By Rajesh Pandey

Quantity i of this sequence serves as a textbook for semester i of thesubject engineering arithmetic. compatible figures and diagrams havebeen used to make sure a simple knowing of the recommendations concerned. to stress program of the subjects mentioned, appropriate examplesare integrated through the e-book. Solved examples, within the bookinclude ideas of questions from prior u. P. T. U. Examinations. past years query papers were incorporated as to exposestudents to the trend and kind of questions they could face in anexamination. This ebook is particularly necessary for measure, honours andpostgraduate scholars of all indian universities and for ias, pcsand different aggressive examinations. in regards to the writer dr. Rajesh pandey he has greater than fourteen years experiencein instructing scholars of undergraduate, postgraduate and engineeringlevel. He bought his b. Sc and m. Sc measure in 1991 and 1993respectively from gorakhpur college and phd in 12 months 2003 andparticipated in a variety of seminars & meetings of nationwide andinternational point. For graduate & postgraduate, the authorhas additionally written books on increase calculus, vectors, numericalanalysis, summary algebra, mechanics, fluid mechanics and so forth. shortly, he's operating as an assistant professor/reader inmathematics, sherwood collage of engineering, study andtechnology, lucknow. desk of contents easy effects and ideas successive differentiation and leibnitz's theorem partial differentiation curve tracing growth of functionality jacobian approximation of mistakes extrema of features of a number of variables lagranges approach to undetermind multipliers matrices a number of integers beta and gamma features vector differential calculus vector critical calculus exam papers of uptu from 2001-2009

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**Example text**

111) . ay ax ay ayay az ay av av ax av ay av az . and Vz = - = - - + - - + - - .................. (IV) az ax az ay az az az with the help of (i), equations (ii), (iii) and (iv) gives V = av (2)+ av (0)+ av (-2) ay az x ax or V = 2 (av _ av) x ax az => 6Vx = 12( ~~ - ~~) .................. (v) Now V = av (-3) + av (3) + av (0) Y ax ay az or V =3(- av + av) y ax ay => 4V, = 12(-~~ + ~~) .................. (vi) av av av and V7 = ax(O)+ ay(-4)+ az(4) or V =4(- av + av) z ay az => 3Vz = 12( - ~~ + ~~) .................

Or Y2 (1 +x2) + Xyl - m 2y -0 - ................... 2 or (1 +X2)Yn+2 +(2n+ 1) Xyn+l + (n2 - m2)Yn =0 pu tting x =0 we get (yn+2)O = (m2 - n2) (yn)O ........................... (iv) If n is odd, putting n = 1,3, 5, 7 ............. (n-2), we get When n = 1, (Y3)O = (m2 -12) (Yl)O And from (i) and (ii), putting x =0, we get 2 (y)o = 1; (Yl)~ = m (y)~ = m 2 or (Yl)O = m :. (Y3) 0 = (m2-12) m when n = 3, from (iv) we get (ys)o = (m2 -32)(Y3)O or (ys)o = (m2 - 32) (m2 -1) m when n =5, from (iv) we get (Y7)O = (m2 -52) (ys)o or (Y7)O = (m2 -52) (m2 -32) (m2 -12)m.

If x = r cosS, y = r sinS, prove that (i) ar = ax ax (ii) ar .!. ax = r as r as ax az az 11. If z = eax+by f(ax - by), Show that b - + a - = 2abz ax ay 2 12. If Y = f(r) and r2 = x2+y2 +z2, Prove that Yxx+ Yyy +Y zz = f"(r) + - f'(r) r 13. Find p and q, if x = fa (sin u + cos v), y = fa (cos u - sin v), z =l+sin(u - v) az az . - respectively. ax ay [ Hint. X2+y2 =2az :. z = x2 +y2 2a ] r Ans. p = ~, q = a a 41 A Textbook ofEngineerin~ Mathematics Volume - I au ax au ay 14. 2001) 15. , March 2004) x + 2 a a 16.