df/dx.
Tam sim no nws tsim nyog xav txog cov khoom tseem ceeb ntawm cov khoom siv derivative. Tsuas muaj peb leeg xwb:
- Tus derivative ntawm cov lej lossis qhov sib txawv tuaj yeem sawv cev raws li cov lej lossis qhov sib txawv ntawm cov txiaj ntsig: (a + b)'=a' + b' thiab (a-b)'=a'-b'.
- Cov cuab yeej thib ob yog cuam tshuam nrog kev sib npaug. Lub derivative ntawm ib yam khoom yog cov sum ntawm cov khoom ntawm ib tug muaj nuj nqi thiab lub derivative ntawm lwm tus: (ab)'=a'b+ab'.
- Qhov sib txawv ntawm qhov sib txawv tuaj yeem sau tau raws li qhov sib npaug hauv qab no: (a/b)'=(a'b-ab')/b2.
Txhua yam khoom no yuav muaj txiaj ntsig rau kev nrhiav kev daws teeb meem rau thawj qhov sib npaug sib npaug.
Tseem muaj ib feem derivatives. Cia peb hais tias peb muaj ib txoj haujlwm z uas nyob ntawm qhov sib txawv x thiab y. Txhawm rau xam cov txiaj ntsig ib nrab ntawm qhov kev ua haujlwm no, hais tias, nrog rau x, peb yuav tsum coj qhov sib txawv y ua qhov tsis tu ncua thiab tsuas yog sib txawv.
Integral
Lwm lub tswv yim tseem ceeb yog qhov tseem ceeb. Qhov tseeb, qhov no yog qhov ncaj qha opposite ntawm qhov derivative. Muaj ntau ntau hom kev sib xyaw ua ke, tab sis txhawm rau daws qhov sib npaug sib txawv yooj yim, peb xav tau qhov tsis tseem ceeb tshaj plaws uas tsis muaj qhov sib txawv.
Yog li qhov tseem ceeb yog dab tsi? Cia peb hais tias peb muaj qee qhov kev vam flos ntawm x. Peb muab qhov tseem ceeb los ntawm nws thiab tau txais cov haujlwm F (x) (feem ntau hu ua antiderivative), qhov sib npaug ntawm qhov sib npaug ntawm qhov tseem ceeb. Yog li F(x)'=f(x). Nws kuj ua raws los ntawm qhov no tias qhov sib piv ntawm qhov sib npaug yog sib npaug rau qhov kev ua haujlwm qub.
Thaum daws qhov sib npaug sib txawv, nws tseem ceeb heev kom nkag siab lub ntsiab lus thiab kev ua haujlwm ntawm qhov tseem ceeb, vim koj yuav tau coj lawv ntau zaus los nrhiav kev daws teeb meem.
Qhov sib npaug sib txawv nyob ntawm lawv qhov xwm txheej. Hauv seem tom ntej, peb yuav txiav txim siab txog hom kev sib npaug ntawm thawj qhov sib npaug, thiab tom qab ntawd kawm yuav ua li cas daws lawv.
chav sib txawv sib npaug
"Diffury" muab faib raws li qhov kev txiav txim ntawm cov derivatives koom nrog hauv lawv. Yog li, muaj thawj, thib ob, thib peb thiab ntau qhov kev txiav txim. Lawv kuj tuaj yeem muab faib ua ob peb chav kawm: zoo tib yam thiab ib nrab derivatives.
Nyob hauv kab lus no peb yuav txiav txim siab qhov sib txawv sib txawv ntawm thawj qhov kev txiav txim. Peb kuj tseem yuav tham txog cov piv txwv thiab txoj hauv kev los daws lawv hauv ntu hauv qab no. Peb yuav txiav txim siab tsuas yog ODEs xwb, vim tias cov no yog hom sib npaug tshaj plaws. Cov txheej txheem tau muab faib ua subspecies: nrog sib cais sib txawv, homogeneous thiab heterogeneous. Tom ntej no, koj yuav kawm seb lawv txawv li cas ntawm ib leeg, thiab kawm yuav ua li cas daws lawv.
