ALGEBA Popeties of Asote Ve Fo e mes :, + + Tige Ieqit Popeties of Itege Epoets is Assme tt m e positive iteges, tt e oegtive, tt eomitos e ozeo. See Appeies B D fo gps fte isssio. + ( ) m m m m m m m m m m m Spei Pot Foms + A B ( A B) A B + + A+ B A AB B + A B A AB B 3 3 ( A+ B) A + 3AB+ 3AB + B 3 + 3 3 3 A B A 3A B 3AB B Ftoig Spei Biomis A B A B A B ( + ) + + + A B A B A AB B 3 3 A + B A+ B A AB B 3 3 Qti Fom Te sotios of te eqtio + + e: ± 4 Diste Fom Te iste etee to poits, Mipoit Fom + +, (, ) is: + ( ) Sope of Lie Hoizot ies ve sope. m Veti ies ve efie sope. Pe Pepei Lies Give ie it sope m: sope of pe ie m sope of pepei ie m Foms of Eqtios of Lie St Fom: + Sope-Iteept Fom: m +, ee m is te sope is te -iteept ( ) is poit o te ie () t + t Poit-Sope Fom: m, ee m is te sope, Veto Fom: v, ee is fie veto v is ietio veto Popeties of Logitms Let,,, e positive e mes it, et e e me. See Appei B fo gps fte isssio. og og og og og ( ) og ( ) og + og og og og og ( ) og og og og e eqivet ge of se fom Tigoometi Hpeoi Ftios: Defiitios, Gps, Ietities See Appei.
GEOMETY A e, imfeee, SA sfe e o te e, V vome etge ie Tige Peogm Tpezoi A A p p A A A + etg Pism Spee etg Pmi V SA + + V 4 3 π SA 4p V 3 3 V igt ie igt i ie oe Ae of Bse V p SA p + p V π SA π 3 + π + Defiitio of Limit Let f e ftio efie o ope itev otiig, eept possi t itsef. We s tt te imit of f s ppoes is L, ite im f L, if fo eve me e > tee is me > s tt f L <ε eeve stisfies < < δ. Bsi Limit Ls ± ± Sm/Diffeee L: im f g im f im g ostt Mtipe L: im kf kim f Pot L: im f g im f im g f im f Qotiet L: im, im g g povie im g Sqeeze Teoem If g f otiig, eept possi t itsef, if img im L, te im f L s e. fo i some ope itev LIMITS otiit t Poit Give ftio f efie o ope itev otiig, e s f is otios t if im f f. L Hôpit s e Sppose f g e iffeetie t poits of ope itev I otiig, tt g fo I eept possi t. Sppose fte tt eite o Te im f im g ± ± im f im g. ( ) f f im im, g g ssmig te imit o te igt is e me o o -.
Te Deivtive of Ftio Te eivtive of f, eote f, is te ftio ose ve t te poit is povie te imit eists. f + f f im, Eemet Diffeetitio es ostt e: k ostt Mtipe e: Sm/Diffeee e: Pot e: Qotiet e: Poe e: i e: kf ( ) kf f ± g f g ± f g f g f g + f f g f g g g f ( g ) f g g Deivtives of Tigoometi Ftios ( si) os ( os)si ( t) se ( s)s ot ( se) se t ( ot)s Deivtives of Ivese Tigoometi Ftios si os ( t ) + s se ( ot ) + Te Me Ve Teoem If f is otios o te ose itev, DEIVATIVES Deivtives of Epoeti Logitmi Ftios e e ( ) ( og ) Deivtives of Hpeoi Ftios ( si ) os ( os ) si ( t ) se ( s )s ot ( se )se t ( ot )s Deivtives of Ivese Hpeoi Ftios si + ( os ), > ( t ), < s + ( se ), ( ot ), > < < Te Deivtive e fo Ivese Ftios If ftio f is iffeetie o itev (, ), if f fo (, ), te f ot eists is iffeetie o te imge of te itev (, ) e f, eote s f, () i te fom eo. Fte, if,, te f f, f if f (, ), te f f f. ( ) [ ] iffeetie o (, ), te tee is t est oe poit ( ) f f f ( )., fo i
INTEGATION Popeties of te Defiite Iteg Give te itege ftios f g o te itev, ostt k, te fooig popeties o. [ ]. f. f f 3. k k 4. kf ( ) k f ± ± 5. f g f g 6. f f f +, ssmig e iteg eists 7. If f g o [, ], te f g. 8. If m mi f M m f, te m f M. Te Fmet Teoem of s Pt I Give otios ftio f o itev I fie poit I, efie te ftio F o I F f () t t. F f fo I. Te Te Sstittio e If g itev I, if f is otios o I, te is iffeetie ftio ose ge is te ( ) f g g f. Hee, if F is tieivtive of f o I, f ( g ) g ( ) F( g )+. Itegtio Pts Give iffeetie ftios f g, f g f g g f. If e et f v g, te f v g ememee iffeeti fom te eqtio tkes o te moe esi v v v. Pt II If f is otios ftio o te itev, tieivtive of f o,, [ ] te f F F. [ ] if F is Smmtio Fts Foms SEQUENES AND SEIES ostt e fo Fiite Sms: ostt Mtipe e fo Fiite Sms: Sm/Diffeee e fo Fiite Sms:,fo ostt,fo ostt ( ± ) ± i i i i i i Sm of te Fist Positive Iteges: Sm of te Fist Sqes: Sm of te Fist es: ( + ) ( + ) ( + ) 3 ( + ) i i i 6 4 Geometi Seies Fo geometi seqee { } it ommo tio : Pti Sm: s Ifiite Sm:, if,, if < Biomi Seies Fo e me m - < < : m ( + ) m ( ) mm mm m + m + +! 3! mm ( ) ( m + ) + +! 3 +
