equal ЪобэШЧ ў уъичшо ў(хфягх)ў Approximately Congruent Is congruent to Modulo Times, cross жбш кяяэ Ё ЬЯЧС x y z + -

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1 Шгу ЧЧссх ЧсбЭуф ЧсбЭэу

2 ЧснЧ Alph ШЪЧ Bt лчуч Gmm ЯсЪЧ Dlt ХШгсцф Epsilo вэъч Уц ЯэлЧуЧ Z Zt ХЪЧ H Et ЪэЪЧ Tht эцъч I Iot ТЧШЧ K Kpp счуяч Уц счушяч Lmbd уэц M Mu фэц N Nu ТгЧэ Xi Ууэпбцф O Omicro ШЧэ Pi бц Rho гэлуч Sigm ЪЧц Tu УцШгсцф Upsilo нчэ Phi ТЧэ Chi ШгЧэ Psi УцуэлЧ Omg

3 3 уычс кбшэ ХфЬсэвэ ЧсксЧуЩ Lss th qul Уелб Уц эгчцэ y y Уелб Уц ЪгЧцэ Grtr th qul УТШб Уц эгчцэ b b УТШб Уц ЪгЧцэ < Lss th >3 Уелб 4 > Grtr th <3 УТШб Approimtly Cogrut ЪобэШЧ ў уъичшо ў(хфягх)ў ABC A B C Proportiol F F k уъфчгш Is cogrut to Modulo ЪпЧнц ) (mod 5 уъичшощ Not qul 3 сч эгчцэ Б Plus-mius вчэя фчое Б Equl c) ( b)&( b эгчцэ c з Tims, cross жбш кяяэ Ё ЬЯЧС жбш уъьхэ з 3 6 A i + y j + k z B bi + by j + bzk i j k A з B y z b b b y z + Plus 3 + Ьук 5 - Mius 3 ибэ Ё фчое

4 4 : Divisio Dividd by Ъогэу Уц Уц / ї 6ї 3 % Prct 50% ЧсфгШЩ ЧсуэцэЩ ЧсфгШЩ нэ ЧсЧсн Pr thousd. Dot жбш ЯЧЮсэ A B A B cos! 5 4з 3з з з 5! кчусэ Уц нчтъцбэс Fctoril 0 T A [ ] Squr root Trspos Brckt m Ьаб ЪбШэк Ьаб ЪпкэШ Ьаб фцфэ фчос Уц эшчяс енцн ц УкуЯЩ нэ уенцнщ Ьаб фчос ЬвС еэээ T B A b ji ij [ ] y [ y ] & Mtri уенцнщ ( ) 5 )) (4 + 3( хсчсчф Ё оцгчф Prthss { } St Brcs Squc уьуцкх уъъчсэх [,] (), clos itrvl op-itrvl ЬвС Тгбэ нъбщ улсощ нъбщ унъцэщ [ 0,0] (, 0)

5 5 [,) (,] clos-op op-clos нъбщ улсощ уф Чсибн ЧсЧэгб нъбщ улсощ уф Чсибн ЧсЧэуф ( 5, ] [ 0,3) Covolutio нэ ЪЭцэсЧЪ нцбээх уснцн { ( )* ( )} { ( )} з { ( )} F g f F g F f ЧсоэуЩ ЧсуисоЩ Absolut vlu, > 0, < 0 Dtrmit Summtio Product Itrsctio уэяящ уьуцк жбш ЪоЧик з з з зз + 3 A A A A 0 0 Uoio ЪЭЧЯХ 0 0 A A A A 0 0

6 6 Itgrl ЪпЧус d 3 (4 ) ЪпЧус ЫфЧээ Doubl itgrl (, ) f y ddy Tripl itgrl Li itgrl Cotour itgrl ЪпЧус ЫсЧЫэ ЪпЧус Юиэ (,, ) g y z ddydz dl C ЪпЧус гиээ Surfc itgrl d A ЪпЧус ЭЬуэ Volum itgrl d V Thrfor Хаф Bcus счф Eist сьуэк ЪцЬЯ b упуу цьцяэ, b / Not ist b сьуэк сч ЪцЬЯ упуу лэб ЬцЯэ, / b For ll b ЪцЬЯ сьуэк Тсэ упуу, b Ќ фоэж Уц фнэ Propositiol Уц ( p) p if th ХгЪфЪЧЬ уф Чсибн ЧсЧэгб p q p r q r ХгЪфЪЧЬ уф Чсибн ЧсЧэуф