Ntxiv rau, cov kev sib npaug no tuaj yeem ua ke, yog li tom qab peb tau txais cov kab ke sib txawv ntawm qhov kev txiav txim thawj zaug. Peb kuj tseem yuav txiav txim siab txog cov kab ke no thiab kawm paub daws lawv li cas.
Vim li cas peb thiaj txiav txim siab tsuas yog thawj qhov kev txiav txim? Vim tias koj yuav tsum pib nrog ib qho yooj yim, thiab piav qhia txhua yam ntsig txog qhov sib txawvKev sib npaug, hauv ib kab lus yooj yim ua tsis tau.
sib piv sib txawv sib txawv
Cov no yog kab tias qhov yooj yim tshaj plaws thawj-qib sib npaug sib npaug. Cov no suav nrog cov piv txwv uas tuaj yeem sau tau zoo li no: y'=f(x)f(y). Yuav kom daws tau qhov kev sib npaug no, peb xav tau ib qho qauv los sawv cev rau qhov sib piv ntawm qhov sib txawv: y'=dy / dx. Siv nws, peb tau txais cov kab zauv hauv qab no: dy/dx=f(x)f(y). Tam sim no peb tuaj yeem tig mus rau txoj kev daws cov qauv piv txwv: peb yuav faib cov kev hloov pauv mus rau hauv ntu, piv txwv li peb yuav hloov txhua yam nrog qhov sib txawv ntawm y mus rau qhov chaw uas dy nyob, thiab peb yuav ua tib yam nrog rau qhov sib txawv x. Peb tau txais ib qho kev sib npaug ntawm daim ntawv: dy / f (y)=f (x) dx, uas yog daws los ntawm kev noj cov khoom ntawm ob qho tib si. Tsis txhob hnov qab txog qhov tas li uas yuav tsum tau teeb tsa tom qab noj qhov tseem ceeb.
Txoj kev daws teeb meem rau ib qho "diffurance" yog kev ua haujlwm ntawm kev vam khom ntawm x ntawm y (hauv peb rooj plaub) lossis, yog tias muaj tus lej, ces cov lus teb yog nyob rau hauv daim ntawv ntawm tus lej. Cia peb txheeb xyuas tag nrho cov kev daws teeb meem uas siv cov piv txwv tshwj xeeb:
y'=2ysin(x)
Tsiv hloov pauv mus rau qhov sib txawv:
dy/y=2sin(x)dx
Tam sim no peb muab cov ntsiab lus. Tag nrho cov ntawm lawv tuaj yeem pom nyob rau hauv ib lub rooj tshwj xeeb ntawm kev sib xyaw. Thiab peb tau txais:
ln(y)=-2cos(x) + C
Yog xav tau, peb tuaj yeem qhia "y" ua haujlwm ntawm "x". Tam sim no peb tuaj yeem hais tias peb qhov sib npaug sib txawv raug daws yog tias tsis muaj qhov xwm txheej tau muab. Ib qho xwm txheej tuaj yeem muab tau, piv txwv li, y(n/2)=e. Tom qab ntawd peb tsuas yog hloov tus nqi ntawm cov kev hloov pauv no rau hauv kev daws teeb meem thiabnrhiav tus nqi ntawm qhov tsis tu ncua. Hauv peb qhov piv txwv, nws yog sib npaug rau 1.
