Apuntes Calculo III - Fernando Chamizo

176
Maxwell Gauss Stokes

Transcript of Apuntes Calculo III - Fernando Chamizo

Page 1: Apuntes Calculo III - Fernando Chamizo

MaxwellGauss Stokes

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Page 10: Apuntes Calculo III - Fernando Chamizo

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Page 11: Apuntes Calculo III - Fernando Chamizo

FÏ! 0! >Ë 0 Ð6 Ø!È0!grÈ0 Ìg!>!0&6g! ¡6!Ì ` QàX#!06)Ì+È aXrÈQX!rÈg¤0!!È0×rÌg¤ > V aÌg!Ì0#!ÄÌgÄÌgrÈ '0!!È0È06gÐ!rÈD !g!rÈ0 !Ï! !g!!rÈ0 ¨ !Ì 0 g!0¼O ¶ ×rÌ X#a!ÆÌÆgÌrÈ &6Ð6 K = Ä ? 0Ðg!¾×rÌg6÷g!0! !rÈ ¨ ÌgÆ ×VÌÈ Ð ¶ +g!0! a!VÈ r 6g a!rÈ0 K = Ä ?MT Ìg 0Ìág <g !0 +ÈÌg!& !' !È0 a!VÈ>0 a6' ×rÌg>'#0á!'Ìg!!ä0ÌË®0BgË®!0!rÈ!9!0! !rÈ '×rÌg0 ágÃgË 0 a£g0ÈrÈ£0zg Ìg! !0! !rÈ g£ËÍÌg!& <!0g!vÌgrÈ r+ȾРÎ!È0 !0È ! & Ð6 ×rÌg 9g 9È0Ð!rÈ0! .QXg ÈÐ!rÈ0! !È0D!ÁÌgÁÌgrÈ ×rÌXÁ!È0!0 a ¤ÌgÁ !!XrÈD N !X ![gc¿.+È 0£X0È !X!rÈ0!!!g!Øc!0&6g0!!×rÌc Ø Ì+Ì!'Q! r< ' BÌg aXrÈ g%×rÌgT.&a&60'#6ág! LÇ!0È>!Ì0 0! ágÐ &6 £g6 ÌgÃÐ á+È0!È ££g!0#!& l!l#0 #0á! lX0QX!È Ä V0È !ÁX[!Ì0 0 zÇÁÌgÐ6g g0g!×rÌgcg!0#gc ¸Ì !Ìg6VÈ rÌg a!0È !g0! Ï×rÌg !Ïg gO!6! cg!×rÌg 0!! O Ï!0! !rÈ %-#0á!- z -ÉaË®Ìg!& Ç[0È0a!-g!agÉËÍ!0!X

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Page 12: Apuntes Calculo III - Fernando Chamizo

ÖØ×ÒÜ_ÝàâÔaÜ¡ ÝÕÒ

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¼ K = e ·¦¥ ? Á K = e ? ¼ Ì ¼ K = e ·¦¥ ? Á K = e ? Á K = e ? ¥ ¼ · ¼ K = e ? ¥ ¼

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à È0rÈ ¢ ü§£ ¼ K = e ·¨¥ ? Á K = e ? ¼ ¤ å !0È ! ¢ ü§£ K = e ·©¥ ?#¤ K = e ? ×rÌg!X!& zÉ rÈ0rÌg

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Page 13: Apuntes Calculo III - Fernando Chamizo

Ïaýgü! Ð íQ.rÌgá!0l gË®!X!! Ð K = e ? K = e ? !VÈ !!Og60ÃX!0È !!È ð¤ ^ È 6¢ & !ÆË®!0!rÈ!#¾) !+ÐÌ È00ÐÌ = X ¸Ë 0 a¼ o ¼ · ¼ u ¼r« ¼ oÍÁÎu ¼ ? !0!0á! ¥ ¤ê» ø»þð¤nå

¼ K = e ·¦¥ ? Á K = e ? Á K = e ? ¥ ¼ · ¼ K = e ·¥ ? Á K = e ? Á K = e ? ¥ ¼¼ ¥ ¼ « ¼ K = e ? Á K = e ? ¼

¼ ¼

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Page 14: Apuntes Calculo III - Fernando Chamizo

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Ç[!!È ×rÌg !Î!-! £ÌgÃ#0á 0-ª g !Ìg6Î OP!Ìg ª!È0 !g!& ![ ×rÌg-ÌOÌg ) )L! 0 È0 ÈP 0 ! !@*!0 !%Ìgc!+È0á ×rÌg¤0¤0!gÈ!c!gOg! +È0!& 9¤È %Ë 0 % .g#Æ×rÌg!rÐ!06 äXª r 0!0ÌÈ Ï 0! g6ªOÐÌg ÏP!Ìg Ï g!XÈ ÏÆ!×rÌg 0ÐÌgÉQ!0 È0ªa!rÈ0!g!0 -agª6XÈ! !rÈ0ÉÈ g6-6-Ë®Ìg! !-×VÌÉ0É XÌ006³

o&bAmbAu htfDhSibQj|xQ_tÔê Ñ¡ÒÝ K Ö Áö e ì e ® Ö îÄ ? Ñ@Ô K ÒÜ#Õá æÜCàLÝ+æàÒÒæàâá æiÕÒÜÀ K = e ? ÒÜçaÝMÝ+ãçaÔÃÕÝÕÔá+æéçHÔaæ¡ÒÝ+çæ× çaÝî ? Ñ@Ô K ÒÜ°×Ræ@Ý;Ý+ã@çaÔÕÝÕÔá+æéçaÔaæiÒÝ+ç Òæàâá æiÕÒÜÀ K = e ?¤ K îÁ ? ÑiÒÝ æ K ©ª K «ª !! ª K e çaÝ+ÜÕáØá+ßCÓ@ÒæÝÓ Ý+ÜÓ@Ò K ë@ÒæàLá+æ¡ÕbÒÜ K ÒÜÓRÔdxÒßCÒæ¡ÕÔaÝÞRçÃÒiÜÔØìÜá+çaáoÜÔàâáÓRÝ+Ü

