MATHEMATICAL PHYSICS

MATHEMATICAL PHYSICS

MATHEMATICAL PHYSICS [For the Students of B.Sc. (Honours) and M.Sc. (Physics)] H.K. DASS M.Sc. Diploma in Specialist Studies (Mathematics) University of Hull England Assisted by Dr. RAMA VERMA M.Sc. (Gold Medalist), Ph.D. HOD (Mathematics) Mata Sundri College Delhi University Secular India Award - 98 for National Integration and Communal Harmony given by Prime Minister Shri Atal Behari Vajpayee on 12th June 1999. S. CHAND & COMPANY PVT. LTD. (AN ISO 9001 : 2008 COMPANY) RAM NAGAR, NEW DELHI-110055

S. CHAND & COMPANY PVT. LTD. (An ISO 9001 : 2008 Company) Head Office: 7361, RAM NAGAR, NEW DELHI - 110 055 Phone: 23672080-81-82, 9899107446, 9911310888; Fax: 91-11-23677446 Shop at: schandgroup.com; e-mail: info@schandgroup.com Branches : AHMEDABAD : 1st Floor, Heritage, Near Gujarat Vidhyapeeth, Ashram Road, Ahmedabad - 380014, Ph: 27541965, 27542369, ahmedabad@schandgroup.com BENGALURU : No. 6, Ahuja Chambers, 1st Cross, Kumara Krupa Road, Bengaluru - 560 001, Ph: 22268048, 22354008, bangalore@schandgroup.com BHOPAL : Bajaj Tower, Plot No. 2&3, Lala Lajpat Rai Colony, Raisen Road, Bhopal - 462 011, Ph: 4274723, 4209587. bhopal@schandgroup.com CHANDIGARH : S.C.O. 2419-20, First Floor, Sector - 22-C (Near Aroma Hotel), Chandigarh -160 022, Ph: 2725443, 2725446, chandigarh@schandgroup.com CHENNAI : No.1, Whites Road, Opposite Express Avenue, Royapettah, Chennai - 600014 Ph. 28410027, 28410058, chennai@schandgroup.com COIMBATORE : 1790, Trichy Road, LGB Colony, Ramanathapuram, Coimbatore -6410045, Ph: 2323620, 4217136 coimbatore@schandgroup.com (Marketing Office) CUTTACK : 1st Floor, Bhartia Tower, Badambadi, Cuttack - 753 009, Ph: 2332580; 2332581, cuttack@schandgroup.com DEHRADUN : 1st Floor, 20, New Road, Near Dwarka Store, Dehradun - 248 001, Ph: 2711101, 2710861, dehradun@schandgroup.com GUWAHATI : Dilip Commercial (Ist floor), M.N. Road, Pan Bazar, Guwahati - 781 001, Ph: 2738811, 2735640 guwahati@schandgroup.com HALDWANI : Bhatt Colony, Talli Bamori, Mukhani, Haldwani -263139 (Marketing Office) Mob. 09452294584 HYDERABAD : Padma Plaza, H.No. 3-4-630, Opp. Ratna College, Narayanaguda, Hyderabad - 500029, Ph: 27550194, 27550195, hyderabad@schandgroup.com JAIPUR : 1st Floor, Nand Plaza, Hawa Sadak, Ajmer Road, Jaipur - 302 006, Ph: 2219175, 2219176, jaipur@schandgroup.com JALANDHAR : Mai Hiran Gate, Jalandhar - 144008, Ph: 2401630, 5000630, jalandhar@schandgroup.com KOCHI : Kachapilly Square, Mullassery Canal Road, Ernakulam, Kochi - 682 011, Ph: 2378740, 2378207-08, cochin@schandgroup.com KOLKATA : 285/J, Bipin Bihari Ganguli Street, Kolkata - 700012, Ph: 22367459, 22373914, kolkata@schandgroup.com LUCKNOW : Mahabeer Market, 25 Gwynne Road, Aminabad, Lucknow - 226018, Ph: 4076971, 4026791, 4065646, 4027188, lucknow@schandgroup.com MUMBAI : Blackie House, IInd Floor, 103/5, Walchand Hirachand Marg, Opp. G.P.O., Mumbai - 400 001, Ph: 22690881, 22610885, mumbai@schandgroup.com NAGPUR : Karnal Bagh, Near Model Mill Chowk, Nagpur - 440032, Ph: 2720523, 2777666 nagpur@schandgroup.com PATNA : 104, Citicentre Ashok, Mahima Palace, Govind Mitra Road, Patna - 800 004, Ph: 2300489, 2302100, patna@schandgroup.com PUNE : Sadguru Enclave, Ground floor, Survey No. 114/3, Plot no. 8 Alandi Road, Vishrantwadi Pune 411015 Ph: 64017298 pune@schandgroup.com RAIPUR : Kailash Residency, Plot No. 4B, Bottle House Road, Shankar Nagar, Raipur - 492 007, Ph: 2443142,Mb. : 09981200834, raipur@schandgroup.com (Marketing Office) RANCHI : Shanti Deep Tower, Opp.Hotel Maharaja, Radium Road, Ranchi-834001 Mob. 09430246440 ranchi@schandgroup.com SILIGURI : 122, Raja Ram Mohan Roy Road, East Vivekanandapally, P.O., Siliguri, Siliguri-734001, Dist., Jalpaiguri, (W.B.) Ph. 0353-2520750 (Marketing Office) siliguri@schandgroup.com VISAKHAPATNAM : No. 49-54-15/53/8, Plot No. 7, 1st Floor, Opp. Radhakrishna Towers, Seethammadhara North Extn., Visakhapatnam - 530 013, Ph-2782609 (M) 09440100555, visakhapatnam@schandgroup.com (Marketing Office) 1997, H.K. Dass All rights reserved. No part of this publication may be reproduced or copied in any material form (including photocopying or storing it in any medium in form of graphics, electronic or mechanical means and whether or not transient or incidental to some other use of this publication) without written permission of the copyright owner. Any breach of this will entail legal action and prosecution without further notice. Jurisdiction : All disputes with respect to this publication shall be subject to the jurisdiction of the Courts, Tribunals and Forums of New Delhi, India only. First Edition 1997 Subsequent Editions and Reprints 2003, 2004, 2005, 2007, 2008 (Twice), 2009 (Twice), 2010, 2011 (Twice), 2012, 2013, 2014 Seventh Revised Edition 2014 ISBN : 81-219-1469-8 Code : 16C 224 PRINTED IN INDIA By Rajendra Ravindra Printers Pvt. Ltd., 7361, Ram Nagar, New Delhi -110 055 and published by S. Chand & Company Pvt. Ltd., 7361, Ram Nagar, New Delhi -110 055.

