Elementr mthemticl epressions Qurtic equtions b b b The solutions to the generl qurtic eqution re (1) b c () b b 4c (3) Tlor n Mclurin series (power-series epnsion) The Tlor series n n f f f n 1!! n! f f (4) gives useful w of pproimting function ner to some specific point, giving power-series epnsion in n for the function ner tht point. The Mclurin series n n f f f n n f f (5) 1!!! is specil cse of the Tlor series where we re epning roun the point. Power-series epnsions of common functions For smll, the Mclurin epnsions of vrious common functions re, to first orer 1 1 / (6) 1 1 1 (7) sin (8) tn (9) cos 1 (1) ep 1 (11) Quntum Mechnics for Scientists n Engineers On-Line Course 1
Sine n cosine ition n prouct formule sin cos 1 (1) sin sin cos cos sin (13) sin sincos (14) cos cos cos sin sin (15) cos cos sin cos 1 1 sin (16) cos 1 cos (17) 1 sin 1 cos (18) 1 cos cos cos cos (19) sin sin cos cos () sin cos sin sin (1) cos cos cos cos () sin sin sin cos (3) coscossin sin (4) sin sin cos sin (5) Quntum Mechnics for Scientists n Engineers On-Line Course
Differentil clculus Prouct rule uvu v v u (6) Quotient rule u v v u u (7) v v Chin rule f g f g (8) g f Derivtives of elementr functions ep n n n1 (9) ep ln sin cos (3) 1 sin 1 1 (31) cos (3) sin (33) 1 1 tn 1 (34) (35) 1 Quntum Mechnics for Scientists n Engineers On-Line Course 3
Integrl clculus Integrtion b prts b b g b f f ( ) f g g (36) where we use the common nottion n, specificll, here Some efinite integrls b h h b h (37) b f g f b g b f g (38) sin n (39) 4nm / sin n sin m, for nm o nm nm, for nm even (4) sin cos / 3 (41) sin cos 4 / 3 (4) 4 3 3 sin (43) 1/ t eptt (44) sin (45) sin (46) ep (47) 1 (48) 1 Quntum Mechnics for Scientists n Engineers On-Line Course 4
Prtil ifferentition For function h, tht is function of two inepenent vribles n, the prtil erivtive, often stte s prtil h b or, more eplicitl, prtil h b t constnt, n written s h h (49) is the erivtive of h with respect to with the vrible hel t constnt vlue. Tht vlue cn lso be eplicitl stte, for emple, s in the nottion h (5) o which woul be the prtil erivtive tken t the specific vlue. Higher prtil erivtives cn be forme similrl, s in the nottions n, for the cross erivtive, h h h h Provie ll the vrious first erivtives n the two cross-erivtives in the two ifferent orers both eist, we cn interchnge the orer of the prtil ifferentitions in the crosserivtive; tht is, h h (53) For smll or infinitesiml chnges in n in, the resulting totl chnge in h or ifferentil or ect ifferentil is written h h h If n re both functions of some other vrible t, then the totl erivtive h / t is given b h h h t t t If n re ech themselves functions of two vribles n b, then we cn write h h h b b b o (51) (5) (54) (55) (56) Quntum Mechnics for Scientists n Engineers On-Line Course 5
Becuse this works for n function of n (for which ll pproprite erivtive eist), we cn write b b b which cn be use to chnge prtil erivtives from one coorinte sstem to nother. (57) Quntum Mechnics for Scientists n Engineers On-Line Course 6
Vector clculus Crtesin coorintes The the opertor, which occurs in vrious ifferent vector clculus opertors, is known s el or nbl, cn be written s i j k (58) z in Crtesin coorintes, with i, j, n k s unit vectors in the,, n z irections respectivel. The grient opertor opertes on sclr function f ( z,, ) to give vector whose mgnitue n irection re the slope or grient of the sclr function t the point of interest. In Crtesin coorintes f f f gr f f i j k (59) z The Lplcin opertor, lso known s el squre, opertes on sclr function, giving sclr result. It is written in Crtesin coorintes s f f f f (6) z The opertor, sometimes lso written s, cn operte on vector function, in which cse, in Crtesin coorintes, we hve F F Fz Fi j k (61) z In Crtesin coorintes, the ivergence of vector F is efine s F F Fz ivff (6) z In Crtesin coorintes, the curl of vector F is efine s F F z F F Fz F curlf F i j k (63) z z or in the equivlent eterminnt shorthn form, i j k F (64) z F F F z Quntum Mechnics for Scientists n Engineers On-Line Course 7
Sphericl polr coorintes In sphericl polr coorintes, which cn be efine s in the following igrm z (,, z) q f r with rsin cos (65) rsin sin (66) z rcos the grient cn be written f 1 ˆ 1 ˆ f f r f ˆ (67) r r rsin the Lplcin cn be written 1 1 1 f r f sin f f r r r r sin r sin the ivergence cn be written 1 1 1 F F rfr F sin (69) r r rsin rsin n the curl cn be written (68) rˆ ˆ ˆ r sin rsin r F (7) r F rf rsin F r Quntum Mechnics for Scientists n Engineers On-Line Course 8
Vector clculus ientities F (71) f (7) FFF (73) FGFGGF (74) (75) Quntum Mechnics for Scientists n Engineers On-Line Course 9