ÒçaçHÝ ÜoçaáøÜá+æ¡î5Â%Ó@Òè3Ý+Ü]Òæ ÒÜCàÒ°ÕÝ+ÜáøçaÝ+Ü%ÕáØá ßCÓÒæ@ÝÓRÝ+Ü%Ó@Ò¤ K = e ? ë Üá+æà K ©R= e ? ë ªR K «R= e ? ë ª X! ª K e = e ? ë ãÝ ßݳàLáÓ á ë ® Ö î

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Page 15: Apuntes Calculo III - Fernando Chamizo

À ? Ñ@Ô N ¤º ë ± ¤ ë e ¤ = Ä © ª Ä « ? ì K = ë ?¤ ¨ © ¨ « Õá+æ ë ¤ =H¨ © ª¨ « ? ÒæàLá+æ¡ÕÒÜ K = e ? ë ¤Ä « ¨ © · Ä © ¨ « î

Ä ? ÑÔ N ¤ ± ¤ ÒæàLá+æ¡ÕÒܧ K = e ? ÒÜoçaÝøÝ ãçaÔÃÕÝÕÔá+æ çaÔaæiÒÝ ç¨ µö K = Ä ? ¨Ó á+æ¡ÓÒ K = Ä ? ÒÜTçaÝÓ@ÒßÔ;Å_ÝÓRݳ×RÜC× Ý+ç Òæ¸× æ@ÝÆÅ_Ý+ßÔaÝÞRçÃÒî

Ïaýgü! Рೡ0È Ä ?¾! árr¡  ?¾0ä0Ð6ÌgäËD! !rÈ0ägä !g!& ßgäËÍ!0!g!Ì+6g ×rÌÉ a!g

K = e ·¥ ? Á K = e ?¤ K = e ? · K =Ç¥ ? Á K = e ?¤ K =;¥ ?N

)L0 á Á ?>> È0!0×rÌg ¨ «© · !! ·¿¨ «e Ì Q ¨d© Q · !! · Q ¨ e Q %è>¢ Ìg!>0P K&È

ËÍ!0!g!á! !0 Ð! K = e ? !zXM!rÌg!

å#Ì ¼ K = e ·¥ ? Á K = e ? Á K = e ? ¥ ¼¼ ¥ ¼ Ì

e

È U ©¼ KYÈ = e ·¥ ? Á K&È = e ? Á K&È = e ? ¥ ¼

¼ ¥ ¼ ª

ÎÈ 6g ¼¢ ÈX+ ágÈ!gî×rÌ K !ä0! aXrÈîgËÍ!0!g!áD Ç[>Xr¢ 0 á0!0&6 ×rÌg<¾gËÍ!0!g!áÆg K X¾×rÌg ¼ ¥ ¼ © = K = e ·4¥ ? Á K = e ? Á K = e ? ¥ ? ö ^!Ìg6g ¥ ö ^ IÉ Á !0 aÉ g!rÈ- !+È ¼¢ aÈ0 !É!g! -Ë®!0!!ágg6 K&È

à³ag0Ì!ágag À ? 0É0!gÌ!É ag ág ×rÌg

¢ ü§£Q = Ä ©· © ? = Ä «8· « ? Á>Ä © Ä « Á = Ä « ©· Ä © « ? Q

¼ ¥ ¼ ¤når

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7Lg aXrÈ Ä ?¤+XÌ!ccg!g!& ¸g!#ÄÌÄË®Ìg!& ÁgÌÄ&60ág!0!0È0!zË 0 = K = Ä · @? Á K = Ä ? Á K = Ä ?-@? Z ö åA

¹ ÈÌ0 !rÈ0-[ËÍÌg! Xg-#0[#06ág![0ÌX!O0! ½6[ ag!g6[×rÌÏ×rÌg0!!! ! !T0!0ÌÈ rÈ0!0 ×rÌaÌX! 0!000¾ ÌgÆ!0!!¾g¾g á>0 #¤à³Æ0!ÐÆg¾z!!gÆ É! È0¾+6á!É×rÌ- !Ì00a!Ì È0!! Ìg !g! É -rÈ06 ÉÈ!0 g! -!gËÍ0!rÈ0 -a00!!!

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Page 16: Apuntes Calculo III - Fernando Chamizo

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³=Í: K ? = e ?f¤ Í¡= K = e ?? :5 K = e ?N

ï &0¶&©0«/r±D»0¬Q¹´!¯ ¹ È0!0¾×r̾6¢ !0XÈää0!Ð6ä¾ä!g!äg60!!¾ ½6g6È0Ì0P×V̾!0ÌÎË 0 aÃÌg a!g0 ÇÏ90 ! a!rÈ0z!z×rÌz gËÍ!0!g! zÌgà a 0! Î! 0! £g Ë®!0!!X-ä! ! ½-gÉ+!-!0 0Ìg!Oá!³)L! 0È06rÈ00 È -!-×rÌgÉ!-6Xgz 0×rÌgÉ g! -g ág6- rÈrÌ!&

_n~z Íz_ CÍzdC n nLz zdzzz Í P zLÍ <~/ |H_C zR|HLz zz Cz_ zL Lz 1Czz _Cz R z Ð6z | |H6z_ |H6nz C Í_Czz n~ z | z Cz| ÀG nÑ C|H6zR|HLzS Cz R|H n nz z_ þ_Cz ]~ z Lz C n ;H6.\])(+" 6.)(+F¢V" K#: 9G'Y /F

