A TEXTBOOK OF ENGINEERING MATHEMATICS

A TEXTBOOK OFENGINEERING MATHEMATICS

A TEXTBOOK OFENGINEERINGMATHEMATICSForB. Tech.I semester(Common to All Branches)(According to the Latest Syllabus Prescribed by APJ Abdul KalamKerala Technological University, Kerala)ByN.P. BaliDr. Remadevi.SFormerly, PrincipalS.B. College, GurugramHaryanaProf. and Head of Department of MathematicsModel Engineering College, CochinKeralaUNIVERSITY SCIENCE PRESS(An Imprint of Laxmi Publications Pvt. Ltd.)An ISO 9001:2008 CompanyBENGALURU CHENNAI COCHIN GUWAHATI HYDERABADJALANDHAR KOLKATA LUCKNOW MUMBAI RANCHI NEW DELHIBOSTON (USA) ACCRA (GHANA) NAIROBI (KENYA)

A TEXTBOOK OF ENGINEERING MATHEMATICS by Laxmi Publications (P) Ltd.All rights reserved including those of translation into other languages. In accordance with the Copyright (Amendment) Act, 2012,no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic,mechanical, photocopying, recording or otherwise. Any such act or scanning, uploading, and or electronic sharing of any part of thisbook without the permission of the publisher constitutes unlawful piracy and theft of the copyright holder’s intellectual property. Ifyou would like to use material from the book (other than for review purposes), prior written permission must be obtained from thepublishers.Printed and bound in IndiaTypeset at Goswami Associates, DelhiFirst Edition : 2016ISBN 978-93-85750-42-7Limits of Liability/Disclaimer of Warranty: The publisher and the author make no representation or warranties with respect to theaccuracy or completeness of the contents of this work and specifically disclaim all warranties. The advice, strategies, and activitiescontained herein may not be suitable for every situation. In performing activities adult supervision must be sought. Likewise, commonsense and care are essential to the conduct of any and all activities, whether described in this book or otherwise. Neither the publishernor the author shall be liable or assumes any responsibility for any injuries or damages arising here from. The fact that an organizationor Website if referred to in this work as a citation and/or a potential source of further information does not mean that the author orthe publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readersmust be aware that the Internet Websites listed in this work may have changed or disappeared between when this work was writtenand when it is read.BranchesAll trademarks, logos or any other mark such as Vibgyor, USP, Amanda, Golden Bells, Firewall Media, Mercury, Trinity, Laxmiappearing in this work are trademarks and intellectual property owned by or licensed to Laxmi Publications, its subsidiaries oraffiliates. Notwithstanding this disclaimer, all other names and marks mentioned in this work are the trade names, trademarks orservice marks of their respective owners.Published in India by&Bengaluru080-26 75 69 30&Chennai044-24 34 47 26, 24 35 95 07&Cochin0484-237 70 04,405 13 03&Guwahati0361-254 36 69,251 38 81&Hyderabad040-27 55 53 83, 27 55 53 93&Jalandhar0181-222 12 72&Kolkata033-22 27 43 84&Lucknow0522-220 99 16&Mumbai022-24 91 54 15, 24 92 78 69&Ranchi0651-220 44 64UNIVERSITY SCIENCE PRESS(An Imprint of Laxmi Publications Pvt. Ltd.)An ISO 9001:2008 Company113, GOLDEN HOUSE, DARYAGANJ,NEW DELHI - 110002, INDIATelephone : 91-11-4353 2500, 4353 2501Fax : 91-11-2325 2572, 4353 2528www.laxmipublications.com info@laxmipublications.comC—Printed at:

CONTENTSPrefaceSyllabusChapters.(vi). (vii)–(viii)PagesMODULE–I(Single Variable Calculus and Infinite Series)(15 Marks)1.Hyperbolic Functions.32.Infinite Series.93.Power Series.44MODULE–II(Three Dimensional Space and Functions of More Than One Variable)(15 Marks)4.Three Dimensional Space.855.Functions of Two or More rtial Derivatives and its Applications)(15 Marks)6.Partial Differentiation and ApplicationsMODULE–IV(Calculus of Vector Valued Functions)(15 Marks)7.Calculus of Vector Valued FunctionsMODULE–V(Multiple Integrals)(20 Marks)8.Multiple IntegralsMODULE–VI(Vector Integration)(20 Marks)9.Vector IntegrationAppendix–I: List of Important FormulaeAppendix–II: Some Important Curves(v)

