Calculus 241, section 12.2 Limits/Continuity & 12.3 Derivatives/Integrals notes by Tim Pilachowski r r r =, with a domain of real ( )

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1 Clculu 4, econ Lm/Connuy & Devve/Inel noe y Tm Plchow, wh domn o el Wh we hve o : veco-vlued uncon, ( ) ( ) ( ) j ( ) nume nd ne o veco The uncon, nd A w done wh eul uncon ( x) nd connuy e he componen uncon o, e ( ), y ( ) z ( ) x, y n Clculu I, we e (ho) de p no dcuon o lm Denon Le [ ] e veco-vlued uncon e dened ech pon n ome open nevl connn, excep poly el A veco L [ L ] he lm o () ppoche (o L he lm o ) o evey ε > hee nume δ > uch h < < δ, hen ( ) L < ε In h ce we we ( ) L lm ex lm nd y h ( ) In -D pce, we cn vulze n open ll [ee Lecue ] o unnel wh cene L nd du ε Then lm ( ) L hee n open nevl ou uch h n o ech nume n he nevl (excep poly ) pon n he ll/unnel Mo ueul o ou pupoe wll e Theoem 4 Le ( ) ( ) ( ) j ( ) Then h lm nd only, nd hve lm Th, lm ( ) lm ( ) lm ( ) j lm ( ) The poo n he ex, o I won duplce hee e ln j 5 e lm ln j 5 5 ( ) Exmple A nd lm ( ) nd ( ) ( ) Sde noe: To ollow n exmple n he ex nd o one o he pcce exece, you ll need o ememe h n lm Only couple moe heoecl hn e needed o nh up econ

2 o ll, popee Theoem 5 Le ( ) nd ( ) e veco-vlued uncon, nd ( ) nd ( ) whch ll lm ex nd pcully lm ( ) lm( )( ) lm ( ) lm ( ) lm ( )( ) lm ( ) lm ( ) lm( )( ) lm ( ) lm ( ) lm ( )( ) lm ( ) lm ( ) lm ( )( ) lm ( ) lm ( ) lm( o )( ) lm ( ) ( ) o ll n n open nevl ou Poo e n he ex, o I won duplce hem hee Second, connuy Denon 6 A veco-vlued uncon connuou pon e el-vlued uncon, o n domn lm ( ) ( ) Theoem 7 A veco-vlued uncon connuou nd only ech o componen uncon connuou Hee en econ Denon 8 Le e nume n he domn o veco-vlued uncon [ ( ) ( ) ] I lm ( ) ( ) ex, we cll h lm he devve o nd we ( ) lm d We ll lo ue he Lenz noon, ( ) d Inomlly ed, ju he devve o y ( x) w deved he lm o lope o ecn lne povdn he lope o he nen o he cuve, he devve ( ) C whch ced ou y veco-vlued uncon ( ) Theoem 9 Le ( ) ( ) ( ) j ( ) e deenle povde u wh veco whch nen o he cuve Then deenle nd only, nd j In h ce, ( ) ( ) ( ) ( ) Exmple B Le H ( ) ( 4) ( ) j ( 6) nd H ( ) A llued n Exmple B, he devve o lne veco-vlued uncon L conn vecovlued uncon pllel o L

3 Exmple C Le ( ) j n co nd π Noe h he poduc ule w needed o he z-componen Almo ll o he deenon ule om Clculu I hve counep o veco-vlued uncon Theoem Le,, nd e deenle, nd le e deenle wh ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) o When hee choce o mehod, I ecommend choon he mple one Exmple D Le ( ) j ln 5 nd ( ) j co nd ( ) ( ) I we ue he ule ove, he h-hnd de would men don wo co poduc In h ce, my e ee o do one co poduc on he le, hen deene Poly he mo ueul o he popee ove wll e he l one, he chn ule

4 Coolly Le e deenle on n nevl I, nd ume hee nume c uch h ( ) c o n I Then ( ) ( ) o n I The ex h hee-lne poo Moe mpon o ou pupoe e he mplcon o Coolly I ( ) conn, hen o ech n he domn o one o he ollown ue ( ) ( ) (e, ( ) j c ) ce ou ccle) ( ) nd ( ) e pependcul (e ( ) Lecue Exmple E eved ven ( ) co co j 5 n, nd ( ) ( ) nd ( ) e pependcul o ll vlue o n he domn (Do h one on you own, o pcce), hen how h The econd devve o veco-vlued uncon dened he devve o he devve o veco vlued uncon ( ) j ( ) j j ( ) Exmple C eved Le ( ) co j n nd ( )

5 When we pply veco-vlued uncon o pplcon nvolvn moon wh epec o me, we e eul ml o hoe ound n Clculu I Poon: ( ) x ( ) y ( ) j z ( ) (dl o du veco) ( ) ( ) (dplcemen veco, om nl poon o cuen one) d dx dy dz Velocy: v ( ) j d d d d Speed: ( ) v Acceleon: ( ) dx dy dz d d d dv d d x d y d z j d d d d d Exmple E nd he poon, velocy nd peed o n ojec hvn cceleon ( ) v j, nd nl poon, nl velocy whee, nd o he exmple ove, we mde ue o Theoem Le ( ) ( ) ( ) j ( ) e connuou on [, ] ( d) ( ( ) d) j ( ( ) d) ( ) d ( ) ( ) d ( ) d ( ) d j ( ) d o nce wh h me de ppled o pplcon nvolvn ojec ujec only o he vy o eh (ex exece 47-49), ee he ex Exmple nd Exmple