Uchungechunge lweFourier: imbali kunye nempembelelo yendlela yokwenza imathematika ekuphuhliseni isayensi

Uchungechunge lweFourier lubonakaliso lomsebenzi obanjwe ngumbandela ngexesha elithile kwindlela yokwahlula. Ngokuqhelekileyo, esi sicombululo kuthiwa ukwandiswa kwento elandelwayo. Ukwandiswa kwemisebenzi kwinqaku le-Fourier kuyisisombululo esinamandla esinokusombulula iingxaki ezahlukahlukeneyo ngenxa yeepropati zenguqu enikeziweyo ekuhlanganiseni, ukwahlukana, kunye nokutshintshwa kwegama lokuxabana kunye ne-convolution.

Umntu ongaqhelekanga kwimathematika ephezulu, kunye nemisebenzi yenzululwazi yaseFransi, uFourier, mhlawumbi akayi kuqonda ukuba luphi uhlobo "lwamanani" kunye nezinto ezifunekayo. Nangona kunjalo le nguqulelo iyanzima kakhulu ebomini bethu. Asetshenziswanga kuphela ngababalo beemathematika, kodwa kunye namafizikiki, iikhemistri, oogqirha, izazi zeenkwenkwezi, i-seismologists, i-oceanographers nabanye abaninzi. Masiqhelane nemisebenzi yesazinzulu esingumFrentshi owenza ukufunyanwa ngaphambili kwexesha lakhe.

Ukuguqulwa komntu kunye neFourier

Iingqungquthela zeFourier enye yeendlela (kunye nokuhlalutya nabanye) kwenguqu yeFourier. Le nqubo ivela rhoqo xa umntu eva isandi. Indlebe yethu iguqula ngokuzenzekelayo isangqa somsindo ibe yindlela yokuzenzekelayo . Iimpawu ezinqabileyo zeengqungquthela zeplastiki kwi-elastic medium zihlukaniswa zibe ziqendu (ngokuhambelana nombakala) weemilinganiselo ezilandelelanayo zesantya sokuphakama kweetoni eziphakamileyo ezahlukeneyo. Ngaphezu koko, ingqondo ijika le data ibe izandi esaziwayo kuthi. Konke okwenzekayo kwongezelela umnqweno wethu okanye ukuqonda, ngaphakathi nangokwakho, kodwa ukuze kuqondwe le nkqubo, kuya kuthatha iminyaka emininzi ukufunda iimathematika eziphezulu.

Okunye malunga nokuguqulwa kweFourier

Utshintsho lweFourier lungenziwa ngokuhlalutya, ukubala kunye nezinye iindlela. Uchungechunge lwe-Fourier lubhekisela kwinkqubo yokwahlukana kwanoma iyiphi na inkqubo ephazamisayo - ukusuka kumanxweme olwandle kunye namaza okukhanya ukuya kwelanga (kunye nezinye izinto zeenkwenkwezi) imijikelezo yomsebenzi. Ukusebenzisa olu buchule lweemathematika, unokwazi ukuxubusha imisebenzi, emele nayiphi na inkqubo ye-oscillatory njengengqungquthela yamacandelo e-sinusoidal, ahamba ukusuka kumncinci ukuya kumphezulu kunye nangemva. Uguquko lwe-Fourier ngumsebenzi uchaza isigaba kunye nobukhulu be-sinusoids ezihambelanayo nesantya esithile. Le nkqubo ingasetyenziselwa ukuxazulula ukulingana okunzima kakhulu okuchaza iinkqubo ezinamandla ezenzeka phantsi kwezenzo zamandla ashushu, ombane okanye ombane. Iingqungquthela zeFourier zenza ukuba kube lula ukuhlukanisa iziganeko eziqhubekayo kwizibonakaliso ezixakekayo, ezenza kube lula ukutolika ngokufanelekileyo ukujonga okufunyenweyo kumachiza, kumakhemikhali nakwi-astronomy.

