Engineering Mathematics-II This page is intentionally left blank. Engineering Mathematics-II Third Semester Rajiv Gandhi Proudyogiki Vishwavidyalaya Paper Code: BE-301 BABU RAM Formerly Dean, Faculty of Physical Sciences, Maharshi Dayanand University, Rohtak Customized by VIJAY GUPTA Associate Professor, Department of Mathematics, UIT, Rajiv Gandhi Proudyogiki Vishwavidyalaya, Bhopal, M.P Copyright © 2012 Dorling Kindersley (India) Pvt. Ltd Licensees of Pearson Education in South Asia No part of this eBook may be used or reproduced in any manner whatsoever without the publisher’s prior written consent. This eBook may or may not include all assets that were part of the print version. The publisher reserves the right to remove any material present in this eBook at any time. ISBN 9788131764176 eISBN 9788131776292 Head Office: A-8(A), Sector 62, Knowledge Boulevard, 7th Floor, NOIDA 201 309, India Registered Office: 11 Local Shopping Centre, Panchsheel Park, New Delhi 110 017, India Contents Preface viii 2 Fourier Transform 2.1 Roadmap to the Syllabus ix 2.1 Fourier Integral Theorem 2.1 Symbols and Basic Formulae x 2.2 Fourier Transforms 2.4 2.3 Fourier Cosine and Sine 1 Fourier Series 1.1 Transforms 2.5 2.4 Properties of Fourier Transforms 2.6 1.1 Trigonometric Series 1.1 2.5 Solved Examples 2.9 1.2 Fourier (or Euler) Formulae 1.2 2.6 Complex Fourier Transforms 2.18 1.3 Periodic Extension of a Function 1.4 2.7 Convolution Theorem 2.19 1.4 Fourier Cosine and Sine Series 1.5 2.8 Parseval’s Identities 2.21 1.5 Complex Fourier Series 1.6 2.9 Fourier Integral Representation of a 1.6 Spectrum of Periodic Functions 1.7 Function 2.23 1.7 Properties of Fourier Coeffi cients 1.7 2.10 Finite Fourier Transforms 2.25 1.8 Dirichlet’s Kernel 1.10 2.11 Applications of Fourier Transforms 2.26 1.9 Integral Expression for Partial Sums 2.12 Application to Differential of a Fourier Series 1.11 Equations 2.26 1.10 F undamental Theorem (Convergence 2.13 Application to Partial Differential Theorem) of Fourier Series 1.12 Equations 2.30 1.11 Applications of Fundamental Theorem of Fourier Series 1.13 Exercises 2.38 1.12 Convolution Theorem for Fourier Series 1.14 3 Laplace Transform 3.1 1.13 Integration of Fourier Series 1.15 1.14 Differentiation of Fourier Series 1.16 3.1 D efi nition and Examples of Laplace 1.15 Examples of Expansions of Functions Transform 3.1 in Fourier Series 1.17 3.2 Properties of Laplace Transforms 3.8 1.16 Method to Find Harmonics of Fourier 3.3 Limiting Theorems 3.24 Series of a Function from Tabular 3.4 Miscellaneous Examples 3.25 Values 1.33 Exercises 3.28 1.17 Signals and Systems 1.35 1.18 Classifi cation of Signals 1.35 1.19 Classifi cation of Systems 1.37 4 Inverse Laplace Transform 4.1 1.20 Response of a Stable Linear 4.1 D efi nition and Examples of Inverse Timeinvariant Continuous Time System Laplace Transform 4.1 (LTC System) to a Piecewise Smooth 4.2 Properties of Inverse Laplace and Periodic Input 1.38 Transform 4.2 1.21 Application to Differential 4.3 Partial Fractions Method to Find Inverse Equations 1.39 Laplace Transform 4.10 1.22 Application to Partial Differential 4.4 Heaviside’s Expansion Theorem 4.14 Equations 1.41 4.5 Series Method to Determine Inverse 1.23 Miscellaneous Examples 1.45 Laplace Transform 4.15 Exercises 1.49 4.6 Convolution Theorem 4.16 vi (cid:132) Contents 4.7 Complex Inversion Formula 4.21 8 Partial Differential Equations 8.1 4.8 Miscellaneous Examples 4.26 8.1 Formulation of Partial Differential Exercises 4.31 Equation 8.1 8.2 Solutions of a Partial Differential Equation 8.4 5 Applications of Laplace Transform 5.1 8.3 Miscellaneous Examples 8.11 5.1 Ordinary Differential Equations 5.1 Exercises 8.14 5.2 S imultaneous Differential Equations 5.14 9 Non-Linear Partial Differential Equations 9.1 5.3 Difference Equations 5.17 5.4 Integral Equations 5.22 9.1 Non-linear Partial Differential 5.5 Integro-differential Equations 5.26 Equations of the First Order 9.1 5.6 Solution of Partial Differential 9.2 Charpit’s Method 9.1 Equations 5.26 9.3 Some Standard Forms of Non-linear 5.7 Evaluation of Integrals 5.30 Equations 9.7 5.8 Miscellaneous Examples 5.33 Exercises 9.14 Exercises 5.38 10 Partial Differential Equations with Constant 6 Second Order Differential Equation Coeffi cient 10.1 with Variable Coeffi cient 6.1 10.1 Linear Partial Differential Equations 6.1 Method of Solution by Changing with Constant Coeffi cients 10.1 Independent Variable 6.1 10.2 Equations Reducible to Homogeneous 6.2 Method of Solution by Changing the Linear Form 10.18 Dependent Variable 6.7 6.3 Method of Undetermined Exercises 10.21 Coeffi cients 6.13 6.4 Method of Reduction of Order 6.14 11 Classical Partial Differential Equations 11.1 6.5 Cauchy–Euler Homogeneous Linear 11.1 C lassifi cation of Second Order Linear Equation 6.22 Partial Differential Equations 11.1 6.6 Legendre’s Linear Equation 6.26 11.2 The Method of Separation of 6.7 Method of Variation of Parameters Variables 11.1 to Find Particular Integral 6.27 11.3 Classical Partial Differential Exercises 6.34 Equations 11.3 11.4 Solutions of Laplace