Studying the Problems of Multiple Integrals with Maple Chii-Huei Yu

Transcription

1 Itetol Joul of Resech (IJR) e-issn: , - ISSN: X Volume 3, Issue 5, Mch 26 Avlble t htt://tetoljoulofesechog Studyg the Poblems of Multle Itegls wth Mle Ch-Hue Yu Detmet of Ifomto Techology, N Jeo Uvesty of Scece d Techology, T Cty, Tw E-ml: chhue@mljuedutw Abstct I ths tcle, we use Mle fo the uly tool to study two tyes of multle tegls The fte sees foms of the two tyes of multle tegls c be obted mly usg boml sees d tegto tem by tem theoem Moeove, some emles e oosed to demostte the clcultos, d we use Mle to clculte the omtos of some multle tegls d the fte sees foms fo vefyg ou swes Key Wods: Mle; multle tegls; fte sees foms; boml sees; tegto tem by tem theoem Itoducto As fomto techology dvces, whethe comutes c become comble wth hum bs to efom bstct tss, such s bstct t sml to the tgs of Pcsso d muscl comostos sml to those of Beethove, s tul uesto Cuetly, ths es uttble I ddto, whethe comutes c solve bstct d dffcult mthemtcl oblems d develo bstct mthemtcl theoes such s those of mthemtcs lso es ufesble Nevetheless, seeg fo ltetves, we c study wht ssstce mthemtcl softwe c ovde Ths study toduces how to coduct mthemtcl esech usg the mthemtcl softwe Mle The m esos of usg Mle ths study e ts smle stuctos d ese of use, whch eble beges to le the oetg techues shot eod By emloyg the oweful comutg cbltes of Mle, dffcult oblems c be esly solved Eve whe Mle cot deteme the soluto, oblem-solvg hts c be detfed d feed fom the omte vlues clculted d solutos to sml oblems, s detemed by Mle Fo ths eso, Mle c ovde sghts to scetfc esech The multle tegl oblem s closely elted wth obblty theoy d utum feld theoy, d c be studed bsed o Stet [] d Ryde [2] Fo ths eso, the evluto d umecl clculto of multle tegls s mott Fo the study of elted multle tegl oblems c efe to Yu [3-] I ths e, we mly study the followg two tyes of -tule tegls d d, () d d d, (2) whee, e el umbes,, s ostve tege,, e o-egtve teges fo ll,,, d some codtos e stsfed We c deteme the fte sees foms of the two tyes of multle tegls mly usg boml sees d tegto tem Avlble ole:htt://tetoljoulofesechog/ P g e 36

2 Itetol Joul of Resech (IJR) e-issn: , - ISSN: X Volume 3, Issue 5, Mch 26 Avlble t htt://tetoljoulofesechog by tem theoem; these e the mjo esults of ths tcle (e, Theoems d 2) I ddto, two emles e used to demostte the oosed clcultos The esech methods doted ths study volved fdg solutos though mul clcultos d vefyg these solutos by usg Mle Ths tye of esech method ot oly llows the dscovey of clculto eos, but lso hels modfy the ogl dectos of thg fom mul d Mle clcultos Theefoe, Mle ovdes sghts d gudce egdg oblem-solvg methods 2 Pelmes d M Results Fst, otto d two mott theoems used ths study e toduced below 2 Notto: Boml sees ([,244]) : Suose tht, e el umbes d, the ( ) ( )! whee ( ) ( ) ( ) fo ll ostve teges, d ( ) 22 Itegto tem by tem theoem ([, 269]): Suose tht g s seuece of Lebesgue tegble fuctos defed o tevl I If I I g s coveget, the g I g Befoe devg the mjo esults ths e, lemm s eeded, Lemm Assume tht, e el umbes,, s ostve tege, d, e o-egtve teges fo ll Cse If,,, the ( ) ( )! Cse 2 If ( Poof If hve Cse If ( ), the ) ( )! (3) (4), the by boml sees we ( ), the ( )! (5) Avlble ole:htt://tetoljoulofesechog/ P g e 37

3 Itetol Joul of Resech (IJR) e-issn: , - ISSN: X Volume 3, Issue 5, Mch 26 Avlble t htt://tetoljoulofesechog ( ) ( )! (bye (5)) ( ) ( )! Cse 2 If, the ( ) ( )! (bye (5)) ed The followg s the fst esult ths tcle, we obt the fte sees fom of the multle tegl () Theoem If, e el umbes,,, s ostve tege, d, e o-egtve teges fo ll d,,, the d! ( ) ( ) ( ) Poof Sce, t follows tht Theefoe, d d (6) ( ) ( ) d d! (by E (3)) ( ) ( ) d d! (by tegto tem by tem theoem) ( ) ( ) d!! ( ) ( ) ( ) ed Net, we deteme the fte sees fom of the multle tegl (2) Theoem 2 Suose tht, e el umbes,,, s ostve tege, d, e o-egtve teges such tht fo ll,,, the Avlble ole:htt://tetoljoulofesechog/ P g e 38