PREFACE TO THE SEVENTH REVISED EDITION The demand of Mathematical Physics by the students and teachers has encouraged me to revise the text book. The entire book is rewritten in such a way that it can cover the syllabus of B.Sc. (H) Physics, B.Sc.(H) Electronics, and M.Sc. (Physics) of various universities. The contents of the book is divided into five units. Each unit is further divided into simpler and short chapters, so that readers can follow the subject matter easily. The text is very lucid and simple. Four Solved Question Papers of Delhi University, 1st, 2nd, 3rd and 4th Semesters, 2012 are included at the end of the textbook. This book should satisfy both average and brilliant students. It would help the students to get high grades in their examination and at the same time would arouse greater intellectual curiosity in them. The misprints that came to my knowledge, have been removed. We are thankful to the Management Team and the Editorial Department of S. Chand & Company Pvt. Ltd. for all help and support in the publication of this book. All valuable suggestions for the improvement of the book will be highly appreciated and gratefully acknowledged too. D-1/87, Janakpuri New Delhi-110 058 Mob. 9350055078 011-28525078, 32985078 e-mail: hk_dass@yahoo.com H.K. DASS Disclaimer : While the authors of this book have made every effort to avoid any mistake or omission and have used their skill, expertise and knowledge to the best of their capacity to provide accurate and updated information. The author and S. Chand does not give any representation or warranty with respect to the accuracy or completeness of the contents of this publication and are selling this publication on the condition and understanding that they shall not be made liable in any manner whatsoever. S.Chand and the author expressly disclaim all and any liability/responsibility to any person, whether a purchaser or reader of this publication or not, in respect of anything and everything forming part of the contents of this publication. S. Chand shall not be responsible for any errors, omissions or damages arising out of the use of the information contained in this publication. Further, the appearance of the personal name, location, place and incidence, if any; in the illustrations used herein is purely coincidental and work of imagination. Thus the same should in no manner be termed as defamatory to any individual. (v)

