Math Notebook for Students

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1 Mth Notebook for Stuets 50 Essetil Mthemticl Formuls Equtios b Peter I. Ktt Petr Books

2 Mth Notebook for Stuets Peter I. Ktt, PhD Correspoece bout this book m be set to the uthor t oe of the followig two emil resses: pktt@lumi.lsu.eu pktt@tet.et.jo The uthor ckowleges the work of Aubislivess which ppers i the imge of Pi o the frot bck covers of this book. Mth Notebook for Stuets: 50 Essetil Mthemticl Formuls Equtios. Writte b Peter I. Ktt. ISBN: 4407 ISBN-: All rights reserve. No prt of this book m be copie or reprouce without writte permissio of the uthor or publisher. 009 Peter I. Ktt

3 Mth Notebook for Stuets To m prets, brothers, sisters

4 Mth Notebook for Stuets 4 Mth Notebook for Stuets 50 Essetil Mthemticl Formuls Equtios

5 Mth Notebook for Stuets 5 Prefce This is little book for stuets who wish to hve the essetil formuls equtios of mthemtics i sigle esil ccessible source. I bout 50 pges, the 50 most essetil mthemticl formuls re liste. Ulike other lrge books o this topic, there is o ee to go through hures of pges thouss of formuls for the stuet to get the bsic equtios. The uthor hs serche severl books o mthemticl formuls tbles selecte ol those equtios which re essetil to the stuet. The mthemticl formuls equtios liste i this book re useful for stuets reserchers i vrious fiels icluig mthemtics, phsics, egieerig, etc. Ol the most elemetr bsic topics re covere icluig formuls for vrious geometric shpes, severl tpes of fuctios (trigoometric, hperbolic, epoetil, logrithmic, etc), the qurtic equtio, ltic geometr, erivtives itegrls, rithmetic series, geometric series, Tlor series. A comprehesive referece list is iclue t the e of the book i itio to umerous web liks to more formuls, equtios, tbles. The uthor hs ecie gist icluig umericl tbles for itegrtio, logrithms, etc becuse the use of such tbles hs bee supersee b the vilbilit of scietific clcultors computer lgebr progrms. Thus, ecisio hs bee reche to keep the book i compct formt tht iclues ol the most essetil bsic formuls. It is hope tht the uthor hs succeee i this regr. Etreme cre hs bee tke to esure the ccurc of the formuls tht re liste i this book. It is hope tht this little book will prove to be vluble source of

6 Mth Notebook for Stuets 6 iformtio for stuets i mthemtics, sciece, egieerig. Fill, the uthor wishes to ckowlege the help support of his fmil members without which he woul ot hve bee ble to prouce this book i its preset form. Peter I. Ktt Mrch 009

7 Mth Notebook for Stuets 7 Cotets Specil Costts 9 Specil Proucts Fctors 9 Biomil Formul Biomil Coefficiets 0 4 Geometric Formuls 5 Trigoometric Fuctios 6 Comple Numbers 8 7 Epoetil Logrithmic Fuctios 0 8 Hperbolic Fuctios 4 9 The Qurtic Equtio 5 0 Ple Altic Geometr 6 Soli Altic Geometr 8 Derivtives Iefiite Itegrls 4 4 Defiite Itegrls 47 5 Series 50 6 Tlor Series 5 7 Iequlities 5

8 Mth Notebook for Stuets 8 8 Coversio Fctors 54 9 List of Prime Numbers 55

9 Mth Notebook for Stuets 9. Specil Costts π.459 Rtio of perimeter of circle to its imeter e.788 Nturl bse of logrithms γ Euler s costt.44 Squre root of.705 Squre root of Squre root of 5. Specil Proucts Fctors ( ) ( ) ( ) ( ) ( ) ( ) 4 6 4