Ïaýgü! й Ìg!0È0Q.g> È0!0-

= X7? ¢ ü¤£¼WÒ = e ·¦¥ ? ¼

¼ ¥ ¼ ¤nå ª ¢ ü¤£

¼ÔÓ = K = e ? · ¥ ? ¼¼ ¥ ¼ ¤nå

gÒ = ë ?f¤ K = ë ? Á K = e ? Á K = e ? = ë0ÁÎe ? ª Ó = ë ?¤ Í¡= ë ? Á =dÍ: K ? = e ? Á Í¡= K = e ?? = ë0Á K = e ??.Õ ×rÌ!0! Ï á

¢ ü¤£¼ÔÖ = e ·¦¥ ? ¼

¼ ¥ ¼ ¤nå Ö = ë ?¤ =dÍ: K ? = ë ? Á =Í: K ? = e ? Á Íi= K = e ?? :5 K = e ? = ë0ÁÎe ?N

à³a!+È0È!Ð árr!-0!! Ö Ò Ó

, zP!Ìg !! !rÈMgÌg!á= X] ? Ö = e ·¥ ?f¤ Í¡= K = e ?? Ò = e ·¦¥ ? · Ó = K = e ·¥ ??N, ÈaE! = 6?[ ¥ ¤ K = e ·×¥ ? Á K = e ?[a!g! g%¼¢ È 0%g!Ìg!%×rÌ%0¤È ÆSÈ å0 ¼ ¥ ¼ = È06VÈ ¼ ¥ ¼ - rÈ0rÌgzg K ?-!-0Ìg!!rÈ0! aXrÈÉ!×rÌg

¼WÒ = e ·¥ ? ¼ ÌÎÆ ¼ ¥ ¼ ¼ÔÓ = K = e ·¥ ?? ¼ ¤ ¼ÔÓ = K = e ? · ¥ ? ¼ ÌÎÆ ¼ K = e ·¦¥ ? Á K = e ? ¼ è>X6g !0Ð6ÌgzÈ00ÐÌ

¼ K = e ·¦¥ ? Á K = e ? ¼ Ì ¼ K = e ·¦¥ ? Á K = e ? Á K = e ? ¥ ¼ · ¼ K = e ? ¥ ¼0Ìg0ÈÈ0Ì!g !0ÈX0ÐÌg6£! = 6 ?

¼ÔÖ = e ·¦¥ ? ¼ ÌÎÆ ¼YQ Í¡= K = e ?? ¼RQ¼ ¥ ¼ · Æ ¼ K = e ·¦¥ ? Á K = e ? Á K = e ? ¥ ¼ · Æ ¼RQ K = e ? ¼RQ¼ ¥ ¼

Page 17: Apuntes Calculo III - Fernando Chamizo

<r! XrÈ0 ¼ ¥ ¼ Æ 0g!06g ×rÌg]Æ-!%60áÈ00 a!rÈ0 !×rÌg !Ìg ¥ ö ^ 0 áÈ0!ÉXMX0ÌÈ0 !0!06

Av! LÌ)! Ì rÈ!0!0rÈ0 0×rÌ'>gª V a<0!!!& 0ágX d.!!0 0ÁÈrÈ ¤ 0 a!rÈ è% ! ¤+!0rª£gcg!¤P 0O ág6Á ¤X0ÌÈ0 rÈ!0 0!

Ç¡¼! ! 0È00<×rÌ K « ö « Æ K =H¨ª¶ ?¤ = Ä · 0X =a¨s¶ ? ª¨`· ¶ ?<!ËÍ!0!g!áÉ!£È gÌrÈ

íg!Ð-Ìg¡!P0È06 Á ?>gäg 0! ì äág6+Èäg ág6×rÌ K ©R=a¨ª¶ ?¤ Ä · 0! =a¨s¶ ? K «=H¨ª¶ ?T¤ ¨`· ¶ ¡gËÍ!0!g!áX 6)  ? ^K « !#6) È0ÆgÈ0¾ ! !0!0ágK © ¤ Íx: õ Í¡=H¨ª¶ ?f¤ ¨6·³¶ õ =a¨ª¶ ?¤ = Ä ª 0! =a¨¶ ??XLàË®ÌX& Í !'! BÈ0rÈ ËÍ!0!g!áD N 6 áX! 0¾!Ì ¾×rÌg õ !Ë®!0!!ág¾ä×rÌg ¡0Ìg< g!rÈXCõ ©R=a¨ª¶ ?Q¤ Ä ×rÌg!ä 0!& îgäg !!!& « Áö ¡ä!r !g!L!Ì&6ág ¤B×rÌ%0%!6  ?[ Ä ?&ÏS a!rÈ0B×rÌ õ «=H¨ª¶ ?¤n0! =a¨s¶ ?[!B¤ 0X& ¨s¶ +! ¨ ×rÌgÉ zËÍ!0!g!á!- À ? Ä ?X

¹ ÈÌ0 !rÈ0 ! ag! ¡ËD 0 Ì Æ 0!Ð6¡gÆ¡!g! ¢ a +!ÐÌÌg£Ë 0 Ìg£0z£6g!!& lX6 K = e ?É! È ágX!lÈÌg06 a!rÈ0 .g! ×rÌP¢

TÏaýgõ÷ðÙØrö÷öLË6öMõûðÍöòDýËh³ðÍõ÷ýgûûªðÍóEþMöòDýÇ[ß!9!Ìg g è%Ð6Xá0 àg!90Æ P¢ ß ÌÁ!0 ag!!g ×rÌÆÌg !6X&

g!v ! Ìgz 6È00aEîv0×rÌ!0£z!£!!& Ãg!v 0!0 P¢ £Ì£ 6È00aE0 È¢ ! a!rÈ0gÈ Ìg!î!á0?Ï! !0Ä!>×rÌgEg6Eg¼Xî ×rÌg¼!0|!0%È ¼!0!! g 6Ð6Xá0Á!%0Ä0! ¢ |! ! 6È00!! nÇ[rÌg!0È0É0ÐÈ0Ì0 !Ìg0ªÉ a0 K = e ? ¢ ×rÌ-0!ªÉ0 g !!v0Ì 6È00aE!ÄrÌX0È0ág6+¤0!Ë®!0'!> g!A èÒ!0È!E !%Ìgc0X!XÈ !Ì<³!>!E!+ÌÈ0 á0ÉÈ !Ì ÉrÈ! ! -!!0Èg!g!&

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¶'ü5² õ = e · p ­ ? Á õ = e ?