PREFACE TO THE FIRST EDITION This is my first effort to write a book on Mathematical Physics. The chief aim of this book is to meet the requirements of students of B.Sc. Honours (Physics) and M.Sc. of various Indian Universities. The subject-matter is presented in a very systematic and logical manner. Every endeavour has been made to make the content simple and lucid as far as possible. While every effort has been made to present the material correctly, no attempt has been made to be absolutely rigorous. The subject matter has been so arranged that even an average student can understand how to apply the mathematical operations to the problems of Physics. All valuable suggestions for the improvement of the book will be highly appreciated and gratefully acknowledged. I am thankful to Shri Rajendra Kumar Gupta, Managing Director, Shri Ravindra Kumar Gupta, Director and other members of the staff of the Publishers, M/s S. Chand & Co. Ltd., New Delhi without whose co-operation, it would not have been possible to put this book in such a fine format and that too in record time. D-1/87, Janakpuri New Delhi-110 058 Tel. 5555078 H.K. DASS

C O N TEN TS UNIT I 1. REVIEW OF VECTOR ALGEBRA 1 14 1.1 Vectors 1; 1.2 Addition of Vectors 2; 1.3 Rectangular Resolution of a Vector 512; 1.4 Unit Vector 2; 1.5 Position Vector of a Point 2; 1.6 Ratio formula 3; 1.7 Product of Two Vectors 3; 1.8 Scalar, or Dot Product 3; 1.9 Useful Results 4; 1.10 Work Done As a Scalar Product 4; 1.11 Vector Product or Cross Product 4; 1.12 Vector Product Expressed As a Determinant 4; 1.13 Area of Parallelogram 5; 1.14 Moment of a force 5; 1.15 Angular Velocity 5; 1.16 Scalar Triple Product 5; 1.17 Geometrical Interpretation 6; 1.18 Coplanarity Questions 8; 1.19 Vector Product of Three Vectors 10; 1.20 Scalar Product of Four Vectors 12; 1.21 Vector Product of Four Vectors 21. 2. DIFFERENTIATION OF VECTOR (POINT FUNCTION, GRADIENT, DIVERGENCE AND CURL OF A VECTOR AND THEIR PHYSICAL INTERPRETATIONS) 15 55 2.1 Vector Function 15; 2.2 Differentiation of Vectors 15; 2.3 Formulae of Differentiation 16; 2.4 Scalar and Vector Point Functions 18; 2.5 Gradient of a Scalar Function 18; 2.6 Geometrical Meaning of Gradient, Normal 19; 2.7 Normal and Directional Derivative 19; 2.8 Divergence of a Vector Function 31; 2.9 Physical Interpretation of Divergence 32; 2.10 Curl 36; 2.11 Physical Meaning of Curl 36. 3. INTEGRATION OF VECTORS 56 107 3.1 Line Integral 56; 3.2 Surface Integral 64; 3.3 Volume Integral 66; 3.4 Green s Theorem 67; 3.5 Area of the Plane Region by Green s theorem 70; 3.6 Stoke s theorem (Relation Between Line Integral and Surface Integral) 72; 3.7 Another Method of Proving Stoke s theorem 73; 3.8 Gauss s theorem of Divergence 88; 3.9 Deduction from Gauss Diversion Theorem 103. 4. ORTHOGONAL CURVILINEAR COORDINATES 108 121 4.1 Curvilinear Coordinates 108; 4.2 Differential of an Arc Length 109; 4.3 Geometrical Significance of h1, h2, h3 109; 4.4 Differential Operator 109; 4.5 Divergence 110; 4.6 Curl 111; 4.7 Laplacian Operator 2 112; 4.8 Cylindrical (Polar) Co-ordinates 112; 4.9 Spherical Polar Co-ordinates 114; 4.10 Transformation of Cylindrical Polar Co-ordinates Into iˆ, ˆj, kˆ 117; 4.11 Conversion of Spherical Polar Co-ordinates (r,, ) into iˆ, ˆj, kˆ 117; 4.12 Relation Between Cylindrical and Spherical Co-ordinates 118. 5. DOUBLE INTEGRALS 122 148 5.1 Double Integration 122; 5.2 Evaluation of Double Integral 122; 5.3 Evaluation of double integrals in Polar Co-ordinates 127; 5.4 Change of order of Integration 132; 5.5 Change of Variables 142. (vii) Created with Print2PDF. To remove this line, buy a license at: http://www.software602.com/