To Seies Mi Seies Give ftio f it eivtives of oes togot ope itev otiig, te poe seies f f f ( ) f + f ( ( )+ ) ( ) ( +!! 3! ) 3 ( ) + is e te To seies geete f ot. Te To seies geete f ot is so ko s te Mi seies geete f. VETO ALULUS Popeties of S Mtipitio Veto Aitio Fo vetos, v, ss : S Mtipitio Popeties ( + v) + v + + ( ) ( ),, Veto Aitio Popeties + v v + + ( v+ ) ( + v)+ + + ( ) Dot Pot Give to vetos,, 3 v v, v, v3, te ot pot v of te to vetos is te s efie v v + v + v 3 3. A simi fom efies te ot pot of to vetos i. Popeties of te Dot Pot Fo vetos, v, s : v v ( v+ ) v+ ( v) ( ) v ( v) Dot Pot te Age etee To Vetos If to ozeo vetos v e epite so tt tei iiti poits oiie, if q epesets te sme of te to ges fome v (so tt q p), te v v os θ. Pojetio of oto v Let v e ozeo vetos. Te pojetio of oto v is te veto poj v v v. v oss Pot Give,, 3 v v, v, v3, i j k v 3 v v v 3 3 3 i j+ k v v v v v v 3 v v 3 ( ) + ( ) i v v j v v k. 3 3 3 3 Popeties of te oss Pot Fo vetos, v, i 3 ss : v v ( v+ ) v+ ( + v) + v ( ) ( v) v ( v ) ( v) ( v ) ( ) v( v) ooite ovesio etiosips ii tesi Spei ii Spei tesi + + z + + z os q si j ρsiϕosθ si q q q ρsiϕsiθ z z z os j z os j Giet Veto Give ftio f,,,, f (,,, ) f,,,, f,,,,, f,,,.
(Dietio Deivtive) t te poit (, ) i te ompttio of Df Assmig te eivtive of f, ietio of te it veto, eists, D f, f,, f,, f,. Moe gee, if f,,, is iffeetie t te poit,,, if,,, is it veto, te D f ( ) f ( ). Popeties of te Giet Assme f g e ot iffeetie ftios tt k is fie e me. Te te fooig s o. Sm/Diffeee L: ( f ± g) f ± g ostt Mtipe L: ( kf ) k f Pot L: ( fg) f g + g f Qotiet L: f g f f g, povie g g g Te Fmet Teoem fo Lie Itegs (Giet Teoem) Assme tt f is iffeetie ftio ose giet f is otios og ve tt is efie te smoot veto ftio t (), t. Te, ( ) ( ) f f f. Divegee (F Desit) of Veto Fie Te ivegee, o f esit, of veto fie F( z,, ) P, Q, is te s ftio iv F P + Q + z. I gee, e eote te ivegee of veto fie F s te ot pot of te e opeto F. of Veto Fie Te of veto fie iv F F F( z,, ) P( z,, ), Q( z,, ), ( z,, ) is te veto ftio F Q P Q P,,. z z Te of veto fie F e ememee s te oss pot of te e opeto F. F F Gee s Teoem (Tgeti- Fom) Let e positive oiete, pieeise smoot, simpe ose ve i te pe, et e te egio eose. If F, P,, Q, pti eivtives o ope egio otiig, te P Q ve otios F Ts F P+ Q Q P A. sig te Eteig F to F z,, P,, Q,, ft tt Q P Q P k k F k F k, e ite tis vesio of te fom s F Ts F ka. Gee s Teoem (Nom-Divegee Fom) Let e positive oiete, pieeise smoot, simpe ose ve i te pe, et e te egio eose. If F, P,, Q, pti eivtives o ope egio otiig, te P Q ve otios F s PQ P + Q A F A. Stokes Teoem Assme F is veto fie it otios pti eivtives i ope egio of spe otiig pieeise smoot sfe S. Assme tt te o of S is simpe, ose, pieeise smoot ve, tt is positive oiete it espet to S. Te F Ts F σ. S Te Divegee Teoem Assme F is veto fie it otios pti eivtives i ope egio of spe otiig D 3, ssme D is eite simpe io o fiite io of simpe egios. Let S eote te sfe of D, et e te ot-poitig fie of it vetos om to S. Te S F σ FV. D