4 Itetol Joul of Resech (IJR) e-issn: , - ISSN: X Volume 3, Issue 5, Mch 26 Avlble t htt://tetoljoulofesechog d d ( ) ( ) ( )! ( ) (7) Poof mles tht d d ( ) ( ) d d! (by E (4)) ) ( )! d d Thus, ( (by tegto tem by tem theoem) ( ) ( ) d! ( ) ( ) ( )! ( ) 3 Emles ed Fo the multle tegl oblems dscussed ths e, we oose two emles d use Theoems d 2 to obt the fte sees foms Moeove, Mle s used to clculte the omtos of some multle tegls d the solutos to vefy ou swes Emle Usg E (6) Theoem yelds the double tegl / 6 ( 3) d d ( ) ( / 6) 7/6!3 (2 6)(4 4) 2 (8) Net, we use Mle to vefy the coectess of E (8) >evlf(doublet(^5*^3/(^2*^4+3)^(7/6 ),=,=),4); >evlf(sum((-)^*oduct(+/6-j,j=(- ))/(!*3^(+7/6)*(2*+6)*(4*+4)),=fty ),6); Emle 2 O the othe hd, by E (7) Theoem 2, we obt the followg tle moe tegl / 2 ( 3 3/ 4) d d 2 d ( 3/ 4) ( 7 / 2)!( 6 24)( 2 5)( 5 35/ 2) 3 (9) We lso emloy Mle to vefy the coectess of E (9) >evlf(tlet(^2*^3*3^4/(^6*^2*3 ^5+3/4)^(9/2),=fty,=fty,3= fty),4); >evlf(-sum((-3/4)^*oduct(+7/2-j,j= (-))/(!*(-6*-24)*(-2*-5)*(-5*-35/2)), =fty),6); Avlble ole:htt://tetoljoulofesechog/ P g e 39

5 Itetol Joul of Resech (IJR) e-issn: , - ISSN: X Volume 3, Issue 5, Mch 26 Avlble t htt://tetoljoulofesechog 4 Cocluso As metoed, the evluto d umecl clculto of multle tegls s mott Ths e ovdes some techues (e, boml sees d tegto tem by tem theoem) to obt the fte sees foms of two tyes of multle tegls I fct, the lctos of the two theoems e etesve, d c be used to esly solve my dffcult oblems; we edevo to coduct futhe studes o elted lctos O the othe hd, Mle lso lys vtl ssstve ole oblem-solvg I the futue, we wll eted the esech toc to othe clculus d egeeg mthemtcs oblems d emloy Mle to solve these oblems Refeeces: []F Stet, O Multle Itegl Geometc Itegls d The Alctos to Pobblty Theoy, Cd Joul of Mthemtcs, vol 22, 5-63, 97 [2]L H Ryde, Qutum Feld Theoy, 2d ed, New Yo: Cmbdge Uvesty Pess, 996 [3]C -H Yu, A Study o the Multle Imoe Itegl Poblems, ( Chese) Joul of Hs Sheg, vol 2, 75-94, 23 [4] C -H Yu, Alcto of Mle: Tg the Double Imoe Itegls s Emles, ( Chese) Poceedgs of 23 Ifomto Educto d Techology Alcto Sem, Oveses Chese Uvesty, Tw, -5, 23 [6]C -H Yu, Alcto of Mle o Multle Imoe Itegl Poblems, ( Chese) Poceedgs of 22 Otoelectocs d Commucto Egeeg Wosho, Ntol Kohsug Uvesty of Aled Sceces, Tw, , 22 [7]C -H Yu, Evlutg Multle Imoe Itegl Poblems, ( Chese) Poceedgs of Ye Geel Educto Symosum, Ntol Pgtug Uvesty of Scece d Techology, Tw, -7, 22 [8]C -H Yu, Usg Mle to Study the Multle Imoe Itegl Poblem, Poceedgs of IIE As Cofeece 23, Ntol Tw Uvesty of Scece d Techology, Tw, vol, , 23 [9] C -H Yu, Techues fo Evlutg Some Tye of Multle Imoe Itegl, Itetol Joul of Resech, Vol 3, Issue 4, 5-54, 26 []C -H Yu, Alcto of Comle Fucto Theoy o Double Itegl Poblems, Itetol Joul of Resech, Vol 3, Issue 4, 24-27, 26 [] T M Aostol, Mthemtcl Alyss, 2d ed, Msschusetts: Addso-Wesley, 975 [5]C -H Yu, Evluto of Two Tyes of Multle Imoe Itegls, ( Chese) Poceedgs of 22 Chghu, Yul d Chy Colleges Uo Symosum, D-Yeh Uvesty, Tw, M-7, 22 Avlble ole:htt://tetoljoulofesechog/ P g e 4