6. APPLICATION OF THE DOUBLE INTEGRALS (AREA, CENTRE OF GRAVITY, MASS, VOLUME) 149 160 6.1 Introduction 149; 6.2 Area in Cartesian Co-ordinates 149; 6.3 Area in Polar Co-ordinates 152; 6.4 Volume of Solid by Rotation of an area (Double Integral) 155; 6.5 Centre of Gravity 157; 6.6 Centre of Gravity of an Arc 159. 7. TRIPLE INTEGRATION 161 172 7.1 Introduction 161; 7.2 Triple Integration 163; 7.3 Integration by Change of Cartesian Coordinates Into Spherical Coordinates 166. 8. APPLICATION OF TRIPLE INTEGRATION 173 197 8.1 Introduction 173; 8.2 Volume = dx dy dz. 173; 8.3 Volume of Solid bounded by Sphere or by Cylinder 175; 8.4 Volume of Solid bounded by Cylinder or Cone 177; 8.5 Volume Bounded by a Paraboloid 182; 8.6 Surface Area 185; 8.7 Calculation of Mass 189; 8.8 Centre of Gravity 191; 8.9 Moment of Inertia of a Solid 191; 8.10 Centre of Pressure 195. 9. GAMA, BETA FUNCTIONS 198 240 9.1 Gamma Function 198; 9.2 Prove that 200; 9.3 Transformation of Gamma Function 201; 9.4 Beta Function 202; 9.5 Evaluation of Beta Function 202; 9.6 A Property of Beta Function 203; 9.7 Transformation of Beta Function 204; 9.8 Relation Between Beta and Gamma Functions 204; 9.9 Show that 205; 9.10 Duplication formula 211; 9.11 To show that 212; 9.12 To show that 213; 9.13 Double Integration 219; 9.14 Dirichlet s integral (Triple Integration) 221; 9.15 Liouville s Extension of Dirichlet theorem 221; 9.16 Elliptic Integrals 228; 9.17 Definition and Property 228; 9.18 Error Function 232; 9.19 Differentiation Under the Integral Sign 233; 9.20 Leibnitz's Rule 234; 9.21 Rule of Differentiation Under the Integral Sign When the Limits of Integration are Functions of the Parameter 237. 10. THEORY OF ERRORS 241 248 10.1 Numbers 241; 10.2 Significant Figures 241; 10.3 Rounding off 241; 10.4 Types of Errors 242; 10.5 Error due to Approximation of the Function 244; 10.6 Error in a series Approximation 245; 10.7 Order of Approximation 246; 10.8 Most Probable Value and Residual 246; 10.9 Gaussian Error 247; 10.10 Theoretical Distributions 247. 11. FOURIER SERIES 249 288 11.1 Periodic Functions 249; 11.2 fourier Series 249; 11.3 Dirichlet s Conditions for A Fourier Series 250; 11.4 Advantages of Fourier Series 250; 11.5 Useful Integrals 250; 11.6 Determination of Fourier Coefficients (Euler s formulae) 251; 11.7 Fourier Series for Discontinuous Functions 255; 11.8 Function Defined in Two or More Sub-ranges 256; 11.9 Discontinuous Functions 257; 11.10 Even Function and Odd Function 261; 11.11 Half-range Series, Period 0 to p 265; 11.12 Change of Interval and Functions having Arbitrary Period 268; 11.13 Half Period Series 271; 11.14 Parseval s formula 280; 11.15 Fourier Series in Complex form 284; 11.16 Practical Harmonic Analysis 285. (viii)