ÓRá+æiÓ@Ò p ­fÒÜWÒç+¯ Á ÒÜÔaè;áÝÅ+ÒÕàLá+ßWÓÒçHÝMÞRÝ+ÜCÒÕÝ æÝá+æ@ÔÃÕÝ;óàâáÓRÝ+ÜfÜC×RÜQÕáØá+ßbÓÒæ@ÝÓ Ý ÜfÜá+æ%æ×RçHÝ ÜQÒåÕÒãiàLáçaÝ#¯ Á ÒÜÔHèMݳÖØ×@ÒÒÜ°× æ@á+ôCî

Page 18: Apuntes Calculo III - Fernando Chamizo

ï &0¶&©0«/r±D»0¬Q¹´!¯ , g£!0&£g60! !¤£VÌ! 0Ìl r 0!g6gÉ Á !0 a£ ! ½6B×rÌ%¤!0&¤gÏÈ ¤¤rg¤ õ È = Ä © ª !! ª Ä ­ © ª¨8ª Ä ­AÞ© ª !! Ä Ö ?[! ¨ ¤ Ä ­ EÇ[[g!! á0È0l+Ì0È0ÈÌgÈ g66&60ág! ! ¯ Á !0 Á |X0QX!È ll!Ìg6Ïg!0# Ç+ÈÉr0& ag É!0&6gÉg60!g0- Ì!0È00É Ìgg !c. 0É!Égª6!È0! 0 ×rÌX! áÌg00 Ï!0!0á! ¼¢ aÈ0!

೸ È!& g60 ä!0&6gä0!! Ú õ Z Ú ¨­ !ä!&0! Î!0Ȫ¸È06½0Ì a!rÈ0!rÈ0!g×rÌgË+¢ ! a!rÈ006!ªg!0+E! > È! î g!0 °­ õ Ïè !0%g! á0ÉÈ !z Ïág Ï##E!6 Ì/. Ï aÈ0! È0 -È0È06£gÉ!rÈ0- È06X& !&0X 0×rÌg%r Ìg!6Ð6ÌgB +0ÈX! 8) B)! XÌ 0%!0!Ð!!gaËÍÌg! K =H¨ = »c? ª¶= »_?? 0É0Ì!É!0!0ág

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Page 19: Apuntes Calculo III - Fernando Chamizo

è>X6g ß 0!& 6 Ä ?ä Í Ä ?ä  ?ä æ 0¡È0!g¡!½Èg!0 a ê¼ aÈ0!! (Í = Ä ­ ?¤ õIÈ = e ? p ­ ×rÌgÉ!×rÌ#ÉaÐ6ÌgzágÌ0!gg

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¹ X-Ë®!0!!ágÉ 0×rÌg .g!/. çK gM0×rÌg!0a!Ï VÈrÌgaXz! 0Ð!

º ü5² K = » « ª »c?f¤

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» ¬ · C» ¬ ¤ ð¤ K = å ª å+?.

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Page 20: Apuntes Calculo III - Fernando Chamizo

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Page 21: Apuntes Calculo III - Fernando Chamizo

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Page 22: Apuntes Calculo III - Fernando Chamizo

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Page 23: Apuntes Calculo III - Fernando Chamizo

Q ¨ Q Z ¨ « ·¸¶ « !0黧 Èì! « Á ¦ = å ª å ? ² !zË!!Æ×rÌg K !z rÈVÌA) g! !0È0Ìg9a!r0È0!!¾g60!6!9g0!!! ! ÷ 0 È0! = 6 0 !0 Ì ! !+Q!X<g¡6z0!ÐÌg? í ¤ = ÷ ©ª ÷ « ?Æ!rÈ g!! âö K = ^ ?G¤ Í = å+?Æ Í¡= »_?¤ K = » ? LígÉÈ!g

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Page 24: Apuntes Calculo III - Fernando Chamizo

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Page 36: Apuntes Calculo III - Fernando Chamizo

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!î!E ag g!g!& Äg KL © Ï!0È0ËD 0 Ìg0!gÌ!×rÌ rÌ6X& îg K! g!!!00£g60£×rÌg!r0È0 = K © ? Ç[Ãg!gÈ& 0È0!O×rÌO0 K Áö òK ®Ûý © !rÈ g!! K È!g¾ÌgÆr!+ägË®!0!!ág¾!¡Ìg¡!rÈ 0 ¨ ¤ Ä 0Lc÷ 0 K = Ä ?Mð¤ åAè>! ½-0É!Ì = KL © ? = »c?f¤ Z K = K © = »_??

Ç[È 0! >ËÍÌg!& är!000 aÐ 0 '!ä#0'&60ág! !9!! g60Ë®Ì! g! õ ® ý © g!Bg e = ÌáXÈ 0Ìg ?B! e LÇ0g! ËD 0 Ìgg60!0&0ÐÌÉ!Ìg agX!g 0É a0!ÐÉ!!g

= õ : õ © ?f¤ õ = õ © = d ?? :5 õ © = d ?¤ õ © = d ?¤ õ = õ © = d ?? © èo. 0O0¤0!0!rÈ0O g!& cg!!!00O×rÌg õ È!gÐOr!00 g ×r̤!% 0 ×rÌg¤!³!È0!0 arȼ á +¤6rÌgD9íÄ! ág60Ð ! % %á0!¢ g!Q!Q! gÏg a!g0& Ì 0×rÌgÏg õ e Áö e È!gÏ+!rÈ 0!B!0!!!rÈ0 !!0!!!rÈ !0ÈÌg'!+Ð >g!0#g ä !Ì +6á! '00! !rÈ0>!r0È0îr!+î! Ð-Ì á!È Ç0È0ä!XBg!gB!+ g60îg ág! = gË0¢ !a?È0 X aä¡ËÍÌg!& r!00Aîà³î Ìg!& 0!ªîË®á0!g¡! k Ìgî!!& rÈ0!È0#×rÌg£0!0Ìg!#Ál!!Ìg!& õ = ë ?I¤ k ! ½g.g6ágªl×rÌcg ág6!!0ÈÁ0!Ð6Ìg0Ø!ØXQ!g!g! k É Ì! ³