10 Mth Notebook for Stuets 0 ( )( ) ( )( ) i i Fctors re comple umbers (cot be fctore with rel fctors) ( )( ) ( )( ) ( )( ) ( )( )( ) ( )( )( )( ) i i 4 4 The lst lie bove iclues comple fctors.. Biomil Formul Biomil Coefficiets ( )... where the biomil coefficiets re give b!( )!! k k k ( ) 4! for iteger,! 0. Note tht k k

11 Mth Notebook for Stuets 4. Geometric Formuls Ares of Commo Two-Dimesiol Shpes Shpe Are Squre of sie Are Rectgle of legth Are b with b Trigle of ltitue h bse b Are bh Trigle of sies, b, Are s s s b s c c where s ( b c) Trpezoi of ltitue h Are h b prllel sies b ( ) Circle of rius r Are π r ( )( )( ) Circle of imeter Are π 4 Sector of circle with rius r gle θ Are r θ, Ellipse of semi-mjor is Are π b semi-mior-is b θ i ris Perimeters of Commo Two-Dimesiol Shpes Shpe Perimeter Squre of sie Perimeter 4 Rectgle of legth Perimeter ( b) with b Trigle of sies, Perimeter b c b, c Circle of rius r Perimeter π r

12 Mth Notebook for Stuets Circle of imeter Sector of circle with rius r gle θ Perimeter π Arc Legth s rθ, θ i ris Volumes of Commo Three-Dimesiol Solis Soli Volume Cube of sie Volume Rectgulr prllelepipe of Volume bc legth, with b, height c Sphere of rius r 4 Volume π r Right circulr clier of Volume π r h rius r height h Right circulr coe of rius r height h Volume π r h Prmi of bse re A Volume Ah height h Ellipsoi of semi-es, b, 4 c Volume π bc Surfce Ares of Commo Three-Dimesiol Solis Soli Surfce Are Cube of sie Surfce Are 6 Rectgulr Surfce Are b c bc prllelepipe of legth, with b, height c Sphere of rius r Surfce Are 4π r Right circulr Lterl Surfce Are π r h clier of rius r height h ( )

13 Mth Notebook for Stuets Right circulr coe of rius r height h Lterl Surfce Are π r r h 5. Trigoometric Fuctios Cosier right trigle with gle θ sies,, r. si θ r cos θ r t θ opposite hpoteuse jcet hpoteuse opposite jcet θ r For gles, we hve the followig reltios: π ris 60 o π ris 80 o 80 ris π o o π 80 o ris

14 Mth Notebook for Stuets 4 We hve lso the followig trigoometric ietities: siθ t θ cosθ cosθ cot θ siθ tθ sec θ cosθ csc θ siθ si θ cos θ sec θ t csc θ cot θ θ We hve the followig results for some commo gles: θ (egrees) θ (ris) si θ cos θ o o o 45 o 60 o 90 π 6 π 4 π π 0

15 Mth Notebook for Stuets 5 Formuls for the egtive of gle si ( θ ) siθ ( θ ) cosθ cos t ( θ ) tθ Co-fuctio formuls π si θ cosθ π cos θ siθ Formuls for sums iffereces of two gles ( α β ) siα cos β cosα si β si si cos ( α β ) siα cos β cosα si β ( α β ) cosα cos β siα si β ( α β ) cosα cos β siα si β cos t t ( α β ) ( α β ) tα t β tα t β tα t β tα t β

16 Mth Notebook for Stuets 6 Formuls for ouble gles ( θ ) siθ cosθ si cosθ cos θ si θ si θ cos θ t tθ t θ θ Formuls for hlf gles θ si ± cosθ θ cosθ cos ± θ cosθ t ± cosθ siθ cosθ cosθ siθ cscθ cotθ

17 Mth Notebook for Stuets 7 Squre of the sie cosie fuctios cos si θ cos cos θ ( θ ) ( θ ) Sums, iffereces, proucts of the sie cosie fuctios α β α siα si β si cos α β α siα si β cos si α β α cosα cos β cos cos α β α cosα cos β si si cos siα si β cos cosα cos β si siα cos β ( α β ) cos( α β ) ( α β ) cos( α β ) ( α β ) si( α β ) β β β β