Thawj-txiav homogeneous sib npaug sib npaug
Tam sim no mus rau qhov nyuaj dua. Homogeneous sib npaug sib npaug ntawm thawj qhov kev txiav txim tuaj yeem sau rau hauv daim ntawv hauv qab no: y'=z(x, y). Nws yuav tsum raug sau tseg tias txoj haujlwm zoo ntawm ob qhov sib txawv yog homogeneous, thiab nws tsis tuaj yeem muab faib ua ob qhov kev vam meej: z ntawm x thiab z ntawm y. Tshawb xyuas seb qhov sib npaug yog homogeneous lossis tsis yog qhov yooj yim heev: peb ua qhov hloov pauv x=kx thiab y=ky. Tam sim no peb tshem tag nrho k. Yog tias tag nrho cov ntawv no raug txo, ces qhov sib npaug yog homogeneous thiab koj tuaj yeem ruaj ntseg mus daws nws. Saib tom ntej, cia peb hais: lub hauv paus ntsiab lus ntawm kev daws cov qauv no kuj yooj yim heev.
Peb yuav tsum hloov pauv: y=t(x)x, qhov twg t yog qee qhov haujlwm uas tseem nyob ntawm x. Tom qab ntawd peb tuaj yeem nthuav qhia cov txiaj ntsig: y'=t'(x)x + t. Hloov tag nrho cov no rau hauv peb qhov kev sib npaug qub thiab ua kom yooj yim rau nws, peb tau txais ib qho piv txwv nrog cov sib txawv sib txawv t thiab x. Peb daws nws thiab tau txais qhov dependence t(x). Thaum peb tau txais nws, peb tsuas hloov y=t (x)x rau hauv peb qhov kev hloov dhau los. Tom qab ntawd peb tau txais kev vam khom ntawm y ntawm x.
Yuav kom meej meej, cia peb saib ib qho piv txwv: xy'=y-xey/x.
Thaum kuaj nrog hloov, txhua yam raug txo. Yog li qhov sib npaug tiag tiag homogeneous. Tam sim no peb hloov lwm qhov uas peb tau tham txog: y=t(x)x thiab y'=t'(x)x + t (x). Tom qab ua kom yooj yim, peb tau txais cov kab zauv hauv qab no: t'(x)x=-et. Peb daws qhov ua piv txwv uas tau muab cais tawm thiab tau txais: e-t=ln(Cx). Peb tsuas yog yuav tsum hloov t nrog y / x (tom qab tag nrho, yog tias y=tx, ces t=y / x), thiab peb tau txaisteb: e-y/x=ln(xC).
Thawj Order Linear Differential Equations
Nws yog lub sijhawm rau lwm lub ntsiab lus loj. Peb yuav txheeb xyuas qhov sib npaug sib npaug sib npaug ntawm thawj qhov kev txiav txim. Lawv txawv li cas ntawm ob qho dhau los? Cia peb xav txog nws. Cov kab zauv sib txawv ntawm thawj qhov kev txiav txim hauv daim ntawv dav dav tuaj yeem sau tau raws li hauv qab no: y' + g(x)y=z(x). Nws tsim nyog qhia meej tias z(x) thiab g(x) tuaj yeem yog qhov qub.
Thiab tam sim no ib qho piv txwv: y' - yx=x2.
Yuav muaj ob txoj hauv kev los daws nws, thiab peb yuav daws nrog ob qho tib si. Thawj qhov yog tus qauv ntawm kev hloov pauv ntawm qhov tsis sib xws.
Yuav kom daws tau qhov sib npaug ntawm txoj kev no, koj yuav tsum ua ntej sib npaug ntawm sab xis mus rau xoom thiab daws qhov kev sib npaug tshwm sim, uas tom qab txav cov khoom yuav siv daim ntawv:
y'=yx;
dy/dx=yx;
dy/y=xdx;
ln|y|=x2/2 + C;
y=ex2/2yC=C1e x2/2.
Tam sim no peb yuav tsum hloov qhov tas mus li C1 nrog cov haujlwm v(x) uas peb yuav tsum nrhiav.
y=vex2/2.
Cia peb hloov pauv qhov derivative:
y'=v'ex2/2-xvex2/2.
