U Vikipediyi ye statti pro inshi znachennya cogo termina Rotor znachennya Ro tor dvo chi trivimirnogo vektornogo polya v matematici vektor koordinati yakogo viznachayutsya viznachnikom tretogo poryadku pershij ryadok yakogo orti koordinatnih osej drugij operatori chastinnogo diferenciyuvannya v takomu zh poryadku yak i orti osej tretij koordinati funkciyi yaka viznachaye vektorne pole rot A i j k x y z A x A y A z A z y A y z i A x z A z x j A y x A x y k displaystyle text rot mathbf A left begin matrix mathbf i amp mathbf j amp mathbf k frac partial partial x amp frac partial partial y amp frac partial partial z A x amp A y amp A z end matrix right left frac partial A z partial y frac partial A y partial z right mathbf i left frac partial A x partial z frac partial A z partial x right mathbf j left frac partial A y partial x frac partial A x partial y right mathbf k Z praktichnoyi tochki zoru rotor vektornogo polya harakterizuye obertalnu zdatnist polya v danij tochci vona najbilsha same v ploshini perpendikulyarnij rotoru Pole dlya yakogo rotor v kozhnij tochci ye nulovim vektorom nazivayut potencialnim Zmist 1 Poznachennya 2 Priklad 3 Fizichne znachennya 4 DzherelaPoznachennya Redaguvatirot displaystyle operatorname rot nbsp vikoristovuyetsya v rosijskomovnij literaturi takozh v bagatoh krayinah Yevropi abo curl displaystyle operatorname curl nbsp v anglomovnij a takozh yak vektornij dobutok diferencialnogo operatora nabla na vektorne pole dd displaystyle mathbf nabla times nbsp Priklad RedaguvatiObertannya navkolo osi z displaystyle z nbsp zi staloyu kutovoyu shvidkistyu w displaystyle w nbsp trayektoriyi ye kolami z centrom na z displaystyle z nbsp osi v w y w x 0 displaystyle mathbf v langle wy wx 0 rangle nbsp Todi v 0 i 0 j w w k 2 w k displaystyle nabla times mathbf v dots 0 mathbf hat i 0 mathbf hat j w w mathbf hat k 2w mathbf hat k nbsp Otzhe dovzhina rotora dorivnyuye podvoyenij kutovij shvidkosti i napryamok zbigayetsya z vissyu obertannya Fizichne znachennya RedaguvatiRozglyanemo rotor u xy ploshini Mi interpretuyemo rotor yak podvoyenu kutovu shvidkist malenkogo grebnogo kolesa u cij tochci Funkciyu c u r l F displaystyle rm curl mathbf F nbsp mozhna rozglyadati yak vimir tendenciyi F displaystyle mathbf F nbsp stvoryuvati obertannya Interpretuyuchi F displaystyle mathbf F nbsp yak silove pole abo pole shvidkostej F displaystyle mathbf F nbsp primushuvatime ob yekt roztashovanij u tochci P 0 displaystyle P 0 nbsp obertatis iz kutovoyu shvidkistyu proporcijnoyu do c u r l F 0 displaystyle rm curl mathbf F 0 nbsp Shob bachiti ce vikoristayemo grebne koleso z radiusom a displaystyle a nbsp i centrom u x 0 y 0 displaystyle x 0 y 0 nbsp i vertikalnoyu vissyu Nas cikavit yak shvidko obertatimetsya koleso Yakbi koleso malo lishe odnu lopat shvidkist ciyeyi lopati bulo b F t displaystyle mathbf F cdot mathbf t nbsp tobto dorivnyuvala b skladovij sili perpendikulyarnij do lopati spryamovanij uzdovzh dotichnoyi Oskilki F t displaystyle mathbf F cdot mathbf t nbsp ne odnakova uzdovzh usogo kola yaksho b koleso malo lishe odnu lopat to jogo obertannya bulo b nerivnomirnim Ale yaksho lopatej bagato todi koleso krutilos bi iz shvidkistyu sho dorivnyuvala b serednomu znachennyu F t displaystyle mathbf F cdot mathbf t nbsp uzdovzh kola Ce znachennya mozhna znajti shlyahom integruvannya i dilennya na dovzhinu kola shvidkist lopati 1 2 p a C F t d s 1 2 p a C F d r displaystyle frac 1 2 pi a oint C mathbf F cdot mathbf t ds frac 1 2 pi a oint C mathbf F cdot d mathbf r nbsp 1 2 p a R c u r l F 0 d x d y displaystyle frac 1 2 pi a int int R rm curl mathbf F 0 dxdy nbsp zgidno z teoremoyu Grina 1 2 p a c u r l F 0 p a 2 displaystyle approx frac 1 2 pi a rm curl mathbf F 0 pi a 2 nbsp de c u r l F 0 displaystyle rm curl mathbf F 0 nbsp ce znachennya funkciyi u x 0 y 0 displaystyle x 0 y 0 nbsp Obgruntuvannyam ostannogo nablizhennya ye te sho yaksho kolo utvorene grebnim kolesom malenkij todi c u r l F displaystyle rm curl mathbf F nbsp po vsomu regionu maye znachennya priblizno c u r l F 0 displaystyle rm curl mathbf F 0 nbsp otzhe mnozhachi cyu stalu na ploshu kruga mi otrimuyemo znachennya podvijnogo integrala Z cogo mi vivodimo dotichnu shvidkist grebnogo kolesa a 2 c u r l F 0 displaystyle frac a 2 rm curl mathbf F 0 nbsp Mi mozhemo pozbutisya a displaystyle a nbsp vikoristavshi kutovu shvidkist w 0 1 2 c u r l F 0 displaystyle omega 0 approx frac 1 2 rm curl mathbf F 0 nbsp Dzherela RedaguvatiFihtengolc G M Kurs differencialnogo i integralnogo ischisleniya Moskva Nauka 1966 T 3 656 s ros Mala girnicha enciklopediya u 3 t za red V S Bileckogo D Donbas 2007 T 2 L R 670 s ISBN 57740 0828 2 Otrimano z https uk wikipedia org w index php title Rotor matematika amp oldid 40290958