Umlando

Ubaba oyisekayo wale ngqungquthela ngumbalo weFrench uJean Baptiste Joseph Fourier. Igama lakhe labizwa ngokuba ngulo nguqulelo. Ekuqaleni, usosayensi wasebenzisa indlela yakhe yokufunda nokucacisa iindlela zokuqhuba i-thermal - ukusasazwa kobushushu kwizinto ezinamandla. UFourier wacebisa ukuba ukusabalalisa kokuqala okungaqhelekiyo komswakama obushushu kungabhidlika kwiisono ze-sinusoids ezilula, nganye zazo ziza kuba neqondo lokushisa elincinci kunye nesigaba sayo. Icandelo ngalinye liza kubalwa ukusuka kumncinci ukuya kummandla ophezulu kunye nakwezinye. Umsebenzi weemathematika ochaza iingcipheko eziphezulu kunye nezingezantsi zekhalo, kunye nesigaba ngasinye se-harmonics, ibizwa ngokuba nguguqulelo lweFourier lwamazwi okuhambisa ukushisa. Umlobi wenkolelo uye wanciphisa umsebenzi wokusabalalisa jikelele, obunzima ukuchazwa ngokwemathematika, kwinqanaba leemisebenzi ze- cosine kunye nemisebenzi yeempawu ezilungele ukusingatha, ngesibalo esinikeza ukuhanjiswa kokuqala.

Umgaqo wokuguqulwa kunye nembono yabantu

Abahlalutyo bezenzululwazi-abaphathi beemathematika ekuqaleni kwekhulu leshumi elinesithoba-abazange bamkele le ngqungquthela. Isichaso esona sikhulu sisigxina sokuthi u-Fourier uthi umsebenzi oyekayo uchaza umgca ochanekileyo okanye i-curve yokubamba ingamelwa njengesixa semibono ye-sinusoidal eqhubekayo. Njengomzekelo, sinokuqwalasela "isinyathelo" sikaHeiside: ixabiso layo liyi-zero ngakwesobunxele bexesha lokuyeka kwaye enye ngakwesokudla. Lo msebenzi uchaza ukuxhomekeka kwendawo yamandla ngexesha lokutshintsha xa isekethe ivaliwe. Abahlalutyi beengcamango ngelo xesha abazange bahlangabezane neemeko ezifanayo, xa inqaku eliyekayo liza kuchazwa ngokuhlanganiswa kwemisebenzi eqhelekileyo, efana nokubonisa, i-sinusoid, i-linear okanye i-quadratic.

Yintoni eyayihlazisa ama-mathematizi aseFransi kwimfundiso kaFourier?

Emva koko, ukuba i-mathematici ilungile kwiingxelo zakhe, ngoko, ukuququzelela i-series ye-Fourier engapheliyo, inokufumana ukubonakaliswa okuchanekileyo kwintetho yesinyathelo nangona kunamanyathelo amaninzi anjalo. Ekuqaleni kwekhulu leshumi elinesithoba, loo mazwi yayibonakala ingenangqondo. Kodwa nangona kukho ukungathandabuzeki, abaninzi beemathematika baye bawandisa ubuninzi bokufunda le nkalo, bayithatha ngaphaya kwemiqathango yophando malunga nokuqhuba ukushisa. Nangona kunjalo, ezininzi iinzululwazi zaqhubeka zibuza umbuzo: "Ingaba isahluko se-sinusoidal series singaguqukela kwixabiso elichanekileyo lomsebenzi oyekayo?"

Ukuguquguquka kweeklasi zeFourier: umzekelo

Umbuzo wokuguqulwa kuphakanyiswa rhoqo xa kudingekile ukongeza inani elingenamlinganiselo lamanani. Ukuze siqonde le nto, makhe siqwalasele umzekelo weklastiki. Ngaba unako ukufikelela eludongeni ukuba isinyathelo ngasinye esilandelayo sisisiqingatha sangaphambili? Cinga ukuba unamamitha amabini ukusuka kumgomo, isinyathelo sokuqala sikuzisa kumakishi kwinqanaba elinesibini, elilandelayo - kumakishi amathathu, kwaye emva kwesihlanu uya kunqoba malunga neepesenti ezingama-97 zendlela. Nangona kunjalo, kungakhathaliseki ukuba uthatha amanyathelo amaninzi, awuyi kufezekisa injongo ekujoliswe kuyo kwisimo esiqinileyo semathematika. Ukusebenzisa izibalo zamanani, kunokubonakaliswa ukuba ekugqibeleni kunokwenzeka ukuba ufike kumgama omncinci omiselwe ngaphambili. Olu bungqina lulingana nokubonisa ukuba ixabiso elipheleleyo lekota yesibini, enye yesine, njl, iya kubambisana.