18 Mth Notebook for Stuets 8 Lw of Sies b si A si B c sic b C Lw of Cosies A c B c b bcosc 6. Comple Numbers i, i bi is the geerl form of comple umber where b re rel umbers. bi c i if ol if c b. Aitio subtrctio of comple umbers ( bi) ( c i) ( c) ( b )i ( bi) ( c i) ( c) ( b )i Multiplictio ivisio of comple umbers ( bi)( c i) ( c b ) ( bc)i

19 Mth Notebook for Stuets 9 bi c i bi c i c i c i c b bc i c c ( c b ) ( bc ) c i Polr form of comple umber ( cosθ isiθ ) bi r where (,b) r b r θ t b θ Multiplictio ivisio of comple umbers i polr form ( cosθ i θ ) z r si ( cosθ i θ ) z z r si [ cos( θ θ ) ( θ θ )] z r r isi z z r [ cos( θ θ ) ( θ θ )] isi r Powers of comple umbers ( cosθ isiθ ) z r

20 Mth Notebook for Stuets 0 z r [ cos ( θ ) isi( θ )] Roots of comple umbers ( cosθ isiθ ) z r θ kπ θ kπ z z r cos isi where k is iteger. 7. Epoetil Logrithmic Fuctios Rules for epoets... ( times),, re rel umbers. m m, m,, re rel umbers. m m m m ( ) 0, 0, 0 ( )

21 Mth Notebook for Stuets m m Defiitio of logrithms log if ol if, Properties of logrithms log log 0 ( ) log log log log log ( ) log log log Chge of bse for logrithms log logb,, b log b

22 Mth Notebook for Stuets log,, log Specil logrithms log0 log loge log e 0.44 l 0 l l log log0 l Other formuls θ e i cosθ isiθ θ e i cosθ isiθ Reltio betwee epoetil trigoometric fuctios e siθ iθ e cosθ iθ e e iθ iθ Perioicit of the epoetil fuctio ( θ kπ ) iθ i e e, k is iteger More o the polr form of comple umbers iθ ( cos θ i θ ) re z i r si

23 Mth Notebook for Stuets Multiplictio ivisio of comple umbers i polr form z re iθ z r e iθ z z z z r r e i ( θ θ ) ( θ ) i θ r e r Powers roots of comple umbers i polr form iθ z re iθ z r e, is rel umber z r e θ kπ i,, k re itegers Logrithms of comple umbers i polr form iθ ( re ) l r iθ kπi l z l, k is iteger

24 Mth Notebook for Stuets 4 8. Hperbolic Fuctios e sih e cosh e e e e sih th e e cosh e e cosh coth e e sih th sec h e e csc h e e cosh sech coth sih th csch cosh sih Reltio betwee hperbolic trigoometric fuctios ( i) isih si ( i) cosh cos

25 Mth Notebook for Stuets 5 ( i) i th t ( i) isi sih ( i) cos cosh ( i) i t th 9. The Qurtic Equtio The solutio of the qurtic equtio b c 0 is give b the qurtic formul s follows (, b, c re rel) b ± b 4c D b 4c is clle the iscrimit. We hve the followig three cses:. D > 0 implies there re two rel but uequl solutios.. D 0 implies there re two rel equl solutios.. D < 0 implies there re two comple cojugte solutios.

26 Mth Notebook for Stuets 6 0. Ple Altic Geometr The istce betwee two poits with coorites (, ) ( ) is give b, ( ) ( ) The slope of the lie joiig two poits with coorites (, ) (, ) is give b m tθ where θ is the gle of iclitio of the lie. The equtio of the lie joiig two poits with coorites (, ) (, ) is give b The equtio of the lie with slope m pssig through poit with coorites (, ) is give b m ( ) The equtio of the lie with slope m -itercept b is give b m b

27 Mth Notebook for Stuets 7 The equtio of the lie with -itercept -itercept b is give b, 0, b 0 b The geerl equtio of lie is give b A B C 0 where A, B, C re costts., to the lie give b the equtio A B C 0 is give b The istce from poit with coorites ( ) A B C ± A B The gle θ betwee two lies with slopes m m is give b θ t m m mm The re A of trigle with vertices t the three poits with coorites (, ), (, ), ( ), is give b A ± where the plus or mius sig is chose to give positive re.