Ç[ 0ÈrÈ0¾g00È0 á0¾!LgÌrÈ 0 ! !L<600Ë rÈ0! C% !0!g!Lg!È0 0! ÄcË®Ìg!& Ár!+c !ÄËD 0 Ìg õ © = d ?°¤ õ = õ © = d ?? ©

Ä!Ìg6B0XÌ!ÉË! !rÈ0¾0!Ða¾!g!aÌg¾E0á! -×rÌ á a! á !r0È! 0 g!Ì0 ÄágR !!0Èo.< È!0%×rÌ0! a!VÈ õ © õ © ÈX!Ä0!rÈ0 X!!0È >áXÈ %Ç>ÌgÄ0!+ÌÈ06 ËÍÌgg6 !rÈ0 aXrÈ aÈ0! È0 ×rÌÌg°)Ì0È0g!6X& 0ÐÌ 0 = [ág0!6BË! !XÈ ?î0!!ÄËD 0 ÌgÄ×rÌg !ÌgÄÌgļ¢ g!Ä!¸ ág T'P!Ìg È0L!00!06#i)Ìg!9×rÌ'!0È 0Ì!g'0P¢ XÌ !¾ÌXÌ0 0!ÐÌg ! !L×rÌa0¾ gÌ!! 0 ! /. ×rÌg! ¡!T0Ð QX 0a0Ì ¾×rÌg¾Ì X0ÈÌg

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f Cz¡~z _C Lz L z|HzR z_|HzWz |Hz \Cz n +~ ÍC n\znzW z RCzz Rn Ñ Cz8z_|H z RnÍz z ~z n x z zJS C n z nC~ W |H <~ R n <~ Rn nW z z_ gS z_ Rn Rz z Czþ R|Hz~ G1~C|H z z_ T z nz n zzþnz ] n x fÃz |H C~nF¡ |Hz_~ø/ |HLz| ` R z]z nR|H z_0 ~z °~ _ \zW z zWz|HzWCR\CzQ n ~ #³~ CzWz_ z T n| z ;.\])(+" 6.)(+F¢V" K#: 9G'?ÍY/F

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rBprsleÁ¿Q_tÔê ÑiÒÝ õ Ù ùnúÚ e Áö e ë õ ®$ý © =ÃÙ ? îÑ¡× ãsá+æñ Ý+è;á+Ü]ÖØ×@Ò°àâÔÃÒæ¡Ò#×Ræ@ÝøÔaænÅ+ÒßÜÝçaáÕÝ+çþÕá+æàLÔaæ× Ý×6 Áö ñ ìêÖØ×ÒÝ õ = ë ? ÒÜ]ÔHænÅ ÒßbàâÔÃÞ çÃÒ°ãÝ ßÝhàâáÓRá ë ®Ûñ%î¹å æàâá+æ¡ÕÒÜCëQÓ@Ò$¡ÒbÕi$áë 6Å® ý © =; ? ì`ÜCÒÕ× è;ã@çÃÒçaÝË­á ßèI× çaÝ

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Ïaýgü! L= !M! a+? Ð XØ!Ï! Ìg !g0 = '!6+c!Ïág g³î!Ì ¸ ? ËÍ!0!g!áz0Ég0Ì!ága gÌg 6g!!Ìg6 !rÈ0É!M !!rÈÉ!0! !rÈ

j ¤ ¼ 6M= ~ · ç ? Á 6M= ~ ? Á õ =Ç6M= ~ ?? ©ç ¼

¼ ç ¼

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<X e ¤ 6M= ~ ? ¥ ¤ 6M= ~ · ç ? Á 6M= ~ ? !VÈ !!

j ¤ ¼ ¥ Á =; õ = e ?? © = õ = e ·¦¥ ? Á õ = e ?? ¼¼ õ = e ·¥ ? Á õ = e ? ¼

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j © ¤ ¼ ¥ ¼¼ õ = e ·¦¥ ? Á õ = e ? ¼ j « ¤ ¼ =Ç õ = e ?? © = õ = e ·¦¥ ? Á õ = e ?? Á ¥ ¼

¼ ¥ ¼

) rÈ0rÌg ¥ ö ^ !Ì çéö ^ L<Ég!0ÐÌg6

jÍ« Ì ¼RQ =; õ = e ?? © ¼RQ ¼ õ = e ·¥ ? Á õ = e ? Á õ = e ? ¥ ¼¼ ¥ ¼

ÃÄgËÍ!0!g!á¸g õM 00ÐÌ lk+ü¤£ jÍ« ¤ åA£rÌg! ÄgËÍX!g!á ÆSÈ å 0 ¼ ç ¼ 0Ìg!!rÈ0! a!VÈÉ!×rÌg 0ÉÈ!

j ©© ¤ ¼ õ = e ? ¥;· õ = e ·¦¥ ? Á õ = e ? Á õ = e ? ¥ ¼¼ ¥ ¼ « õ = e ? ¥

¼ ¥ ¼ Á Æ+

¥ Z ¼ ¥ ¼ !0È!g!!î ¸Ë® VÈ!0 î á Ìg ×rÌgî!äÌ !È !rÈ g!! õ = e ? ¥ Z ¼ ¥ ¼ !E! Ì ¡¢ N 0 6Ð-Ìg ¥ ¤N ð¤nå 0×rÌ õ = e ?v! 0ÐÌ