Umbuzo wokuguquguquka: ukuza okwesibini, okanye iComputer yeNkosi kaKelvin

Ngokuphindaphindiweyo lo mbuzo wanyuswa ekupheleni kwekhulu leshumi elinesithoba, xa iinqununu zeFourier zazama ukufaka ubungqina bemigangatho yamanzi kunye namanxuwa ukuba aqikelele. Ngeli xesha, iNkosi uKelvin yaqulunqa isixhobo esiyinkqubo yokusebenzisa i-analog evumela ukuba oomatshini bemikhosi yemikhosi kunye nabarhwebi balandele le nto yendalo. Le ndlela izimisele iiseti zezigaba kunye nama-amplitudes ukusuka kwitheyibhile yokuphakama kwamatye kunye neengongoma ezifanelekileyo ngokulinganiswayo kwi-harbor kunyaka. Ipharamitha nganye yayiyinxalenye yes sinidyidal yegama lokuphakama kwe-tide kwaye yayingenye yezinto eziqhelekileyo. Iziphumo zeemilinganiselo zangeniswa kwisixhobo sokubala saseKelvin, ukuhlanganiswa kwekhava eqikelele ukuphakama kwamanzi njengomsebenzi wesikhashana kunyaka ozayo. Ngokukhawuleza loo mizuzu yahlanganiswa kuwo onke amachweba ehlabathi.

Yaye ukuba inkqubo iyaphulwa ngumsebenzi onqanyulwayo?

Ngelo xesha kwabonakala kubonakala ukuba isicatshulwa sokuchazela umtsalane ogqithisiweyo kunye nenani elikhulu lokubala izinto lingabalwa inani elikhulu leenxalenye kunye nama-amplitudes kwaye ke unikezele ngezibikezelo ezichanekileyo. Nangona kunjalo, kwavela ukuba le mihla ayitholakali kwiimeko xa i-expressional expression, efuneka ukuba ihlanganiswe, iqulethe ukuqaqa okubukhali, oko kukuthi, kwakunyanqanyulwa. Kwimeko apho ifowuni ingena kwi data ukusuka kwitheyibhile yexesha, libala amanqaku angama-coofficients ambalwa. Umsebenzi oyimvelaphi ubuyiselwa ngenxa yamacandelo e-sinusoidal (ngokuhambelana neentlawulo ezifunyenweyo). Ukwahlukana phakathi kwegama lokuqala kunye nokubuyiswa kunokulinganiswa nakweliphi na indawo. Xa senza izibalo ngokuphindaphindiweyo kunye nokuthelekiswa, kuyacaca ukuba ixabiso lephulo elikhulu liyancishiswa. Nangona kunjalo, bahlala kwindawo ehambelana nenqaku lokuyeka, kwaye nakweyiphi enye into abayithandayo. Ngomnyaka we-1899, esi siphumo sasiqinisekisiwe ngoYoshuwa Willard Gibbs waseYale University.

Ukuguqulwa kweeklasi zeFourier kunye nophuhliso lweemathematika ngokubanzi

Uhlalutyo lweFourier alusebenzi kumazwi aqukethe inani elingapheliyo lokugqitywa kwexesha elithile. Ngokubanzi, uchungechunge lweFourier, ukuba umsebenzi oyimvelaphi uboniswa ngumphumo wesimo sangempela senyama, uhlala uguqulela. Imibuzo yokuguqulwa kwale nkqubo yeeklasi ezithile zemisebenzi yabangela ukubonakala kwamacandelo amatsha kwimathematika, umzekelo, inkolelo yemisebenzi jikelele. Inxulumene namagama anjengoL. Schwartz, J. Mikusinsky kunye neJ. Temple. Esi sikhokelo sale ngqungquthela, isakhelo esicacileyo nesicacileyo sakhiwe malunga namazwi afana nomsebenzi weDirac delta (ichaza indawo yendawo enye ejoliswe kwindawo engapheliyo kwinqanaba) kunye ne "step" yeHeaviside. Ngenxa yolu msebenzi, iinqununu zeFourier zisebenzise ukuxazulula ukulingana kunye neengxaki apho kuvela khona imiqondo enembileyo: i-point point, i-point mass, i-dipol magnetic, kwaye igxininisa umthwalo kwi-boram.