28 Mth Notebook for Stuets 8 Polr coorites: The followig re the reltios betwee polr r,θ rectgulr (Crtesi) coorites coorites ( ) (, ) r cosθ r siθ or ltertivel r θ t The equtio of circle with rius R ceter t the poit is give b with coorites ( ) 0, 0 ( ) ( ) 0 0 R. Soli Altic Geometr The istce betwee two poits with coorites, z, z is give b ( ) ( ),, ( ) ( ) ( z ) z The irectio cosies l, m, of lie pssig through two poits with coorites (,, z ) (,, z ) re give b

29 Mth Notebook for Stuets 9 cosα l cos β m cosγ z z where is the istce betwee the two poits α, β, γ re the gles of iclitio of the lie with the positive,, z es, respectivel. We lso hve the followig reltio betwee the irectio cosies: cos α cos β cos γ or equivletl l m The equtios of lie pssig through two poits with coorites (,, z ) (,, z ) re give s follows i str form z z z z or ltertivel l m z z

30 Mth Notebook for Stuets 0 The equtios of lie pssig through two poits with coorites (,, z ) (,, z ) re give s follows i prmetric form lt mt z z t where t is the prmeter, l, m, re the irectio cosies. The geerl equtio of ple is give b A B Cz D 0 where A, B, C, D re costts. The equtio of ple pssig through three poits with coorites (,, z ), (,, z ), (,, z) is give b z z z z z z 0 The equtio of lie with -itercept, -itercept b, z-itercept c is give b z, 0, b 0, c 0 b c

31 Mth Notebook for Stuets The equtios of lie pssig through poit with coorites ( 0, 0, z0 ) perpeiculr to the ple A B Cz D 0 is give b i str form A B z z C The equtios of lie pssig through poit with coorites ( 0, 0, z0 ) perpeiculr to the ple A B Cz D 0 is give b i prmetric form z z At Bt Ct The istce from poit with coorites (, z ) the ple A B Cz D 0 is give b A ± 0 B A 0 Cz B 0 C Cliricl coorites ( r,θ, z) z D to 0 0, r cosθ r siθ θ z z r where (,, z) re the rectgulr (Crtesi) coorites. Altertivel, we hve 0 z (r, θ, z)

32 Mth Notebook for Stuets z z r t θ Sphericl coorites ( ),θ,ϕ r θ ϕ θ ϕ θ cos si si cos si r z r r where ( ) z,, re the rectgulr (Crtesi) coorites. Altertivel, we hve cos t z z z r θ ϕ The equtio of sphere with rius R ceter t poit with coorites ( ) 0 0 0,, z is give b ( ) ( ) ( ) R z z φ z r θ (r, θ, φ)

33 Mth Notebook for Stuets. Derivtives Defiitio of the erivtive of fuctio. Let f ( ) b be give fuctio. The its erivtive is give f ( ) lim h 0 f ( h) f ( ) h If we let h Δ, the the bove efiitio c be writte equivletl s follows f ( ) lim Δ 0 f ( Δ) f ( ) Δ Rules of ifferetitio I the followig rules of ifferetitio, we ssume c costt, f v g () c 0 ( ) ( c) c costt, u ( ), ( )

34 Mth Notebook for Stuets 4 ( ) ( c ) c ( u v) u v or equivletl ( f ( ) g( ) ) f ( ) g ( ) ( u v) u v or equivletl ( f ( ) g( ) ) f ( ) g ( ) ( cu) u c or equivletl ( cf ( ) ) cf ( ) ( uv) v u v u or equivletl ( f ( ) g( ) ) f ( ) g ( ) g( ) f ( ) u v u v v u, v 0 v