Thiab hloov cov kab lus no rau hauv qhov sib npaug qub:
v'ex2/2 - xvex2/2 + xvex2 /2=x2.
Koj tuaj yeem pom tias ob nqe lus tso tseg ntawm sab laug. Yog hais tias nyob rau hauv ib co piv txwv qhov no tsis tshwm sim, ces koj ua ib yam dab tsi tsis ncaj ncees lawm. Txuas ntxiv:
v'ex2/2 =x2.
Tam sim no peb daws qhov kev sib npaug li niaj zaus uas peb yuav tsum tau cais cov sib txawv:
dv/dx=x2/ex2/2;
dv=x2e-x2/2dx.
Txhawm rau rho tawm qhov tseem ceeb, peb yuav tsum siv kev koom ua ke los ntawm qhov chaw ntawm no. Txawm li cas los xij, qhov no tsis yog lub ntsiab lus ntawm peb tsab xov xwm. Yog tias koj txaus siab, koj tuaj yeem kawm paub ua li cas koj tus kheej. Nws tsis yog qhov nyuaj, thiab nrog kev txawj txaus thiab kev saib xyuas tsis siv sijhawm ntau.
Cia peb tig mus rau txoj kev thib ob ntawm kev daws qhov sib npaug sib npaug: Bernoulli txoj kev. Txoj kev twg sai dua thiab yooj yim dua yog nyob ntawm koj.
Yog li, thaum daws qhov sib npaug los ntawm txoj kev no, peb yuav tsum hloov pauv: y=kn. Ntawm no k thiab n yog ib co x-dependent functions. Ces tus derivative yuav zoo li no: y'=k'n+kn'. Hloov ob qho kev hloov pauv rau hauv kab zauv:
k'n+kn'+xkn=x2.
Group:
k'n+k(n'+xn)=x2
Tam sim no peb yuav tsum sib npaug rau xoom dab tsi hauv cov kab ke. Tam sim no, yog tias koj muab ob qhov sib npaug sib npaug, koj tau txais qhov system ntawm thawj qhov sib txawv sib npaug uas koj yuav tsum tau daws:
n'+xn=0;
k'n=x2.
thawj qhov sib npaug yog daws zoo li qhov sib npaug ib txwm. Txhawm rau ua qhov no, koj yuav tsum cais cov hloov pauv:
dn/dx=xv;
dn/n=xdx.
Tau qhov sib xyaw thiab tau txais: ln(n)=x2/2. Tom qab ntawd, yog tias peb nthuav qhia n:
n=ex2/2.
Tam sim no peb hloov qhov kev sib txig sib luag rau hauv qhov sib npaug thib ob ntawm qhov system:
k'ex2/2=x2.
Thiab hloov pauv, peb tau txais kev sib npaug tib yam li hauv thawj txoj kev:
dk=x2/ex2/2.
Peb yuav tsis mus rau theem ntxiv. Nws yog tsim nyog hais tias thaum xub thawj qhov kev daws teeb meem ntawm thawj qhov sib txawv sib npaug ua rau muaj teeb meem loj. Txawm li cas los xij, thaum koj nkag mus tob rau hauv lub ncauj lus, nws pib zoo dua thiab zoo dua.
qhov sib npaug sib txawv siv nyob qhov twg?
Differential equations yog siv heev nyob rau hauv physics, vim hais tias yuav luag tag nrho cov kev cai yooj yim yog sau nyob rau hauv txawv daim ntawv, thiab cov qauv uas peb pom yog cov kev daws ntawm cov kab zauv. Hauv chemistry, lawv tau siv tib yam: cov kev cai lij choj tau muab los ntawm lawv. Hauv biology, kev sib npaug sib txawv yog siv los ua qauv ntawm tus cwj pwm ntawm cov kab ke, xws li predator-prey. Lawv kuj tuaj yeem siv los tsim cov qauv tsim tawm ntawm, hais, ib pawg ntawm cov kab mob.
kev sib npaug sib txawv yuav pab tau li cas hauv lub neej?
