TARY TREATISE DIFFERENTIAL AND INTEGRAL AL C CULUS, W ITH EXA MP LES A ND AP PLICA TIONS. B . N S. GEOEGE A . OSB OR E , , PROFESSOR OF MATH EMATIC S IN TH E M ASSACHU SETTS INSTITUTE O F T ECHNOLO GY . H HE TH CO . PUBL S ERS . A D. C , I BOSTON NEW YORK C HICA GO 1903 C OPYR IGHT 1891 , , B B Y GEORGE A . O S ORNE REFA E P C . T IS k n n as a x -b k co and scien H wor , i te ded te t oo for lleges tific c ba o n m m as m s hools , is sed the ethod Of li its , the ost rigorou s and most intelligible form o f presenti ng the first Th e m o f m h a principles Of the s ubject . ethod li its s also the important advantage o f being a familiar method ; for it is n o w so generally introduc ed in the study o f the more ele m n ar anc ma ma c a u n ma be e t y br hes Of the ti s , th t the st de t y assumed to b e fully c onvers ant with it o n beginning the Differential Calculus . o mu a f n a n in The rules r for l e for di fere ti tio Chapter III . in o n e c m i n m a x -b k in b n differ respe t fro those si il r te t oo s, ei g x in m u n a Of x u b n an unc n e pressed ter s Of i ste d , ei g y f tio T are u c a cab all x n o f x. hey th s dire tly ppli le to e pressio s, withou t the aid o f the u sual theorem c oncernin g a function o f a function . A ac u n c f n a n u n fter q iri g the pro esses of di fere ti tio , the st de t i n a V . n uc n a n a n as a Ch pter is i trod ed to the differe ti l ot tio , c onvenient abbreviation o f the c orresponding expression s by n T n a n h a m an dvan differential c oefficie ts . his ot tio s ifest a tages i n the stu dy Of the Integral Calculus an d in its n applicatio s . I n u n u In Chapter X . a d subseq e t pages I have introd ced for 8 Pa a n a o n n a n — c has c n rti l Differe ti ti the ot tio 7 whi h re e tly 6a: n c ome into such ge eral u s e . The chapters On Maxima and Minima h ave been pl aced a a ca n cu as c n a n a fter the ppli tio s to rves , the o sider tio Of th t s ubject is muc h simplified by representi ng the function by n u M x m M e ak n a Of a c . a a and n ma ma b the ordi te rve i i i y t e , ua a an a mm a a C a if desired, with eq l dv t ge i edi tely fter h pter III X . u k m In a X . n a Ca c u a a n b Ch pter , I tegr l l l s, I h ve t e the pro le o f n n M m n n a o f a an a a as a b fi di g the o e t of I erti pl e re , etter illustration o f double i ntegration than that Of finding the The u n m a c m n n area itself. st de t ore re dily o prehe ds the i de n Of a: an d in ub n a pendent variatio y the do le i tegr l, 2 x dx d an in doc d . ( yfi y, th y A a o f a XII n a Ca cu u are d few p ges Ch pter , I tegr l l l s, evoted to a description o f the Hyperbolic Functions together with f n a and a c m a n ma c o r their dif ere ti ls, o p riso is de with the responding Circular Functions . E A . O B R . G . S O N BO TON 1895. S , NT C ONTE S. DIFFERENTIAL CALCULUS. CHAPTER I . FUNCTIONS. Definitio n and Classificatio n o f Functio ns No tatio n Of Emotio n Exam les s . p CHAPTER II . DI ERENTIA E I IE T FF L CO FF C N . mi In r m n Li t . c e e t - m l Diff r n i l ffi i nt . Ex a es 8 10 . e e t a Co e c e p I CHAPTER I I. E DIFFER NTIA TION. l r i Fun tio ns Exam e —2 Differentiatio n o f A geb a c c . pl s 10 1 Diff renti ti n o f Lo arithmic and Ex o nential Functio e a o g p ns . Examples iffer n i io n Of Tri o no metric Fun ctio ns Exam les D e t at g . p . iff r f n r ri n m ri F n i n Ex D e entiatio n o I ve se T go o et c u ct o s . — amples 32 3 7 21 22 Differentiatio n o f In erse Functio n and Functio n Of a , . v Ex m l 3 -40 Functio n . a p es 7 CHAPTER IV. E I E DI ERENTIATI N SUCC S S V FF O . 23 24 . Definiti n nd N t i n , o a o at o 25 . The h i r ffi i Ex m l nt D ffe ential Co e c ent . a p e s ’ 26 . Leibnitz Th r m Ex m l s eo e . a p e s vi TE T CON N S . CHAPTER V. DI EREN I FF T ALS. A M E. Difi erentials as relate d t o Differential Co efficients Differentiatio n b y Differentials Successi e Differentials Ex am les v . p I CHAPTER V . IMP I IT UN TI NS L C F C O . iff r nti tio n o f Im licit Fun i E - D e e a p ct o ns . xamples 52 54 CHAPTER VII . EXPA SI N OF UN TI S N O F C ON . ri ’ Th o r m Exam l Maclau n s e e . p es ’ Ex m l Taylo r s Th eo re m. a p es ’ Rigo ro us Pro o f Of Taylo r s Theo rem ’ ’ Remainder in Taylo r s and Maclaurin s Th eo rems CHAPTER VIII . I DETERMINATE RMS N FO . u f Fra tio n Limiting Val e o 3. c Evaluatio n o f 3. Examples E m le E aluatio n o f 0 co co 00 . xa s v g, , p rm Exam les Evaluatio n o f Exp o nential Fo s . p CHAPTER IX. N PA RTIAL DIFFERENTIATIO . tial Differential C o effi cients Of First Order . Exam 9 6 . Par 5 , 0 — Hi h er rd r Ex m Differential Co efficients o f O e s . a 61 63 . Partial g — ples 80 82 Fun tio ns Of Se eral Variables To ta l Difi erential Of c v . Examples Ex m le s Co nditio n fo r an Exact Differential . a p n tio ns Differentiatio n Of Implicit Fu c . ’ Taylo r s Th e o rem fo r Several V ariables S ii CONTENT . v CHAPTER X . CHANGE OF VARIABLES IN DIFFERENTIAL E COEFFICI NTS. Ch angi ng fro m so to y Ch anging fro m y to z h an in fro m to z Ex am les C g g ac . p CHAPTER XI . REPRESENTATI OF VA RI US URVES ON O C . 4- 85 Rectan ul ar Co -o rdinates 7 . g — - 86 93 . Po lar Co o rdinates CHAPTER XII . D E R E TAN EN D IR TI F U V T AN RMA . C ON O C . G NO L A SYMPTOTES. - i 4 97 Direct o n o f urve ub tan ent and ubno rmal . 9 . C . S g S Example s 98 Differential o effi i n h A r , C c e t o f t e c 99 E uatio n o f th e Tan ent and No rmal Exam les . q g . p 100- 106 A s m to tes Exam les . y p . p CHAPTER XIII . D E N IR TI OF URVA TURE. P I TS OF I F EX I C O C O N N L ON. 10 —109 Directio n Of ur ur 7 . C vat e 110 . Po ints o f Infi exi o n Ex am l . p es CHAPTER XIV. URVATURE. IR E OF URVATURE EV C C CL C .