35 Mth Notebook for Stuets 5 or equivletl g ( ) 0 f g ( ) ( ) g ( ) f ( ) f ( ) g ( ) ( g( ) ), ( u ) u u or equivletl ( f ( ) ) Chi rule u u ( ) f ( ) ( ) f ( ) or equivletl ( f ( g( ) )) f ( g( ) ) g ( ) u u u u Derivtives of trigoometric fuctios ( si ) cos ( cos ) si

36 Mth Notebook for Stuets 6 ( t ) sec ( cot ) csc ( sec ) sec t ( csc ) csc cot ( u) si cosu u or equivletl ( si ( f ( ) )) cos( f ( ) ) f ( ) ( u) cos siu u or equivletl ( cos( f ( ) )) si( f ( ) ) f ( ) ( u) t sec u u or equivletl ( t ( f ( ) )) sec ( f ( ) ) f ( ) ( u) cot csc u u or equivletl ( cot( f ( ) )) csc ( f ( ) ) f ( )

37 Mth Notebook for Stuets 7 ( u) sec secu tu u or equivletl sec f sec ( u) ( ( ( ))) ( f ( ) ) t( f ( ) ) f ( ) csc cscu cot u u or equivletl csc f csc ( ( ( ))) ( f ( ) ) cot( f ( ) ) f ( ) Derivtives of epoetil fuctios ( e ) e ( ) l u ( e ), costt e u u or equivletl f ( ) f ( ) ( e ) e f ( ) u ( ) u u l, costt or equivletl f ( ) f ( ) ( ) l f ( )

38 Mth Notebook for Stuets 8 Derivtives of logrithmic fuctios ( l ), 0 log e ( log ), 0, u ( l u), u 0 u or equivletl ( ( f ( ) )) f ( ) 0 log u e u ( log u), u 0 or equivletl ( ( f ( ) )) f ( ) 0, 0, ( ) ( ) l f, f log e log f ( ), f ( ) Derivtives of hperbolic fuctios ( sih ) cosh ( cosh ) sih ( th ) sech

39 Mth Notebook for Stuets 9 ( coth ) csch ( sech ) sech th ( csch ) csch coth ( u) sih cosh u u or equivletl ( sih ( f ( ) )) cosh( f ( ) ) f ( ) ( u) cosh sihu u or equivletl ( cosh ( f ( ) )) sih( f ( ) ) f ( ) ( u) th sech u u or equivletl ( th ( f ( ) )) sech ( f ( ) ) f ( ) u ( coth u) csch u coth f csch or equivletl ( ( ( ))) ( f ( ) ) f ( ) ( hu) sec sechu thu u

40 Mth Notebook for Stuets 40 or equivletl sech f sech ( hu) ( ( ( ))) ( f ( ) ) th( f ( ) ) f ( ) csc cschu cothu u or equivletl csch f csch ( ( ( ))) ( f ( ) ) coth( f ( ) ) f ( ) Higher erivtives f ( ) Seco erivtive f ( ) Thir erivtive iv 4 4 f iv ( ) Fourth erivtive Differetils f ( ) Rules for ifferetils ( u v) u v ( u v) u v

41 Mth Notebook for Stuets 4 ( uv) u v v u u v u u v, v 0 v v ( u ) u u ( si u) cosu u ( cosu) siu u ( t u) sec u u Prtil erivtives Defiitio of prtil erivtives f f lim 0 Δ lim 0 Δ f f ( Δ, ) f (, ) Δ (, Δ) f (, ) Δ Seco prtil erivtives f f f f

42 Mth Notebook for Stuets 4 f f f f The ifferetil of fuctio f f f. Iefiite Itegrls I the followig formuls, ote tht the costt of itegrtio is ot show. Furthermore, ote tht costt costt. l, ( ) f ( ) f