Equation 11.19 7 Series Solution of Ordinary 11.5 Telephone Equations of a Transmission Differential Equations 7.1 Line 11.21 11.6 Miscellaneous Examples 11.25 7.1 Solution in Series 7.1 7.2 Bessel’s Equation and Bessel’s Exercises 11.30 Function 7.11 7.3 Fourier–Bessel Expansion of a Continu- 12 Vector Differentiation 12.1 ous Function 7.19 12.1 Differentiation of a Vector 12.3 7.4 Legendre’s Equation and Legendre’s 12.2 Partial Derivatives of a Vector Polynomial 7.19 Function 12.10 7.5 Fourier–Legendre Expansion of a 12.3 Gradient of a Scalar Field 12.11 Function 7.26 12.4 G eometrical Interpretation 7.6 Miscellaneous Examples 7.27 of a Gradient 12.12 Exercises 7.30 12.5 Properties of a Gradient 12.12 Contents (cid:132) vii 12.6 Directional Derivatives 12.13 13.3 Work Done by a Force 13.5 12.7 Divergence of a Vector-point 13.4 Surface Integral 13.7 Function 12.18 13.5 Volume Integral 13.12 12.8 Physical Interpretation of 13.6 Gauss’s Divergence Theorem 13.14 Divergence 12.18 13.7 Green’s Theorem in a Plane 13.21 12.9 Curl of a Vector-point Function 12.20 13.8 Stoke’s Theorem 13.25 12.10 Physical Interpretation of Curl 12.20 13.9 Miscellaneous Examples 13.30 12.11 The Laplacian Operator ∇2 12.20 Exercises 13.35 12.12 Properties of Divergence and Curl 12.23 12.13 Miscellaneous Examples 12.28 Exercises 12.31 Solved Question Papers Q.1 Index I.1 13 Integration of Vector Functions 13.1 13.1 Integration of Vector Functions 13.1 13.2 Line Integral 13.2 Preface All branches of engineering, technology and science require mathematics as a tool for the description of their contents. Therefore, a thorough knowledge of the various topics in mathematics is essential to pursue courses in these fi elds. The aim of this book is to provide students with a sound platform to hone their skills in mathematics and its multifarious applications. This edition has been prepared in accordance with the syllabus requirements of Engineering Mathematics-II, a compulsory paper taught in the third semester in Rajiv Gandhi Proudyogiki Vishwavidyalaya. A roadmap to the syllabus has been included for the benefi t of the students. D esigned for classroom and self-study sessions, the book uses simple and lucid language to explain concepts. Several solved examples, fi gures, tables and exercises have been provided to enable students to enhance their problem-solving skills. Five solved university question papers have been appended to the book for the benefi t of the students. Suggestions and feedback for improving the book fur- ther are welcome. Acknowledgements I would like to thank my family members for providing moral support during the preparation of this book. I would like to acknowledge my son, Aman Kumar, Software Engineer, Goldman Sachs, and daughter-in-law, Ritu, Software Engineer, Tech Mahindra who offered wise comments on some of the contents of the book. I am also thankful to Sushma S. Pradeep for excellently typing the manuscript. Special thanks are due to Thomas Mathew Rajesh, Anita Yadav, and Vipin Kumar at Pearson Education for their constructive support. Last but not the least, I would like to thank Dr. Vijay Gupta for his contribution in customizing this book for Rajiv Gandhi Proudyogiki Vishwavidyalaya. BABU RAM Roadmap to the Syllabus Paper Code: BE-301 Engineering Mathematics-II Unit I Fourier Series Introduction of Fourier series, Fourier series for discontinuous functions, Fourier series for even and odd function, Half range series, Fourier transform: Defi nition and properties of Fourier transform, Sine and cosine transform Refer to Chapters 1 and 2 Unit II Laplace Transform Introduction of Laplace transform, Laplace transform of elementary functions, Properties of Laplace transform, Change of scale property, Second shifting property, Laplace trans- form of the derivative, Inverse Laplace transform and its properties, Convolution theorem, Applications of Laplace transform to solve the ordinary differential equations Refer to Chapters 3, 4 and 5 Unit III Second Order Linear Differential Equation with Variable Coeffi cients Methods of one integral is known, Removal of fi rst derivative, Changing of independent variable and Variation of parameter, Solution by Series Method Refer to Chapters 6 and 7 Unit IV Linear and Non Linear Partial Differential Equation of First Order F ormulation of partial differential equations, Solution of equation by direct integration, L agrange’s linear equation, Charpit’s method, Linear partial differential equation of second and higher order: Linear homogeneous and non-homogeneous partial differential equation of nth order with constant coeffi cients, Separation of variable method for the solution of wave and heat equations Refer to Chapters 8, 9, 10 and 11 Unit V Vector Calculus D ifferentiation of vectors, Scalar and vector point function, Geometrical meaning of gradi- ent, Unit normal vector and directional derivative, Physical interpretation of divergence and curl, Line integral, Surface integral and volume integral, Green’s, Stoke’s and Gauss divergence theorem Refer to Chapters 12 and 13