ENGINEERING MATHEMATICS - IES Master Publication

1 ENGINEERING MATHEMATICSO ffice : Phone : F-126, (Lower Basement), Katwaria Sarai, New Delhi-110016011-26522064 Mobile :E-mail: Web : 8130909220, 9711853908 , Features : 289 topics under 31 chapters in 8 units 672 Solved Examples for comprehensive understanding 1386 questions from last 25 years of GATE & ESE exams with detailed solutions Only book having complete theory on ESE & GATE Pattern Comprising conceptual questions marked with * to save the time while revising(For ESE & GATE Exam)(CE, ME, PI, CH, EC, EE, IN, CS, IT)First Edition:2017 Typeset at : IES Master Publication , New Delhi-110016 IES Master PUBLICATIONF-126, (Lower Basement), Katwaria Sarai, New Delhi-110016 Phone : 011-26522064, Mobile : 8130909220, 9711853908E-mail : : , rights 2017, by IES Master Publications.

2 No part of this booklet may be reproduced, ordistributed in any form or by any means, electronic, mechanical, photocopying, recording, or otherwiseor stored in a database or retrieval system without the prior permission of IES Master , New are liable to be legally , the IES Master , have immense pleasure in placing the first edition of ENGINEERING MATHEMATICS before theaspirants of GATE & ESE Students, as we all know that in 2016 UPSC included ENGINEERING MATHEMATICS as a part of syllabus of commonpaper for ESE exam as well as of a technical paper for EC/EE branch, while ENGINEERING MATHEMATICS already has 15%weightage in GATE exam. We have observed that currently available books cover neither all the topics nor all previouslyasked questions in GATE & ESE exams.

3 Since most of the books focus on only some selected main topics, studentshave not been able to answer more than 60-65% of 1386 questions that have been asked in GATE & ESE exams sofar. Hence to overcome this problem, we have tried our best by covering more than 289 topics under 31 chapters in8 units. (One should not be in dilemma that 289 topics are more than sufficient. These are the minimum topics fromwhere GATE & ESE have already asked questions). Since we have covered every previous year questions from last 25years of each topic, students can easily decide, how much time to allocate on each chapter based on the number ofquestions asked in that particular exam. Again, we have included only those proofs that are necessary for conceptbuilding of topics and we have stressed on providing elaborate solution to all the is the only book in the market which has complete theory exactly on ESE & GATE Pattern.

4 After each topic thereare sufficient number of solved examples for concept building & easy learning. The book includes such types of 672examples. It also covers all the previously asked questions in which conceptual questions are marked with * sign sothat students can save their time, while incorporated my teaching experience of more than 13 years, I believe this book will enable the students to excelin ENGINEERING source of inspiration is Mr. Kanchan Thakur Sir (Ex-IES). He has continuously motivated me while writing this special thanks to the entire IES Master Team for their continuous support in bringing out the book. I stronglybelieve that this book will help students in their journey of success.

5 I invite suggestions from students, teachers &educators for further improvement in the Puneet Sharma( , )IES Master PublicationsNew DelhiUNIT 1 : , Continuity and 44(i) (ii)Limit of a (iii)Theorem on (iv)Indeterminate (v)L-Hospital (vi)Fundamentals of (vii)Kinds of (viii)Properties of Continuous (ix)Saltus of a (x)Function of Two (xi)Limit of a Function of Two (xii)Continuity of Function of (xiii) Years GATE & ESE 65(i)Parial (ii)Homogeneous (iii)Euler s Theorem on Homogenous (iv)Total Differential (v)Change of (vi) (vii)Chain Rule of (vi)(viii)Functional Years GATE & ESE 78(i)Expansion of (ii)Maclaurin s (iii)Taylor s (iv)Convergence and Divergence of Infinite (v)Methods to Find Convergence of Infinite (vi)Power Years GATE & ESE Value 86(i)Rolle s (ii)Lagrange s First Mean Value (iii)Cauchy s Mean Value (iv)

6 Bolzanos (v)Intermediate Value (vi)Darboux Years GATE & ESE and 114(i)Increasing and Decreasing (ii)Maxima and Minima of Function of Single (iii)Sufficient Condition (Second Derivate Test)..89(iv)Maxima-Minima of Functions of Two (v)Sufficient Condition (Lagrange s Conditions)..91(vi)Lagrange s Method of Undetermined Years GATE & ESE 176(i)Concept of (ii)Some Standard (vii)(iii)Some Important Integration and their (iv)Definite (v)Fundamental Properties of Definite (vi)Fundamental Theorem of Integral (vii)Definite integral as the Limit of a (viii)Double (ix)Double Integrals in Polar (x)Triple (xi)Change of Order of (xii)Use of Jacobian in Multiple (xiii)Multiple Integral using Change of Variables (xiv)Area and Volume in Different Coordinates (xv)Arc Length of Curves (Rectification).