7 7 if d oly if iff ХгЪфЪЧЬ уф Чсибнэф ХаЧ ц нои ХаЧ p q p q q p Mmbrship Elmt of Not mmbr Uio эфъуэ кжц уф сч эфъуэ Уц лэбкжц ХЪЭЧЯ A A A { bc,, }, A { bc,, }, d A { bc,, }, B {, d} {,,, } A B b c d Itrsctio } { ЪоЧик A B (propr) Subst Ьвээх ц C { }, C A suprst ХЭЪцЧС ц Not subst / лэб Ьвээх B } { ЧсуЬуцкх ЧсЮЧсэх Empty st уъуу ЧсуЬуцкЩ ЧсЮЧсэЩ эгчцэ ЧсуЬуцкЩ M ЧсдЧусЩ D X ' Drivtio to ХдЪоЧо ШЧсфгШЩ с Уц d d f ( ) f ( ) df d, Pritil drivtio ЪнЧжс Ьвээ f ( ) f, d d Drivtio ordr th, th ЪнЧжс бъшщ f ( ) 3, df d 6

8 8 Prtil drivtio ordr th ЪнЧжс Ьвээ бъшщ f ( ) 3, f 6 Nbl Lplc фчшсч Уц укус счшсчг + + y z oprtor (Nbl) Squr Lp. Op. Lplci убшк ў(ъбшэк)ў укус счшсчг y z + + AB оикщ угъоэу Li sgmt AB AB Ry Ifiity li Trigl дкчк ў(угъоэу)ў угъоэу лэб уфъх ЧсуЫсЫ ABC, ABC уысы Agl ABC вчцэх ў(эчящ)ў ЧсвЧцэх, ABC вчцэх ў(очэущ)ў Right gl Squr убшк уъцчвэ ЧсЧжск Prlllogrm Circl ЯЧэбх Prpdiculr AB куця AC Prlll AB уцчвэ AC Similr ABC ЪдЧШх A B C Cogrut ABC ЪиЧШо A B C Arc оцг оцг ABC

9 9 А ' " (, y) Dgr Miut Scod Crtsi Coordit ксчущ ЧсЯбЬх ксчущ ЧсЯоэоЩ ксчущ ЧсЫЧфэЩ ХЭЯЧЫэЧЪ ТЧбЪэвэЩ А ' " нэ ЧсенЭЩ,5.7) ( (, y, z) ( r, ) Spc Coo. Polr Coo. ХЭЯЧЫэЧЪ нжчээщ ХЭЯЧЫэЧЪ оишэщ нэ ЧснжЧС (,.4,0) А (9,5 ) T F Vctor Dirct sum Dirct product d Tru Fls AB V W W цv нжчэчф уъьхэчф X X ЪЭсэс ЧснжЧэЧЪ ЧсуЪЬхэЩ Уц Чсвуб X i i Чсь нжчэчъ уъьхэщ ЬвээЩ Уц Чсь вуб ЬвээЩ ЪЭсэс ЧснжЧэЧЪ ЧсуЪЬхэЩ Уц Чсвуб X i i Чсь нжчэчъ уъьхэщ ЬвээЩ Уц Чсь вуб ЬвээЩ уъьхх уьуцк ушчдб ЬЯЧС ушчдб Ё ЬЯЧС гсјуэ ЫЧШЪ Чсцес ў(чскин)ў Ё ц еэ лси p q p q T T T T F F F T F F F F or p q p q ЫЧШЪ Чснес Ё Уц T T T T F T F T T F F F

10 0 ( ) k Prmuttio ЪШЯэс k дэ уф дэ Уц P k P k ( ) k! ( k )! C k ( k ) Combitio C k ЪШЯэс ЧсЧдэЧС угуцэ ц ЪпбЧбхЧ лэб угуцэ ЪцнэоэЩ k дэ уф дэ Уц k! k!( k )! ЧсЪШЯэс ц ЧсЪпбЧб лэб угуцэ i Imgiry umbr i ЧскЯЯ ЧсЮэЧсэ Npir s costt кяя фчшэб кяя Уцэсб Eulr s umbr Pi Gold rtio ЧсфгШЩ ЧсЫЧШЪЩ ЧсфгШЩ ЧсахШэЩ m уъцги Уц цги lim limit si фхчэщ lim 0 0 Ifiity Nturl umbrs Ntrurl with 0 сч фхчэщ уьуцкщ ЧсЧкЯЧЯ ЧсиШэкэх ЧсЧкЯЧЯ ЧсиШэкэх ук 0 lim + + {,,3,4, } { } 0 0,,,3, 4,

