Mathematical Notations and Symbols xi. Contents: 1. Symbols. 2. Functions. 3. Set Notations. 4. Vectors and Matrices. 5. Constants and Numbers

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1 Mthemticl Nottios d Symbols i MATHEMATICAL NOTATIONS AND SYMBOLS Cotets:. Symbols. Fuctios 3. Set Nottios 4. Vectors d Mtrices 5. Costts d Numbers

2 ii Mthemticl Nottios d Symbols SYMBOLS = {,,3,... } set of ll turl umbers = {,,,3,... } = {...,,,,,,... } set of itegers m = m, set of ll rtiol umbers = { ± } set of ll rel umbers (ifiite decimls) { ib,b } = + set of ll comple umbers, where I is imgiry uit, i = if = = if < bsolute vlue of rel umber i imgiry uit, i = z = + ib comple umber z = ib comple cojugte Re( z) Im( z) = rel prt = b imgiry prt rg ( z ), Arg ( z ) rgumet, priciple rgumet z = zz = + b = r bsolute vlue of comple umber z, modulus ifiity, l.u.b. of mius ifiity, g.l.b. of N k... N k= = sigm-ottio for summtio M ci c c cm i= = product ottio for product of M umbers! = fctoril lim k m mi sup limit mimum miimum supreum (lowest upper boud) if ifium (gretest lower boud ) g.l.b. l.u.b. gretest lower boud (ifium) lowest upper boud (supreum)

3 ε >, δ > smll positive rel umbers B > big positive rel umber = equl (used i equtios d i ssigmets) Mthemticl Nottios d Symbols iii ideticlly equl (used i defiitios) pproimtely equl (used for umericl represettio of rel umbers,.4 ) ot equl belogs, elemet of proportiol, similr < less th less or equl > greter th greter or equl sigifictly less sigifictly greter orthogol to DNE prllel pproches, goes to the, follows, therefore, implies, it is ecessrily, it is sufficiet if d oly if, it is sufficiet d ecessrily, equivlet d so o util for ll there eists does ot eist d or squre root th root % percet d icremet, differece betwee two vlues differetil, ifiitesimlly smll icremet k! = k! k! ( ) biomil coefficiets ( + b) = k b k k= k Newto s Biomil Theorem

4 iv Mthemticl Nottios d Symbols FUNCTIONS f ( ), y f ( ) f ( ) = fuctio f g = f g( ) compositio lim f c ( ) iverse fuctio f f ( ) = d ( ) limit f f = df d, y,, f derivtive, first derivtive, ordiry derivtive d f d k d f d f k, y,, f secod derivtive, ( k ) y k th order derivtive prtil derivtive F( ) tiderivtive of the fuctio f ( ) : F ( ) = f ( ) ( ) f d + c idefiite itegrl with rbitrry costt of itegrtio c b ( ) f d defiite itegrl F( ) b, F ( ) b brcket ottio for defiite itegrtio, f ( ) m f S mi f S ( ) ( ) Γ ( ) gmm fuctio logb l mimum of the fuctio f ( ) o set S miimum of the fuctio f ( ) o set S l = logrithm with the bse b lb turl logrithm, logrithm with the bse e si ( ), rcsi( ) iverse sie fuctio, rcus sie, etc erf ( ) error fuctio erfc ( ) complimetry error fuctio, erfc ( ) = erf ( ) > sg( ) = sig fuctio < f + p = f p periodic fuctio with the period p > ( ) ( ) f, bs ( f ) bsolute vlue, orm i imgiry uit, i = f if f f = f if f < b ( ) b = F( b) F( ) d = F Re Im, rel prt imgiry prt Nbl opertor Lplce opertor

5 Mthemticl Nottios d Symbols v SET NOTATIONS,b,c,...,, y,z A,B,...,U,V,W elemets of sets sets A elemet belogs to set A y B elemet y does ot belog to set B empty set A= B equlity of sets A B, A B, A B X i uio i= X i itersectio i= subtrctio c A (,b ) = { < < b} [,b ) = { < b} [,b ] = { b} (, ) = { > } complimet ope itervl semi-ope itervl closed itervl semi-ifiite ope itervl VECTORS AND MATRICES, vectors =, = colum-vector i =,, j= k = stdrd bsis for 3 e =, e =,..., e = stdrd bsis for = orm

6 vi Mthemticl Nottios d Symbols (,,..., ) = row-vector T A m T = = (,,..., ) m m m = Am = V,U,W trspose m mtri vector spces, subspces dim( V ) dimesio of vector spce { } sp,,..., u u u sp of vectors u, u,..., u m set of ll rel m mtrices im( f ) imge of mp f ker ( f ) kerel of mp f rk ( f ) rk of mp f I = det A = dja A ij C ij RREF m m uit mtri, idetity mtri determit djoit mior cofctor row reduced echelo form TrA = trce, the sum of the mi digol etries ϕ ( r ) r ( ) grdϕ sclr field vector field = ϕ grdiet div = divergece curl = curl (rotor) A = λ eigevlue problem λ eigevlue eigevector

Mthemticl Nottios d Symbols vii CONSTANTS AND NUMBERS π = 3.4597... e =.7888... γ =.57757... =.4436... 3 =.7358... = g 9.866 m s ccelertio of grvity c =.99795e + 8 h 6.

7 Mthemticl Nottios d Symbols vii CONSTANTS AND NUMBERS π = e = γ = = = = g m s ccelertio of grvity c =.99795e + 8 h e 34 m s = [ J s] speed of light Plck s costt W σ = 5.67e 8 4 m K R = 6 37 [ m ] Stef-Boltzm costt Erth s rdius D S =.39e+9 [ m ] =.496e+ [ m ] dimeter of the Su distce betwee the Su d the Erth ( stroomicl uit) = S c 353 W m Solr costt N = 6.4e3 molecules mol Avgdro s umber P = 35 [ ] N P, m stdrd tmospheric pressure

8 viii Mthemticl Nottios d Symbols

Students must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3,...

Students must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3,... Appedices Of the vrious ottios i use, the IB hs chose to dopt system of ottio bsed o the recommedtios of the Itertiol Orgiztio for Stdrdiztio (ISO). This ottio is used i the emitio ppers for this course