Cov lus teb rau lo lus nug no yooj yim: tsis muaj txoj hauv kev. Yog tias koj tsis yog kws tshawb fawb lossis engineer, ces lawv tsis zoo li yuav muaj txiaj ntsig rau koj. Txawm li cas los xij, rau kev txhim kho dav dav, nws tsis mob kom paub tias qhov sib npaug sib txawv yog dab tsi thiab yuav daws tau li cas. Thiab tom qab ntawd lo lus nug ntawm tus tub lossis tus ntxhais "dab tsi yog qhov sib npaug sib txawv?" yuav tsis confuse koj. Zoo, yog tias koj yog tus kws tshawb fawb lossis tus kws tshaj lij, ces koj tus kheej nkag siab qhov tseem ceeb ntawm cov ncauj lus no hauv txhua qhov kev tshawb fawb. Tab sis qhov tseem ceeb tshaj plaws yog tias tam sim no lo lus nug "yuav daws tau qhov sib npaug ntawm thawj qhov sib txawv li cas?" koj yeej teb tau. Pom zoo, nws ib txwm zoothaum koj nkag siab dab tsi tib neeg txawm ntshai to taub.
Main learning problems
Qhov teeb meem tseem ceeb hauv kev nkag siab lub ntsiab lus no yog qhov tsis zoo ntawm kev sib koom ua ke thiab kev ua haujlwm sib txawv. Yog tias koj tsis zoo ntawm kev noj cov derivatives thiab integrals, ces koj yuav tsum tau kawm ntau ntxiv, paub ntau txoj hauv kev ntawm kev sib koom ua ke thiab kev sib txawv, thiab tsuas yog tom qab ntawd pib kawm cov khoom uas tau piav qhia hauv kab lus.
Qee tus neeg xav tsis thoob thaum lawv pom tias dx tuaj yeem hloov pauv tau, vim tias ua ntej (hauv tsev kawm ntawv) tau hais tias feem dy / dx tsis pom. Ntawm no koj yuav tsum tau nyeem cov ntaub ntawv ntawm cov derivative thiab to taub tias nws yog qhov piv ntawm infinitesimal quantities uas yuav manipulated thaum daws qhov sib npaug.
Ntau tus tsis paub tam sim ntawd qhov kev daws teeb meem ntawm thawj qhov sib txawv sib npaug feem ntau yog qhov ua haujlwm lossis ib qho tseem ceeb uas tsis tuaj yeem coj mus, thiab qhov kev xav tsis zoo no ua rau lawv muaj teeb meem ntau.
Dab tsi ntxiv tuaj yeem kawm kom nkag siab zoo dua?
Nws yog qhov zoo tshaj los pib nkag mus rau hauv lub ntiaj teb ntawm kev sib txawv ntawm kev suav nrog cov phau ntawv tshwj xeeb, piv txwv li, hauv kev suav lej rau cov tub ntxhais kawm tsis yog lej tshwj xeeb. Tom qab ntawd koj tuaj yeem txav mus rau cov ntaub ntawv tshwj xeeb ntxiv.
Nws yuav tsum tau hais tias, ntxiv rau qhov sib npaug sib txawv, kuj tseem muaj qhov sib npaug sib npaug, yog li koj yuav muaj ib yam dab tsi los siv zog thiab ib yam dab tsi los kawm.
Zoo kawg
Peb cia siab tias tom qab nyeem ntawvKab lus no tau muab tswv yim rau koj txog qhov sib npaug sib txawv thiab yuav daws tau li cas kom raug.
Txawm li cas los xij, lej yuav muaj txiaj ntsig zoo rau peb hauv lub neej. Nws txhim kho kev xav thiab kev xav, yam tsis muaj txhua tus neeg zoo li tsis muaj tes.