11 уьуцкщ ЧсЧкЯЧЯ Itgr umbrs ЧсеЭэЭх {,,,0,,, } Rtiol umbrs уьуцкщ ЧсЧкЯЧЯ ЧсуѕфиоЩ m : m,, 0 Rl umbrs ХЪЭЧЯ уьуцкщ ЧсЧкЯЧЯ ЧсуѕфиоЩ ц Чслэб уьуцкщ ЧсЧкЯЧЯ уѕфиощёў ў(чсгчсшщ ц ЧсуцЬШЩ ц Чсенб)ў ЧсЭоэоэЩ + Positiv Rl mumbrs уьуцкщ ЧсЧкЯЧЯ ЧсЭоэоэЩ ЧсуцЬШЩ ц Чсенб уьуцкщ ЧсЧкЯЧЯ ЧсЭоэоэЩ ЧсуцЬШЩ. Compl umbrs уьуцкщ кяяэщ Ъпцф нэхч ЧсЧкЯЧЯ ШецбЩ уьуцкщ ЧсЧкЯЧЯ ЧсубТШЩ Уц ЧскоЯэЩ + iy ц хпач Ё гсгсщ лэб d so o уфъхэщ : Lft hd sid Ъкбэн Чсибн y : f ( ) ЧсЧэгб уф ЮсЧс is dfid by Чсибн ЧсЧэуф th right hd m{ } sid Mimum },3, 4, m{ фхчэщ кйуь 4 mi{ } Miimum },3,4, mi{ фхчэщ елбь si Si ЬэШ si30 А cos Cosi ЬэШ ЧсЪуЧу cos60 А t Tgt А t 45 йс

12 cot Cotgt А cot 45 йс ЧсЪуЧу sc Sct очик sc cos csc Cosct очик ЧсЪуЧу csc si Arc si Arc si оцг ЧсЬэШ Arcsi 30 А Arc cos Arc cosi оцг ЧсЬэШ ЪуЧу 45 А ( ) 4 rd Rdi бчяэчф : 45 ( А ) 4 Arccos rd Arc t Arc tgt оцг Чсйс Arc cot оцг Чсйс ЪуЧу Arc cotgt Arc sc Arc sct оцг ЧсоЧик Arc csc оцг ЧсоЧик ЧсЪуЧу Arc cosct sih sh ЬэШ ЧсвЧэЯэ Hyprbolic si ў(чсхасцсэ)ў Уц sih cosh Уц ch Hyprbolic cosi ЬэШ ЧсЪуЧу ЧсвЧэЯэ ў(чсхасцсэ)ў cosh + sch Hyprbolic sct очик ЧсвЧэЯэ ў(чсхасцсэ)ў sch + cs ch Hyprbolic cosvt очик ЧсЪуЧу ЧсвЧэЯэ ў(чсхасцсэ)ў cs h c

13 3 th Уц th Hyprbolic tgt йс ЧсЪуЧу ЧсвЧэЯэ ў(чсхасцсэ)ў th + Hyprbolic cotgt йс ЧсЪуЧу ЧсвЧэЯэ ў(чсхасцсэ)ў cothуц coth coth + Arc sih оцг ЧсЬэШ ЧсвЧэЯэ Arc hyprbolic si ў(чсхасцсэ)ў Arc cosh оцг ЧсЬэШ ЧсЪуЧу Arc hyprbolic cosi ЧсвЧэЯэ ў(чсхасцсэ)ў ЯсЪЧ Тбцфпб Krochr dlt ij, i 0, i j j i T jk T ij Уц Tsor T T + T ij j j ц Ъэфгцб Уц уцъб i, i T jk i Ясэс ксцэ ц j ц k ЯсЧэс гнсэх S Squc уьуцк уъъчсэщ Ё S ( + ) Logb Logrithm Log сцлчбэыу 0 00 Log 00 l Nturl logrithm Log ЧссцлЧбэЫу ЧсиШэкэ l powr Уг 0 00 Probbility ХЭЪуЧс цоцк B A ХаЧ ЭЯЫЪ ХЭЪуЧс P ( AB) Fuctio ЯЧсЩ Уц ЪЧШк нэ фйбэщ ЧсЯцЧс съкбэн оэущ ЧсЯЧсЩ Уц ЧсЧдЪоЧо Уц ЧсЪпЧус нэ фоищ Уц фочи укэфщ f + 0

14 4 O Compositio )) f g( ) f ( g( ЪбТэШ sg ЪЧШк ЧсксЧуЩ Уц sig fuctio ЧсЧдЧбЩ, > 0 sg 0, 0, < 0 Td to эгкь фэц фэц ЧсЧгнс Roudd dow Roudd up фэц ЧсЧксь grd Grdit ЪЯбЬ F F F i+ j+ k y z grdf з F div Divrgc ЪШЧкЯ divf F F F F + + y z curl Rottio ЯцбЧф i j k curlf з F y z F F F y z эжу хач ЧсШЭЫ укйу ц сэг ЬуэкхЧ. УЮбь УТЪнэЪ ШЧдхбхЧ. Тасп Шкж Чсбуцв схч ХгЪкуЧсЧЪ дъчс 008

15 уцок ЧсШбэЯ ЧсЧспЪбцфэ :

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