Indlela yeFourier

Iingqungquthela zeFourier, ngokuhambelana nemigaqo yokuphazamiseka, qalisa ngokuqhekeka kweefom eziyinkimbinkimbi zibe lula. Ngokomzekelo, utshintsho ekuphumeni kokushisa lubangelwa ukuhamba kwimizila ngemiqobo eyahlukahlukeneyo evela kummandla wokungafaki ukushisa kombane okanye ngokutshintsha umhlaba-inyikima, utshintsho kwindlovu yomzimba wezulu, ngenxa yeeplanethi. Njengomthetho, ukulingana okufanayo okuchaza iinkqubo eziqhelekileyo zeklastiki zisesisombululo esisisiseko somntu ngamnye. UFourier wabonisa ukuba izisombululo ezilula zinokuthi zifakwe kwisisombululo ukuxazulula iingxaki ezinzima. Ukubonakaliswa ngolwimi lweemathematika, uchungechunge lweFourier yindlela yokubonisa intetho ngokubakho kwe-harmonics-cosine ne-sinusoid. Ngoko ke, olu hlalutyo luyaziwa nangokuthi "ukuhlalutya kwe-harmonic".

Uchungechunge lweFourier luyindlela efanelekileyo phambi kokuba "iikhompyutheni yobudala"

Ngaphambi kokudala iteknoloji yekhompyutheni, indlela yeFourier yayiyona sixhobo esona sikhulu kwi-arsenal yezenzululwazi xa sisebenzisana nomoya onyakazayo wehlabathi lethu. Uchungechunge lweFourier ngendlela edibeneyo lisivumela ukuba sisombulule iingxaki nje ezilula ezinokuthi zisebenzise ngokuthe ngqo kwimithetho ye-Newtonian mechanics, kodwa kunye nokulingana okubalulekileyo. Uninzi lweziphumo zenzululwazi ye-Newtonian yekhulu leshumi elinesithoba zenzeke kuphela ngenxa yendlela yokuFourier.

Uchungechunge lweFourier namhlanje

Ngokuphuhliswa kweekhomputha, iinguqu zeFourier ziye zaphakama kwinqanaba elitsha elifanelekileyo. Le ndlela isekelwe ngokubanzi kwiindawo zonke zesayensi kunye nobuchwepheshe. Umzekelo umqondiso we-digital kunye nevidiyo. Ukuqaphela kwayo kwafumaneka kuphela ngenxa yombono ophuhliswe ngumfundi wesiFrentshi ekuqaleni kwekhulu le-9. Ngaloo ndlela, uchungechunge lweFourier kwifom eyinkimbinkimbi lwenze ukuba kwenzeke ukuqhubela phambili ekufundeni kwithuba elingaphandle. Ukongezelela, oku kwachaphazela ukufundiswa kwe-physics yezinto zokufundisa nge-semiconductor kunye ne-plasma, i-microwave acoustics, i-oceanography, i-radar, i-seismology.

Uchungechunge lweTrigonometric Fourier

Kwiimathematika, uchungechunge lweFourier yindlela yokumela imisebenzi enzima yokuzibandakanya njengesixa-mali sezinto ezilula. Ngokubanzi, inani lala magama alinakuphela. Kule meko, ukuba inani labo lithathwa kwi-akhawunti ekubaleni, ngokuqinisekileyo umphumo wokugqibela ufumaneka. Ngokuqhelekileyo, i-cosgonometric cosine okanye imisebenzi ye-sinus isetyenziswe njengezona zilula. Kule meko, uchungechunge lweFourier lubizwa ngokuba yi-trigonometric, kwaye isisombululo saloo mazwi siyakwandiswa kwe-harmonic. Le ndlela idlala indima ebalulekileyo kwimathematika. Okokuqala, uchungechunge lwe-trigonometric lunikeza iindlela zesifaniso, kunye nokufunda imisebenzi, yinto efanelekileyo yezinto ezikhoyo. Ukongezelela, ivumela ukuxazulula iingxaki ezininzi ze-physics ye-mathematical. Ekugqibeleni, le ngcamango yaba negalelo ekuphuhliseni uhlalutyo lweemathematika, yavelisa ebomini inxalenye ebaluleke kakhulu yesayensi yezemathematika (inkolelo yokuhlanganiswa, imfundiso yeemisebenzi zexesha). Ukongezelela, bekusebenza njengendawo yokuqala yokuphuhliswa kweengcamango ezilandelayo : ukusetha, imisebenzi yokuguquguquka kwenene, uhlalutyo olusebenzayo, kunye nokuqalisa ukuhlalutya kwe-harmoniki.