PREFACEThis book is a modified form of the book ‘‘A Textbook of Engineering Mathematics’’,First Year for Cochin University of Science and Technology which is a part of the original book‘‘A Textbook of Engineering Mathematics’’ (with 28 chapters and covering the syllabi ofEngineering courses of all semesters of all the Indian universities) running its eighth editionand very well received by the students and teachers of all Indian Universities.The present form of the book is divided into 6 modules containing 9 chapters and coversentire portion according to latest syllabus prescribed by Kerala Technological University (KTU)for the B Tech First semester MA 101 Calculus paper.There is no dearth of books on Engineering Mathematics but the students find it difficultto solve most of the problems in the exercise in the absence of adequate number of solvedexamples. Anoutstanding and distinguishing feature of the book is the large number of typicalsolved examples followed by well-graded problems.We have endeavoured to present the fundamental concepts in a comprehensive andlucid manner. We are indebted to all authors, Indian and Foreign, whose works we havefrequently consulted.All efforts have been made to keep the book free from errors. Answers to all exerciseshave been checked. All suggestions for improvement will be highly appreciated and gratefullyacknowledged.—Authors( vi )

SYLLABUSFor B. Tech. Semester IKerala Technological University (KTU)Course No. MA 101L.T.P. Credits3-1-0-4Course ObjectivesIn this course the students are introduced to some basic tools in Mathematics which areuseful in modelling and analysing physical phenomena involving continuous changes ofvariables or parameters. The differential and integral calculus of functions of one ormore variables and of vector functions taught in this course have applications across allbranches of engineering. This course will also provide basic training in plotting andvisualising graphs of functions and intuitively understanding their properties usingappropriate software packages.Module I: Single Variable Calculus and Infinite SeriesIntroduction: Hyperbolic functions and inverses–derivatives and integrals.Basic ideas of infinite series and convergence. Convergence tests-comparison, ratio,root tests (without proof). Absolute convergence. Maclaurins series–Taylor series–radiusof convergence.Module II: Three Dimensional Space and Functions of More than One VariableThree dimensional space: Quadric surfaces, Rectangular, Cylindrical and sphericalcoordinates, Relation between coordinate systems.Equation of surfaces in cylindrical and spherical coordinate systems.Functions of two or more variables: Graphs of functions of two variables–level curvesand surfaces–Limits and continuity.Module III: Partial Derivatives and its ApplicationsPartial derivatives–Partial derivatives of functions of more than two variables-higherorder partial derivatives–differentiability, differentials and local linearity.The chain rule–Maxima and Minima of functions of two variables–extreme value theorem(without proof)–relative extrema.Module IV: Calculus of Vector Valued FunctionsIntroduction to vector valued functions–parametric curves in 3-space. Limits andcontinuity–derivatives–tangent lines–derivative of dot and cross product–definiteintegrals of vector valued functions.Change of parameter–arc length–unit tangent–normal–velocity–acceleration and speedNormal and tangential components of acceleration.Directional derivatives and gradients–tangent planes and normal vectors.( vii )

( viii )Module V: Multiple IntegralsDouble integrals–Evaluation of double integrals–Double integrals in non-rectangularcoordinates–reversing the order of integration.Area calculated as double integral–Double integrals in polar coordinates.Triple integrals–volume calculated as a triple integral–triple integrals in cylindricaland spherical coordinates. Converting triple integrals from rectangular to cylindricalcoordinates–converting triple integrals from rectangular to spherical coordinates–changeof variables in multiple integrals–Jacobians (applications of results only).Module VI: Vector IntegrationVector and scalar fields–Gradient fields–conservative fields and potential functions–divergence and curl–the operator–the Laplacian 2.Line integrals–work as a line integral–independence of path–conservative vector field.Green’s Theorem (without proof–only for simply connected region in plane), surfaceintegrals, Flux integral–Divergence Theorem (without proof), Stokes’ Theorem (withoutproof).