=n ! õ = e ?]¤ ¦ ^ ² ?X N Æ]¤ N Z î+ä!Ìg! jS© Ì Z N à È0rÈ lk+ü¤£ j ¤ lk+ü§£ jS©mjS« ¤nå ×rÌX&¢ 6 -! 0È0

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Page 41: Apuntes Calculo III - Fernando Chamizo

[ÉÌ!Ä .#!rÈ!g g6gg¤X0È>¼¢ g!6 ×rÌg¤+¤!0! 0Ì!á×rÌg¤o.> ÈX0rÈ!0 0!<0 ag! !rÈ+].!!îgä!0!0ág< ! <£!P! Ð!!0P!<Ï!rÈ0 ¡ ! ág -6g!!Ì

íg! K = ë ?W¤ ëGÁ õ = ë ? · k k ¼ o õ = ^ ?W¤ K ®7ý © !r0ÈÌgäá ÉÊb= ^ ? Ú Ù !Î×VÌ£!È õ = ë ? ð¤ ^ =; õ = ë ?! 0gÐÌ? ¼RQ K ¼RQ ¤ ¼RQ ^ Á õ ¼RQ XÈ0!×rÌ1 ×rÌ!0 gÐ6 ± ¼RQ K ¼RQ Ì Z AM á! ×rÌg K Xg60£!k ® É Êpob« = ^ ?vÌÉË®Ìg!& rÈ0!È# K ÉÊb= ^ ? Á¡ö ÉÊC= ^ ? ! v)! 6rÈ0!0 0!!6 !÷È0 0! aaX÷# a!g ! !0ÈaE!zË 0 aa!!È 0÷a!a r 0!g K8H áÈ0!!g

= S] ? ¼ K = Å ? Á K = ? ¼ Ìe

­ U ©¼YQ K =b[ ? ¼RQ¼ ÅøÁ ¼ Ë

¼ Å`Á¡ ¼

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É Êpob« = ^ ?!r0È0Ì Ìg ë ® ÉþÊC= ^ ?¤È[×rÌg ë ¤ ë Á õ = ë ? · k !+È ! õ = ë ?°¤ k í !0È .! % g0È0Ìg ÌgË®Ìg!& 6 É Êpob« = ^ ? Áö ÉþÊC= ^ ?Ï×rÌg¤XÌ õ =;6M= k ??¤ k g60È k ®ÅÉ Êob« = ^ ? äà¤+ÐÌ!rÈ0!!0Ð6Ìg6!0Ìg!áÁ×rÌ 6 ! rÈ0rÌg g.g!/. à³+c.gÈESLà³ag a!0!-0 ! aXrÈÉÈ06gÐ6Ìg-a+!Ð6Ìgga!-<6 Ð = A +?

¼ k © Á¡k « ¼r«g¼ 6M= k © ? Á 6M= k « ? ¼ Á ¼ 6M= k © ? Á õ =;6M= k © ?? Á 6M= k « ? Á õ =;6M= k « ?? ¼

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ÄÈ0! nígÌ Ð ×rÌ×à !0ȪÁg !È0aEX6gÁ ¯V Á¡ö 1 ní V = d ?0¤ e d ¤ = | ª ÷? !rÈ g!!z V a V z0Ì gË®!X! ß aÐ! g!0!!Ȫ6gÐÌ ì | ª | · -c| í O ì ÷ ª ÷ · -÷ í !Ég r 6g a!rÈ0a!Tg60! Ð0 Ú ý = Ã8?ÉÐ6!g!0 Ï!!È 0! ;© V = d ?-`|l ;« V = d ?-÷Evà³ aX 3 ! a ! !g6g!Ï!gÌ !X! 3 = ë ?9X9¾! !g6z¾g60È&¢ !̾×rÌg<!+ȾX ë LÇƾÌgÆg<È0! !3Ìg ×rÌg9+6 g!ÈX g6ªÉÌg Ìg agÉ Ë 0 aÉ g60!!P¢ ! !g0!!!& 3 = e ? íÌg gÐ6 -×rÌ !g0£!-Ì = !MX6 g!MÐÌ+? !rÈ g!!- 6+agÉ!0È0 Ì r!gÉ 0 !M0Ð -!M gÌ!È rÈ

3 = e ? µ ;© V = d ?ª-c| O ;« V = d ?ª-÷ ¤ 3 = V = d ?? µ ;© V = d ?`O ;« V = d ? -c|ã-÷O Î!È00! aXrÈ ` ¤ ;© V = d ?(O ;« V = d ?!³- 0 6D! e ×rÌ 0g!060! -0!!! ¾06ä¾ ×VÌÌgY1 0ä!ÐÈ& ( Xî QX!& ¡6VÈ!0 <g!ä×rÌXÌg !rÈ00OXcE¤g¤0 9íg³r !cXÌ0È %!×rÌ1 %0¤È0!g¤!cX³¼¢ È0¤×rÌ!Ì !

¤ 3 µ`

gÉXáQ! -!g

3 µ` ¤ 3 = V = | ª ÷?? µ ;© V = | ª ÷?`O ;« V = | ª ÷? À | À ÷

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K ¤ K =a¨ª¶ªmr ? À ¨ À ¶ À r

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K ¤ O 0OÆ!³ Ì !c !!³BÇ[Ïg!! +6áP¢ % 0¤!g¤!³ Ìg aX#à

A = à6?f¤

à³ È0rÈXÐ60!0-6g!É! ! Ì&¢ 6g aÈ!rÈ0!00!È06! g!Ð6 Ø*È0!6`¼!0X660! ( > Ð

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B =;ý ?¤ úù

¼ þ = »c? ¼ À » Ù6= Ã8?¤ ¼ © V = | ª ÷?`O « V = | ª ÷? ¼ À | À ÷

Ç Ð¢ ¤ ¤rÈ0!ÐQÏ Ìg a! 0%Ìg!%Q!g0[!¤ 60Ï!Ìg0#[a0Ì!0!! î!g0 = g!6P£0Ì!0! 0!0!!È0# !rÈ?< +ÈrÈzg >ÌgË®Ìg!& K T<É!0È0aË 0 a0ÉXÉarÈ0!Ð06ÉÌgaË®Ìg!& z!0X66 K 0Ð ý