43 Mth Notebook for Stuets 4 ( f ( ) g( ) ) f ( ) g( ) ( f ( ) g( ) ) f ( ) g( ) Itegrtio b Prts usig this ottio u f, v g, u f, v g ( ( ) ( ) ( ) ( ) u v uv v u ) Other itegrtio formuls f w ( ) f ( w) ( f ( ) ) ( w) ( ) g f f g w, w ( ) w w w, w l w w Iefiite itegrls of epoetil fuctios e w w e w w w w l, > 0,

44 Mth Notebook for Stuets 44 Iefiite itegrls of trigoometric fuctios si ww cos w cos w w si w ( cos ) t ww l w ( si ) cot ww l w ( sec w t ) sec ww l w ( csc w cot ) csc ww l w sec w w t w csc ww cot w t cot si ww ww ww t w w cot w w w ( w si wcos ) w cos w w sec wt ww secw ( w si wcos )

45 Mth Notebook for Stuets 45 csc wcot ww cscw si ( ) cos( ) cos ( ) si( ) Iefiite itegrls of hperbolic fuctios sih w w cosh w cosh w w sih w ( cosh ) th ww l w ( sih ) coth ww l w sec h ww si w ( th w) t ( e ) w csc h ww l th coth sech ww th w csch ww coth w th ww w th w w ( e )

46 Mth Notebook for Stuets 46 coth sih ww ww w coth w w ( sih wcosh w ) cosh ww w ( sih wcosh w ) sec h wth ww sech w csc h wcoth ww csch w Itegrtio b substitutio F F w ( b) F( w) ( b ) wf( w) w, w b, w b F ( ) F( cos w) cos ww, w si ( ) F( w) ( ) F( t w) F sec sec w w, w t F F ( ) ( w) e F w w, sec wt ww, w sec w e

47 Mth Notebook for Stuets 47 ( ) F( w) w F l e w, w l 4. Defiite Itegrls Defiitio of the efiite itegrl b f ( ) lim[ f ( ) Δ f ( Δ) Δ f ( Δ)... f ( ( ) Δ) Δ] Δ where the itervl [ b], is subivie ito equl prts of b legth Δ. Properties of the efiite itegrl b f g b ( ) ( )] g( b) g( ) where f ( ) g( ) g ( ). b [ f ( ) g( ) ] f ( ) g( ) b [ f ( ) g( ) ] f ( ) g( ) b b b b

48 Mth Notebook for Stuets 48 b ( ) c f ( ) cf b f ( ) 0 b ( ) f ( ) f b c b ( ) f ( ) f ( ) f Me Vlue Theorem b ( ) ( b ) f ( c) b c f, where c is betwee b f ( ) is cotiuous i [ b],. Approimtio of efiite itegrls Subivie the itervl [, b] ito equl prts b the poits 0,,,,...,, b where ( ), i 0,,,,...,( i f i ), Rectgulr rule of pproimtio b f ( ) h( ) 0... b h.

49 Mth Notebook for Stuets 49 Trpezoil rule of pproimtio b h 0 ( ) (... ) f Simpsos rule of pproimtio b h 0 4 ( ) ( ) f Selecte efiite itegrls π 0 si ( m) si( ) 0, π, m m π 0 cos ( m) cos( ) 0, m π, m π 0 si ( m) cos( ) 0 m m, m eve, m o π 0 0 si π π cos 0 si π 4

50 Mth Notebook for Stuets si cos π 4 t π 5. Series Arithmetic series ( ) ( )... ( ( ) ) where l ( ). Emples of rithmetic series... ( ) ( ) 5... Geometric series ( l) r r r... r Emple of geometric series r r, r

51 Mth Notebook for Stuets 5,... < < r r r r r r Emples of other tpes of series ( )( ) 6... ( ) ( ) l π π... 4 π π

52 Mth Notebook for Stuets 5 6. Tlor Series f ( ) f ( ) f ( )( ) f ( )( ) f! ( ) where the remier! ( ) ( )( ) R... R is give b the followig equtio R f ξ! ( ) ( )( ) ξ is betwee. Also, the fuctio f must be cotiuous with cotiuous erivtives. The ifiite series give bove is clle the Tlor series for f bout if lim R 0. ( ) If 0, the the series is clle Mcluri series. Emples of Tlor Mcluri series 4..., < <..., < e..., < <!!

