The GFSBODONI fot fmily Atois Tsolomitis Lbortory of Digitl Typogrphy d Mthemticl Softwre Deprtmet of Mthemtics Uiversity of the Aege 9 Mrch 2006 Itroductio The Bodoi fmily of the Greek Fot Society ws mde vilble for free i utum 2005. This fot existed with commercil licese for my yers before. Support for LTeX d the bbel pckge ws prepred severl yers go by the uthor d I. Vsilogiorgkis. With the free vilbility of the fots I hve modified the origil pckge so tht it reflects the chges occured i the ltest releses by GFS. The pckge supports three ecodigs: OT, T d LGR to the exted tht the fot themselves cover these. OT d LGR should be firly complete. The greek prt is to be used with the greek optio of the Bbel pckge. The fots re loded with \usepckge{gfsdidot} or\usepckge[defult]{gfsdidot}. Th ltter oe sets Bodoi s the defult documet fot. The former defies the evirometbodoi d the commd\textbodoi. For exmple, while i Greek lguge \textbodoi{dokim h} produces δοκιμή. x height is djusted so tht it mtches with the x height of thetxfots pckge. This is doe to help with documets requirig mthemtics. I this cse lodtxfots but beforegfsbodoi.sty. 2 Istlltio Copy the cotets of the subdirectory fm i texmf/fots/fm/gfs/bodoi/ Copy the cotets of the subdirectory doc i texmf/doc/ltex/gfs/bodoi/
Copy the cotets of the subdirectory ec i texmf/fots/ec/dvips/gfs/bodoi/ Copy the cotets of the subdirectory mp i texmf/fots/mp/dvips/gfs/bodoi/ Copy the cotets of the subdirectory tex i texmf/tex/ltex/gfs/bodoi/ Copy the cotets of the subdirectory tfm i texmf/fots/tfm/gfs/bodoi/ Copy the cotets of the subdirectory type i texmf/fots/type/gfs/bodoi/ Copy the cotets of the subdirectory vf i texmf/fots/vf/gfs/bodoi/ I your istlltio s updmp.cfg file dd the lie Mp gfsbodoi.mp Refresh your fileme dtbse d the mp file dtbse (for exmple, o Uix systems ru mktexlsr d the ru the updmp sys (or updmp o older systems) script s root). You re ow redy to use the fots. 3 Usge As sid i the itroductio the pckge covers both eglish d greek. Greek covers polytoic too through bbel (red the documettio of the bbel pckge d its greek optio). For exmple, the premple \documetclss{rticle} \usepckge[eglish,greek]{bbel} \usepckge[iso-8859-7]{iputec} \usepckge[defult]{gfsbodoi} will be the correct setup for rticles i Greek usig Bodoi for the mi fot. 3. Trsformtios bydvips Other th the shpes provided by the fots themselves, this pckge provides slted smll cps shpe usig the stdrd mechism provided by dvips. Get slted smll cps with\scslshpe. For exmple, the code \textsc{smll cps \textgreek{pezokefl i} 023456789}{\scslshpe \textgreek{pezokefl i 023456789}} will give SMALL CAPS 023456789 023456789 The commd\textscsl{} re lso provided. 2
3.2 Tbulr umbers Tbulr umbers (of fixed width) re ccessed with the commd\tbums{}. Compre 0 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 \tbums{ 0 2 3 4 5 6 7 8 9 } 0 2 3 4 5 6 7 8 9 3.3 Text frctios Text frctios re composed usig the lower d upper umerls provided by the fots, d re ccessed with the commd\textfrc{}{}. For exmple,\textfrc{-22}{7} gives -²² ₇. Precomposed frctios re provided too by\oehlf,\oethird, etc. 3.4 Additiol chrcters \textbullet \textprgrph \creof \umero \estimted \textlozege \eurocurrecy \textdgger \textdggerdbl \yecurrecy Euro is lso vilble i LGR ecodig.\textgreek{\euro} gives. 4 Problems The ccets of the cpitl letters should hg i the left mrgi whe such letter strts lie. T E X d LAT E X do ot provide the tools for such feture. However, this seems to be possible with pdft E X As this is work i progress, plese be ptiet... 5 Smples The ext two pges provide smples i eglish d greek with mth. 3
Addig up these iequlities with respect to i, we get c i d i p + q = () sice c p i = d q i =. I the cse p=q=2 the bove iequlity is lso clled the Cuchy Schwrtz iequlity. Notice, lso, tht by formlly defiig ( b k q ) /q to be sup b k for q=, we give sese to (9) for ll p. A similr iequlity is true for fuctios isted of sequeces with the sums beig substituted by itegrls. Theorem Let < p< d let q be such tht /p+/q=. The, for ll fuctios f, g o itervl [, b] such tht the itegrls f (t) p dt, g(t) q dt d f (t)g(t) dt exist (s Riem itegrls), we hve ( ) /p ( /q f (t)g(t) dt f (t) p dt g(t) dt) q. (2) Notice tht if the Riem itegrl f (t)g(t) dt lso exists, the from the iequlity b f (t)g(t) dt f (t)g(t) dt follows tht ( f (t)g(t) dt ) /p ( /q f (t) p dt g(t) dt) q. (3) Proof: Cosider prtitio of the itervl [, b] i equl subitervls with edpoits = x 0 < x < < x = b. Let x=(b )/. We hve f (x i )g(x i ) x i= = f (x i )g(x i ) ( x) p + q i= ( f (x i ) p x) /p ( g(x i ) q x) /q. (4) i= 4
Εμβαδόν επιφάνειας από περιστροφή Πρόταση 5. Εστω γ καμπύλη με παραμετρική εξίσωση x = g(t), y = f (t), t [, b] αν g, f συνεχείς στο [, b] τότε το εμβαδόν από περιστροφή τηςγ γύρω από τον xx δίνεται B=2π f (t) g (t) 2 + f (t 2 )dt. Αν ηγδίνεται από την y= f (x), x [, b] τότε B=2π f (t) + f (x) 2 dx Ογκος στερεών από περιστροφή Εστω f : [, b] R συνεχής και R={ f, Ox, x=, x=b} είναι ο όγκος από περιστροφή του γραφήματος της f γύρω από τον Ox μεταξύ των ευθειών x=, και x=b, τότε V=π f (x)2 dx Αν f, g : [, b] R και 0 g(x) f (x) τότε ο όγκος στερεού που παράγεται από περιστροφή των γραφημάτων των f και g, R={ f, g, Ox, x=, x=b} είναι V=π { f (x)2 g(x) 2 }dx. Αν x=g(t), y= f (t), t=[t, t 2 ] τότε V=π t 2 { f (t) 2 g (t)}dt για g(t t )=, g(t 2 )=b. 6 Ασκήσεις Άσκηση 6. Να εκφραστεί το παρακάτω όριο ως ολοκλήρωμα Riem κατάλληλης συνάρτησης lim k= e k Υπόδειξη: Πρέπει να σκεφτούμε μια συνάρτηση της οποίας γνωρίζουμε ότι υπάρχει το ολοκλήρωμα. Τότε παίρνουμε μια διαμέριση P και δείχνουμε π.χ. ότι το U( f, P ) είναι η ζητούμενη σειρά. Λύση: Πρέπει να σκεφτούμε μια συνάρτηση της οποίας γνωρίζουμε ότι υπάρχει το ολοκλήρωμα. Τότε παίρνουμε μια διαμέριση P και δείχνουμε π.χ. ότι το U( f, P ) είναι η ζητούμενη σειρά. Εχουμε ότι k= ek = e+ e2 + + e = e + e 2 + + e 5