ý K ¤ úùK = þ = »_?? ¼ þ = »c? ¼ À »/

Õ arÈ!Ð0MÉÌgËÍÌg!& zX0!6 K X£Ã

K ¤ K = V = | ª ÷?? ¼ ;© V = | ª ÷?`O ;« V = | ª ÷? ¼ À | À ÷

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á0! 9!Æg r a¾0!!!& Là³ 0 +ÈÌg6X& !Ì00<09rÈ0!Ð06X9¼¢ g!AdzÈ0áR !&Ìg£0Ð ×rÌg !!g OX!!& £!c×rÌ 0 0! 0O!Ìg& !Ìg6³r!g O ÏO0 aXÈ0aE!!& = ϼX ag Ð0#!g6c í¼!Ð ( È0ágR Ìl/./.> l0v 0È XÌ Ìg á1¼!È !6 î È È!g! ×rÌg¥¼ ág6 (!0 0 È006ágR g60ä!!#0 PÌrÈ g0Èg0Ð6Ìg!g 0 È00#!!È 0+? %à³!!X!& ÁgÌgÄg0!!!& ¸ 0 a = !¸+Ì!0!X? ÄÈgÐ6!rÈ0 = !¸!Ìg&?0 0!rÈ06! = ¾!rÈ0g! Ìg¡!g!& 0ÐÌ 0+?X¡Ç îg!È0! 0' 0!rÈ06! Ä×r̤0¤X×rÌ!0¤ !¤ cO 00!0 g!rȤrÌ!+È0Og60 !È0aE!!& 0 ! a!VÈÉ0É! ágÌz+Ð 6M0!0ÌgÈ06

, Æ+!Ð6Ìg á! a ½6- ËÍÌg !-×rÌÉÈ&¢ g! a!VÈ6-0 a!È00aE!! !- Æ *!!CE !Ìgá0!È !Ìg0# 0Ì!0!-×rÌ-g !È0aEX6³) )! ag g60 !È0aE!+*!& Ã_à © gz #þ = å ª ù ? Áö « Äþ = »_?\¤ = '» ª 0! »_?ÉÈ0!glà © Á ¦ = ª å+? ² Ð6!³ )L! ! !0È0!6 .#Ãg ág! î 0×rÌÄ aÐ!¸g!0gÊà © !¸Ì @*)ÌrÈ g 1 !cÌg !g0 !! ( !l!0!rÈ0 gO×rÌOg£ÐÌg6rÈ!Ð0 X = å ª ù ?×rÌgî! ì å ª ùí Ø>!VÈ!g! þâ !0È0 ÌÈ0 rÈ0!0# ß0îXÌá0]à © = Î÷ Ë®6 g!& Æ<g !È0aE!! Æ!Æ 9!rÈ00! ×r̳1 a!Æ ( ?XL<¾ 0 Ë 0 aV = å ªù ?O = å ª ù ? Áö 1 QV = q ª ?¤ = q 0! q ª 0!QÏ0! q ª q ?³0 a!È00aE!Wà « 0 aÈ0z!T¼ aX ßCÒÒædÔÃÕ$(6¶ ¤nåATè%M!rÈ0!g!0a ì å ªùòí O ì å ª ùí QXg! -r!!È0rg6ƾ g!& ÆÐ V ¤n¾!ÆÌgä ¼ÌVÈ g< !g¾!! ×rÌ!9ËÍ!0!rÈ0<;. 0agÉrÈ0!Ð

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[¼! B!×rÌ%È0!! [È0!B Ð!! Ìg0 ¤á+È0!!& ª g60¤g[ gErÌ!0È00-rÈÌg! X B

z° Wâ~ Rz |H S LC|Hz |H _z zg| `¾1 n#C °z n »szL% ½n Wzþ Q |Hz³~ |H Qn`n Gâzâz| z Ð]` Q³~ z_z_Wz Cz z| n8iÅBnnz Cz_ R zP z\ Lz nR z R|Hzx zR|Hz%LzAGBzLz zR zR|H CR |Hzx ô8z Rz W Cz z ÍnÍ~CzLzSCzS8zn C Q nR| d n »zLQz_ Qz<Í~CzL dzCz | nR\z \Í Ð\zC z#z _Cz R| Ã R _ T³~ d@~ ;H6+\](+" 6+)(+F¡¢r" K#: `¾'Ym/F

Ç¡¼! d-6!Ì

3 µ6` g 3 =a¨8ª¶ªmr ?f¤ =a¨ « ·¸¶ « ·ur « ? 1 ob« =a¨8ª¶ªmr ?

ÁÃß!ÏO0Ì!0! !+Ë&!0! = !!rÈ0!£! 0Ð!@?Ï 0 £ 0!rÈ! cO 0 aM!rÈ0!0

Av Ä6X¤ Ë 0 agø.Ä Ì!& iÄ0 !0Ä aX aXP6g!Ä0!ÐÌg ½ 0ÐE0Ì+È0% 0È06g!rÈ!>!È06g0áQ!%rÈ!0gXÈ% á! a TÇz á -! -> ÈX0×rÌgÉ0gÌ!gX0!0á- 6-á0!! !rÈ 3 = ë ?¤ ë[Z ¼ ë ¼ 1

í , , 0 zßg60 !È0aE!!& nÈ0Ì0ì z6£ r g!g6£!0Ë&X!6£ X ¤ ] !g!!cV = q ª s?g¤ = c+! q ª 0!Qc0! q ª q ? , ìPXÌ 0Ì!ág ×rÌg ;© V O ;« V !