MODULE–ISingle Variable Calculusand Infinite Series(15 Marks)

1Hyperbolic Functions1.1. HYPERBOLIC FUNCTIONSDefinition. The hyperbolic functions have similar names to the trigonometric functions,but they are defined in terms of the exponential functions. Hyperbolic sine and cosine functionsare defined assinh x ex e x2cosh x ex e x2tanh x sinh xex e x xcosh xe e xcoth x 1ex e x xtanh xe e xsech x 12 xcosh xe e xcosech x 12 xsinh xe e x223112–11–111–11–1–1–2–2tanh xcosh xsinh x222111–11–11–1–2sech x–1–2cosech xHyperbolic functions3–11–1–2coth x

4A TEXTBOOK OF ENGINEERING MATHEMATICS1.2. RELATION CONNECTING TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS ISResults: (i)(ii)(iii)sin ix i sinh xcos ix cosh xtan ix i tanh xProof. (i) We have sin ei e i 2iPut ixsin ix (ii) cos e x ex ex e x ei( ix ) e i( ix ) (–i ) i i sinh x222i ei e i 22Put ix,(iii) tan cos ix e( i )x e i22x e x ex cosh x2sin cos Put ix,tan ix sin ix i sinh x i tanh x. cos ixcosh xNotes:(i) sinh x and cosh x are periodic functions of period 2 i.(ii) cosh x is an even function.(iii) sinh x is an odd function.(iv) sinh 0 0, cosh 0 1, tanh 0 0(v) The range of sinh x is ( – , )(vi) The range of cosh x is (1, ).1.3. FUNDAMENTAL FORMULAS(i) cosh2 x – sinh2 x 1Proof. We have sin2 cos2 1Put ix, (sin ix)2 (cos ix)2 1(i sinh x)2 (cosh x)2 1 – sinh2 x cosh2 x 1 cosh2 x – sinh2 x 1(ii) sech2 x tanh2 x 1(iii) coth2 x – cosech2 x 1(iv) sinh (x y) sinh x cosh y cosh x sinh y(v) tanh 3x 3 tanh x tanh3 x1 3 tanh3 x[Using results (i) and (ii)]

5HYPERBOLIC FUNCTIONSProof. Same as (i)First write down the respective result in trigonometry in , then put ix and applyresults in 1.2.Inverse Hyperbolic FunctionsIf sinh u x, then u is called the hyperbolic sine inverse of x and is written asu sinh–1 xSimilarly we can define cosh–1 x, tanh–1 x etc.ILLUSTRATIVE EXAMPLESx 2 1 ).Example 1. Prove that sinh–1 x log (x Sol. Letu sinh–1 x. x sinh u 2x eu –1e u e u2e 2u 1 e2u – 2xeu – 1 0eueuTreating this equation as a quadratic equation in eu, we geteu 2x 4x 2 4 x 2x2 1Taking the positive signeu x x2 1sinh–1 x log (x Example 2. Prove that tanh–1 x Sol. Letu tanh–1 x x tanh u u log (x x 2 1 ).1 1 x log .2 1 x eu e ueu e u Using Componendo-Dividendo rule,1 x2e u e2u u1 x2e u 1 1 x log .2 1 x .x2 1 )

A Textbook Of EngineeringMathematics Sem-I (KerlaTechnological University) Kerla40%OFFPublisher : Laxmi PublicationsISBN : 9789385750427Author : N. P. Bali, DrRemadeviType the URL : http://www.kopykitab.com/product/12185Get this eBook

B. Tech. I semester (Common to All Branches) (According to the Latest Syllabus Prescribed by APJ Abdul Kalam Kerala Technological University, Kerala) By N.P. Bali Dr. Remadevi.S Formerly, Principal Prof. and Head of Department of Mathematics S.B. College, Gurugram Model Engineering College, Cochin Haryana Kerala UNIVERSITY SCIENCE PRESS