UNIT II 12. DIFFERENTIAL EQUATIONS OF FIRST ORDER 289 325 12.1 Definition 289; 12.2 order and Degree of a Differential Equation 289; 12.3 Formation of Differential Equations 289; 12.4 Solution of a Differential Equation 292; 12.5 Geometrical Meaning of the Differential Equation of the First order and First Degree 292; 12.6 Differential Equations of the First order and First Degree 292; 12.7 Variables Separable 293; 12.8 Homogeneous Differential Equations 295; 12.9 Equations Reducible to Homogeneous form 297; 12.10 Linear Differential Equations 299; 12.11 Equations Reducible to the Linear form (Bernoulli Equation) 302; 12.12 Exact Differential Equation 309; 12.13 Equations Reducible to the Exact Equations 313; 12.14 Differential Equations Reducible to Exact form (by Inspection) 317; 12.15 Equations of First order and Higher Degree 318; 12.16 Orthogonal Trajectories 320; 12.17 Polar Equation of the Family of Curves 327; 13. LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER 326 356 13.1 Linear Differential Equations 326; 13.2 Non Linear Differential Equations 326; 13.3 Linear Differential Equations of Second order with Constant Coefficients 326; 13.4 Dimension of space of Salution 327; 13.5 Non-Homogeneous 327; 13.6 Homogeneous 327; 13.7 Superposition or Linearity Principle 327; 13.8 Linear Independence and dependence 328; 13.9 Wronskian 328; 13.10 Existence of linearly Independence 328; 13.11 Structure Theorem 328; 13.12 Super position Principle 329 13.13 Abels formula 331; 13.14 Complete Solution = Complementary Function + Particular Integral 334;13.15 Method for finding the Complementary Function 235; 13.16 Rules to find Particular Integral 338; 13.17 1 ax 1 ax e e f( D) f ( a) 339; 13.18 1 n 1 n 1 sin ax x [ f( D)] x. f ( D) 341; 13.19 sin ax 2 2 342; f( D ) f ( a ) 1 ax ax 1 13.20. e ( x) e..() x 1 n f( D) f( D a) 347; 13.21 To find the Value of x sin ax. 353; f( D) 13.12 General Method of finding the Particular Integral of any Function f (x) 263. 14. CAUCHY EULER EQUATIONS, METHOD OF VARIATION OF PARAMETERS 357 381 14.1 Cauchy Euler Homogeneous Linear Equations 357; 14.2 Legendre's Homogeneous Differential Equations 364; 14.3 Method of Variation of Parameters 367; 14.4 Method Undetermined Coefficients 377. 15. DIFFERENTIAL EQUATION OF OTHER TYPES 382 405 d y 15.1 Introduction 382; 15.2 Equation of the Type f ( x) n 382; 15.3 Equation of dx d n y the Type f ( y) n 383; 15.4 Equations which do not contain y directly 386; 15.5 dx Equations which do not contain x directly 385; 15.6 Equations whose solution is known 389; 15.7 Normal form (Removal of first derivative) 394; 15.8 Method of Solving (ix) n

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