h 6 ç c q 0Xc q Á 0! q

Á 0Xc0! q c+! q å¤ « 0! q = c0! q ª 0!Qc0! q ª q ?N

ÇË!L ag ág = !¼¢ ¼0¾ <)! Áq ¤ ¤ ù Z +?<×r̾ä 0 6L!<ä!rÈ!0 ¤!³!rÌg! Xg ág! A ¹ ÈX0¤×rÌ ¼ V = q ª ? ¼ ¤ ] 0×r̤0 a!È00aE!Õà !Ì 3 = V¸?f¤V Z 1 M0Ì0È0È0Ì-!zaËD Ì

3 µ` ¤«ú²

ú²

1 V = q ª ? µ « +! q = c0! q ª 0!Qc0! q ª q ? À q À ª

×rÌg.g!! -L6!Ì -!#

3 µ` ¤«ú²

ú² 0! q À q À g¤nÓ ù

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!0Ë&!0!!!rÈ0ga!£! Ð6! ! ½ ¼ 3 = ë ? ¼ ¤ ¼ ë ¼ Z ¼ ë ¼ 1 ¤ Z « -ÈrÈ

3 µ` ¤ ¼ 3 ¼¼ ` ¼ ¤ « ¼ ` ¼ ¤

« Ù°= Ã8?f¤ « µ Ó ù « ¤nÓ ù

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Ç¡¼! ì<! 0È0×rÌ£09X ¤ K = q ?X£ÌgÃ!Ì0&Ã!Î 0! K ®ý © å#Ë q Ë ù !VÈ !!-+Ìz ÐÈ0ÌzV!Ég6g

B =;ý ?¤«ú² = K = q ?? « ·Å= K = q ?? « À q

à³ßg !È0aEX6! n 0!+ g!rÈ0 ! !0ÈX0z! Ð06!6£ßzËD 0 Ìg ¨ ¤X[ q1 ¶ ¤AX[0! q

þ = q ?¤ = K = q ? q ª K = q ?g0! q ? ª È06VÈ

B =Çý ?¤«ú² ¼ þ = »_? ¼ À »S¤

«ú² = K = q ? q Á K = q ?g0! q ? « ·9= K = q ?g0! q · K = q ?g q ? « À q

Õ !0gÌ-!-g !0-0É!Ð6M0!+ÌÈ06 !0!

Ç¡¼! !Ì6>rÈ0!Ð06Eg¼¢ g! õ =H¨ª¶ ?¤ = ¨ « · ¶ « ª ¨s¶ ?> 60Ð xà ©0! 0!z!M+!rÈ 0È0 = rÈ00 a6-ÐÌR)-!M0! x?

£ag !È0aE!! árr_þ = »_?f¤ = '» ª +! »_?-0ÉÈ!

ý 3 ¤«ú² = v

« » · 0! « » ª v0!S»Ï Ø»_? µ = Á 0!S» ª '»_? À »/

Ç Ìg¡ËD!S.![# ¤gO!+ÈcrÈ!Ð0 .g!! P!Ìg O!Ìg 0¿.!!0 È06g ×rÌg>!rÈ!Ð0 !9 ag' 0 = BQ!0 g!g@?'rÈXÐ60'! ì å ª ùí

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×rÌgX ì Á ù ªùòí TÇz!Ìg×rÌ! X6 0 ágÈ!g

ý 3 ¤når

Ç¡¼! d-6!Ì-!0! = ÈXa?-!M r ¤ ¨ « ·¸¶ « å#Ë r Ë ) g! -g60 !È0aE!-!0È0a0Ìg!g!É

V = X ª q ?f¤ = X[ q ª X[0! q ª X'? å#ËØX%Ë ª q ®ì å ª ùí , zP!Ìg g0Ì!ág

;© V O ;« V ¤h 6 ç

q 0! q Á X[0! q X[ q å

¤ = Á X[ q ª Á X[0! q ª XØ?.

) È0rÈ

Ù°= Ã8?¤«ú²

©² X « « q · X « 0! « q · X « À X À q ¤ ½

«ú²

©² X À X À q ¤ ù ½ r

è ¤ ag ág6X& c ! %g !!!%Ð6 Ø!È0!6 !rÈ0¤ Èg ×rÌg¤6³g!0 ! g!vXrÌg! ágO! = 0Èg 0ÌlÐ!!0È0aEb? 0 áÈ0!gÌgÃ+!!È !0!Ìg'g<g X#¤ ½ « · « ¤ ½ Ȫ×rÌg< gÐ6ÈÌgä<0Ìä0 !9 ù Lè%0ÐÌ !rÈ Ð a!È0&¢ !! !rÈMg0Ì!á£×rÌÉ0̺0!! ù ½ A

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3 µ6` ¤ ý +Ç+ 3 µ`

g ýW©_© !>!0 ¨ ¤ Ç+ÈgÌ!g0!%g !È0aEX6g V = | ª ÷?Iµö = ª | ª ÷? ×rÌgO ! ` ¤ ;© V O ;« V ¤ = ª å ª å ? = !0È !Ð6 Ø!È0! a!rÈ0O& Ð ? è%[0!¤

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0 6 gÌ!0È0a×rÌÉ -g!!M!rÌg! 0!0ÌgÈ0

3 µ` ¤ Á©²

©² = ª å ª å ? µ = ª å ª å ? À | À ÷;¤ Á

à !Ì g¤rÌg! 0Ìg! ágÈ!g!%X!È !rÈ0Ä >×r̤ 0%!³gáÌ 00!0 @*!rÈD

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Page 163: Apuntes Calculo III - Fernando Chamizo

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Page 164: Apuntes Calculo III - Fernando Chamizo

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Page 165: Apuntes Calculo III - Fernando Chamizo

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Page 166: Apuntes Calculo III - Fernando Chamizo

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Page 167: Apuntes Calculo III - Fernando Chamizo

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Page 168: Apuntes Calculo III - Fernando Chamizo

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Page 175: Apuntes Calculo III - Fernando Chamizo
Page 176: Apuntes Calculo III - Fernando Chamizo