7 139(xvi)Intrinsic Equation of a (xvii)Volumes of Solids of (xviii)Surfaces of Solids of (xix)Beta and Gamma (xx)Properties of Beta & Gamma (xxi)Leibnitz Rule of Differentiation under the sign of (xxii)Improper Years GATE & ESE and their 191(i)Cartesian (ii) (iii)Types of (iv) (v)Types of (vi)Some Basic (vii)Graph Years GATE & ESE (viii)UNIT 2 : VECTOR & their Basic 202(i) (ii)Addition or Subtraction of Two (iii)Dot (iv)Angle between Two (v)Triangle (vi)Cross (vii)Area of a (viii)Area of a (ix)Direction Cosines of a (x)Direction Ratio of a Line joining Two Years GATE & ESE , Divergence and 229(i)Partial Derivatives of (ii)Point (iii)Del (iv)The Laplacian (v) (vi)Directional (vii)Level (viii) (ix)Curl (Rotation).

8 211(x)Vector Years GATE & ESE 248(i)Line (ii)Surface (iii)Volume (ix)(iv)Gauss Divergence (v)Stoke s (vi)Green s Years GATE & ESE 3 : COMPLEX 264(i)Complex Numbers as Ordered (ii)Euler (iii)Algebraic (iv)Geometrical (v)Modulus and (vi)Complex Conjugate (vii)Cube Roots of Years GATE & ESE 280(i)Neighbourhood of Complex Number (ii)Function of a Complex (iii)Types of Complex (iv)Continuity of Complex (v)Differentiability of Complex (vi)Analytic (vii)Cauchy-Riemann (viii)Polar Form of Cauchy-Riemann (ix)Harmonic Years GATE & ESE 310(i) (ii)Elementary Properties of Complex (iii)Cauchy Integral (x)(iv)Cauchy s Integral (v)Taylor Series of Complex (vi)Laurent s (vii)Zeros of an Analytic (viii)Singularities of an Analytic (ix)Types of Isolated Singular (x) (xi)Cauchy-Residue (xii)

9 Methods of Evaluating Years GATE & ESE 4 : DIFFERENTIAL Differential 341(i)Definition of Differential (ii)Order and Degree of Ordinary Differential (iii)Non-Linear Differential (iv)Solution of Differential (v)Formation of Differential (vi) (vii)Linearly dependent & Linearly independent (viii)Methods of Solving Differential (ix)Differential Equations of First Order and First (x)Exact Differential (xi)Equations Reducible to Exact Years GATE & ESE Differential Equations of Higher Orderwith Constant 384(i)Complementary (ii)Rules for Finding the Complementary (iii)Particular (xi)(iv)Methods of Evaluating Particular (v)Cauchy s Homogeneous Linear Differential (vi)Legender s Homogeneous Linear Differential (vii)

10 Simultaneous Linear Differential (viii)Variation of (ix)Differential Equation of First Order and Higher (x)Clairaut s Years GATE & ESE Differential 395(i)Order and Degree of Partial Differential (ii)Linear Partial Differential (iii)Classification of 2nd Order Linear in Two (iv)One Dimensional Wave (v)One Dimensional Heat (vi)Two Dimensional Heat (vii)Laplace Years GATE & ESE 5 : NUMERICAL Solutions of Linear 404(i)Gauss-Jacobi (ii)Gauss-Seidel Years GATE & ESE Solution of Algebraic and Transcendental 431(i)Graphical (ii)Bisection (iii)Regula-Falsi (iv)The Secant (v)Newton-Raphson Years GATE & ESE (xii) 440(i)Interpolation and (ii)Methods of (iii)Notation of Finite Difference (iv)Newton-Gregory Forward Interpolation Formula (for equal intervals).