53 Mth Notebook for Stuets 5 ( l ) ( l ) l..., < <!! 4 4 ( )..., < l 5 7 si...,! 5! 7! < < 4 6 cos...,! 4! 6! < < 5 7 sih...,! 5! 7! < < 4 6 cosh...,! 4! 6! < < 7. Iequlities Trigle iequlit......

54 Mth Notebook for Stuets Coversio Fctors Legth kilometer 000 meters meter 00 cetimeters meter 000 millimeters cetimeter 0.0 meter millimeter 0.00 meter ich.540 cetimeters foot 0.48 cetimeters mile.609 kilometers cetimeter 0.97 ich meter 9.7 iches kilometer 0.64 mile Volume liter 000 cubic cetimeters cubic meter 000 liters cubic foot 8. liters

55 Mth Notebook for Stuets 55 Mss kilogrm 000 grms kilogrm.046 pous pou 45.6 grms For etile coversios of other uits, visit the website for immeite olie coversios. 9. List of Prime Numbers The followig tble lists the first 0 prime umbers:

56 Mth Notebook for Stuets 56 Refereces. Spiegel, M. R., Schum s Mthemticl Hbook of Formuls Tbles, McGrw-Hill, Seco Eitio, Spiegel, M. R., Lipschutz, S. Liu, J., Schum s Outlie of Mthemticl Hbook of Formuls Tbles, McGrw-Hill, Thir Eitio, Jeffre, A. Di, H. H., Hbook of Mthemticl Formuls Itegrls, Acemic Press, Fourth Eitio, Abrmowitz, M. Stegu, I. A., Hbook of Mthemticl Fuctios: With Formuls, Grphs, Mthemticl Tbles, Dover Publictios, Zwilliger, D., CRC Str Mthemticl Tbles Formuls, Chpm & HllCRC, st Eitio, Tllri, R. J., Pocket Book of Itegrls Mthemticl Formuls, Chpm & HllCRC, Bllst, D., Architect s Hbook of Formuls, Tbles, Mthemticl Clcultios, Pretice Hll, Burigto, R. S., Hbook of Mthemticl Tbles Formuls, McGrw-Hill, Fifth Eitio, Luerer, B., Nollu, V. Vetters, K., Mthemticl Formuls for Ecoomists, Spriger, Thir Eitio, Spiegel M. R. Liu, J., Schum s Es Outlie of Mthemticl Hbook of Formuls Tbles,, McGrw-Hill, 00.

57 Mth Notebook for Stuets 57. Boljovic, V., Applie Mthemticl Phsicl Formuls Pocket Referece, Iustril Press, Brtsch, H. J., Hbook of Mthemticl Formuls, Acemic Press, 974. Iteret Liks. S. O. S. Mthemtics: Tbles Formuls The Worl of Mth Olie Mth.com Mth Referece Tbles Mesuremet Formuls: Geometr emesure.htm 4. Mth Formuls, Mth Tbles s_mth_tbles.htm 5. The Euctiol Ecclopei: Mthemticl Formuls rmuls.htm

58 Mth Notebook for Stuets Mthemticl Formuls for Ares Volumes 7. Commol Use Mthemticl Formuls muls.html

59 Mth Notebook for Stuets 59 Notes

60 Mth Notebook for Stuets 60 Notes

Sharjah Institute of Technology

Sharjah Institute of Technology For commets, correctios, etc Plese cotct Ahf Abbs: hf@mthrds.com Shrh Istitute of Techolog echicl Egieerig Yer Thermofluids sheet ALGERA Lws of Idices:. m m + m m. ( ).. 4. m m 5. Defiitio of logrithm: