Calculus of One Variable

Transcription

1 Clculu of Oe Vrble Summry Th em wll ddre fudmetl topc from gle-rble clculu cludg the cocept of cotuty d dfferetblty d the eplct clculto of lmt derte d tegrl. Smple Queto Here emple of coceptul queto tht ppered o pror em: Coder the fucto g() = { f rtol 0 f rrtol. I g() cotuou t =0? Ig() dfferetble t = 0? Jutfy your repoe. I order to wer th queto correctly you eed to udertd the oto of cotuty d dfferetblty rgorouly ot merely tutely. Note tht eplct jutfcto requred to upport ech correct repoe. Here re emple of computtol queto tht he ppered o pror em: ( ). l Compute lm e e Compute the rclegth of the grph of f() = coh betwee =0 =. I order to er full credt your repoe to ether queto would he to be jutfed by eplct tepby-tep clculto. Referece Sgle-rble clculu coered my udergrdute tetbook. It good de to reew more th oe book preprto for th em.

2 Clculu of Seerl Vrble Summry Th em wll ddre topc from ector clculu two d three dmeo cludg grdet of clr fucto computto olg the opertor le urfce d olume tegrl d rto of Stoke theorem (cludg Gree theorem d the dergece theorem). Smple Queto Here pror yer em t etrety: Let (r θ) be polr coordte o the ple z = 0. Let Ω deote the rego bouded below by th ple d boe by the urfce Σ o whch z =( r +coθ)e r how: Let C deote the cure log whch the ple z = 0 terect Σ d let F =(y + z co )î +( + z y)ĵ +( co y)ˆk d be two ector feld. G = (y + z)î + yzĵ (yz + z )ˆk. Compute the legth of C.. Compute the cotour tegrl F d the drecto of creg θ. C. Compute the et flu to Ω of the ector feld G. 4. I F coerte? I G? Jutfy your repoe. Note tht rto of Stoke theorem c be ued to mplfy the ecod d thrd queto ubttlly. Referece Vector clculu coered my tetbook cludg Erw Kreyzg Adced Egeerg Mthemtc. It good de to reew more th oe book preprto for th em.

3 ORDINARY DIFFERENTIAL EQUATIONS. Fd the geerl oluto to. Fd the geerl oluto to the ytem of equto:

4 PARTIAL DIFFERENTIAL EQUATIONS. Sole the tl lue problem - < < - < < ug Fourer trform.. Suppoe u d u re oluto of the followg equto. For whch equto u + u oluto? ) b) c) d) e)

5

6 Ph.D. Qulfyg Em Mthemtc Numercl Method Oct. 00 wth ome ector lgebr too. Ge two ector V = 7 0j 8k d V = j + 4k () Fd the gle betwee V d V. (b) Fd the ut ector tht perpedculr to both V d V. (c) Repreet V ug two ector compoet V d V y oe prllel d the other perpedculr to V.. () Clculte the clr trple product of three ector = +6j 7k = +7j 4k d =5 6j + 4k. Determe f the three ector re lerly depedet. (b) Determe the fucto of le L tht pe though pot ( ) wth drecto ector d = + 5j + k (c) For urfce ge by + 4y 7z = the urfce orml ge by = f =<6+4y 4 z >. Obt the tget ple of the urfce t pot ( ).. Elute I 4 0 d by () the trpezodl rule ug two terl. (b) the Smpo / rule ug two terl. Wht the error reducto f the terl ze reduced by hlf? (c) Wht the order of error reducto the trpezodl rule f the terl ze reduced by hlf? Wht the order of error reducto the Smpo / rule f the terl ze reduced by hlf? 4. For the followg ytem of equto wth four ukow d = = = = 0 () Repreet the boe equto the tdrd mtr form [A]{} = {b} d ue Gu elmto to obt the coeffcet mtr [A] the upper trgulr form. Show your work. (b) Ue bck-ubttuto to ole for {}. Show your work.

7 Ler Algebr Q) Ge two rbtrry qure ( ) ogulr mtrce A d B proe det(i-ab) = det(i BA) where I detty mtr d y clr. [Try d ue th ht: Two mtrce M d N re mlr f M = P - NP for ome ertble mtr P]. det td for determt. (5) Q) Let S be ymmetrc mtr d A be kew-ymmetrc mtr. Proe tht trce(sa) = trce(as) = 0 [Trce of mtr defed um of ll elemet o t m dgol (.e. the dgol from the upper left to the lower rght)] (5) Q) Fd the urfce term of bc o whch the followg ytem of equto h o oluto Could there be y lue of bcd for whch the ytem h fte oluto? (Jutfy). (0)

8 Secto: Vector Algebr Decrpto: Algebrc d geometrc properte of ector three dmeo dot d cro product ler depedece of ector clr trple product equto of le d ple. Queto : Ue the followg form for the equto of le: t b b b z y Fd equto for the le tht cot the pot 5 6 d 7 9 Wrte the equto for the d b prmeter tht re tfed f d oly f two le re equlet. (for le ue b b b d for le ue b b b ) Queto : 0 Show whether the followg ttemet true or fle by performg the clculto: Why would you epect the boe to be true or fle? Fd the projecto of oto. Fd the gle betwee d. Fd lue of o tht the ector 4 perpedculr to.

9 Ph.D. Mth Qulfyg Em - Comple Vrble - Sprg 00 Plee wer both queto ) I -D potetl flow theory the comple potetl F = F (z) relted to polr-cyldrcl elocty compoet the followg formul: df dz = r(r θ) θ (r θ) () where r d θ re the r d θ elocty compoet repectely d where z = re θ Note tht for lytc potetl fucto F df dz = F r = F r θ ) Ge F (z) = Γ l reθ π fd the correpodg r d θ elocty compoet. b) the comple potetl lo relted to the o-clled elocty potetl φ d the trem fucto ψ ccordg to F (z) =φ(r θ)+ψ(r θ) () Stremle flow tur correpod to le o whch ψ cott.e. o ech tremle ψ ume cott lue. Sketch few of the tremle the boe flow.

10 ) Coder oclltor e.g. prg-m ytem dre by perodc forcg fucto: ẍ + = bcoωt () where = (t) the tteou poto of the m d b re cott d ω the drg frequecy. ) I order to determe the (cyclclly) tedy moto (t) frt (quckly) how/cofrm tht co ωt = (eωt + e ωt ) (4) b) Net ume oluto of the form (t) = (t)+ (t) where (t) =Ae ωt d (t) =Be ωt. Sole for (t) frt by pluggg the umed form of (t) to Eq. () d by replcg the rght de of () wth the product of b d the frt term o the rght de of Eq. (4). Do the me to obt (t) replcg the rght de of () wth the product of b d the ecod term o the rght of (4). c) Quckly cofrm tht the umed oluto = + tfe the goerg equto Eq. ().

11 probblty d tttc. ) Epl the crcumtce for pplyg two-mple oe d two-ded t tet b) Tble how the meured me cote of two dfferet brd of cr ol. Tet the hypothe tht the two brd re equl t 0% gfcce leel Tble : Meured lue of the me cote of two brd of cr ol Brd A Brd B Stte the tegrl equto for computg the me d rce bed o kowledge of the probblty dety fucto p. For trgulr probblty dety fucto of wdth determe the tdrd deto

12

13

14 4

15 5 Where pproprte the followg equto my be ued The probblty m fucto for boml rdom rble ( p)!!! ) ( p p X P Me N f where N f Vrce N f N f Vrce of the me computed from ormlly dtrbuted mple N Stdrd ormlzed rte u[0] u Poo dtrbuto e r r P r! ) ( me of = rce = Ch-qured ( ) O E E t tet t z c c Fcher F tet F