MATHEMATICAL PHYSICS - Kopykitab

1 MATHEMATICAL PHYSICSMATHEMATICAL PHYSICS [For the Students of (Honours) and ( PHYSICS )] DASS in Specialist studies (Mathematics)University of HullEnglandAssisted byDr. RAMA (Gold Medalist), (Mathematics)Mata Sundri CollegeDelhi UniversitySecular India Award - 98 for National Integration and Communal Harmonygiven by Prime Minister Shri Atal Behari Vajpayee on 12th June CHAND & COMPANY PVT. LTD.(AN ISO 9001 : 2008 COMPANY)RAM NAGAR, NEW DELHI-110055S. 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: ; e-mail: :AHMEDABAD:1st Floor, Heritage, Near Gujarat Vidhyapeeth, Ashram Road, Ahmedabad - 380 014,Ph: 27541965, 27542369, 6, Ahuja Chambers, 1st Cross, Kumara Krupa Road, Bengaluru - 560 001,Ph: 22268048, 22354008, Tower, Plot No.

2 2&3, Lala Lajpat Rai Colony, Raisen Road, Bhopal - 462 011,Ph: 4274723, 4209587. 2419-20, First Floor, Sector - 22-C (Near Aroma Hotel), Chandigarh -160 022,Ph: 2725443, 2725446, Whites Road, Opposite Express Avenue, Royapettah, Chennai - 600014 Ph. 28410027, 28410058, Trichy Road, LGB Colony, Ramanathapuram, Coimbatore -6410045,Ph: 2323620, 4217136 (Marketing Office)CUTTACK:1st Floor, Bhartia Tower, Badambadi, Cuttack - 753 009, Ph: 2332580; 2332581, Floor, 20, New Road, Near Dwarka Store, Dehradun - 248 001,Ph: 2711101, 2710861, Commercial (Ist floor), Road, Pan Bazar, Guwahati - 781 001,Ph: 2738811, 2735640 Colony, Talli Bamori, Mukhani, Haldwani -263139 (Marketing Office) Mob.

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4 114/3, Plot no. 8 Alandi Road ,Vishrantwadi Pune 411015 Ph: 64017298 Residency, Plot No. 4B, Bottle House Road, Shankar Nagar, Raipur - 492 007,Ph: 2443142,Mb. : 09981200834, (Marketing Office)RANCHI:Shanti Deep Tower, Maharaja, Radium Road, Ranchi-834001 Mob. 09430246440 Raja Ram Mohan Roy Road, East Vivekanandapally, , Siliguri, Siliguri-734001,Dist., Jalpaiguri, ( ) Ph. 0353-2520750 (Marketing Office) 49-54-15/53/8, Plot No. 7, 1st Floor, Opp. Radhakrishna Towers,Seethammadhara North Extn., Visakhapatnam - 530 013, Ph-2782609 (M) (Marketing Office) 1997, DassAll rights reserved. No part of this publication may be reproduced or copied in any material form (includingphotocopying or storing it in any medium in form of graphics, electronic or mechanical means and whetheror not transient or incidental to some other use of this publication) without written permission of thecopyright owner.

5 Any breach of this will entail legal action and prosecution without further : All disputes with respect to this publication shall be subject to the jurisdiction of the Courts,Tribunals and Forums of New Delhi, India 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 INDIABy Rajendra Ravindra Printers Pvt. Ltd., 7361, Ram Nagar, New Delhi -110 055and published by S. Chand & Company Pvt. Ltd., 7361, Ram Nagar, New Delhi -110 TO THE SEVENTH REVISED EDITIONThe demand of MATHEMATICAL PHYSICS by the students and teachers has encouraged me torevise the text book.

6 The entire book is rewritten in such a way that it can cover the syllabus (H) PHYSICS , (H) Electronics, and ( PHYSICS ) of various contents of the book is divided into five units. Each unit is further divided into simpler andshort chapters, so that readers can follow the subject matter easily. The text is very lucid and Solved Question Papers of Delhi University, 1st, 2nd, 3rd and 4th Semesters, 2012 areincluded at the end of the book should satisfy both average and brilliant students. It would help the students to gethigh grades in their examination and at the same time would arouse greater intellectual curiosity misprints that came to my knowledge, have been are thankful to the Management Team and the Editorial Department of S.

7 Chand & CompanyPvt. Ltd. for all help and support in the publication of this valuable suggestions for the improvement of the book will be highly appreciated andgratefully acknowledged , DASSNew Delhi-110 058 Mob. 9350055078 011-28525078, 32985078e-mail: : 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. Chanddoes not give any representation or warranty with respect to the accuracy or completeness of the contents of this publicationand are selling this publication on the condition and understanding that they shall not be made liable in any manner and the author expressly disclaim all and any liability/responsibility to any person, whether a purchaser or reader ofthis publication or not, in respect of anything and everything forming part of the contents of this publication.

8 S. Chand shall notbe responsible for any errors, omissions or damages arising out of the use of the information contained in this , the appearance of the personal name, location, place and incidence, if any; in the illustrations used herein is purelycoincidental and work of imagination. Thus the same should in no manner be termed as defamatory to any TO THE FIRST EDITIONThis is my first effort to write a book on MATHEMATICAL chief aim of this book is to meet the requirements of students of Honours ( PHYSICS )and of various Indian subject-matter is presented in a very systematic and logical manner. Every endeavour hasbeen made to make the content simple and lucid as far as every effort has been made to present the material correctly, no attempt has been madeto be absolutely rigorous.

9 The subject matter has been so arranged that even an average student canunderstand how to apply the MATHEMATICAL operations to the problems of valuable suggestions for the improvement of the book will be highly appreciated andgratefully 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 Delhiwithout whose co-operation, it would not have been possible to put this book in such a fine formatand that too in record , DASSNew Delhi-110 058 Tel. 5555078(vii)UNIT I1. REVIEW OF VECTOR ALGEBRA1 Vectors 1; Addition of Vectors 2; Rectangular Resolution of a Vector 512; Unit Vector 2; Position Vector of a Point 2; Ratio formula 3; Product ofTwo Vectors 3; Scalar, or Dot Product 3; Useful Results 4; Work Done Asa Scalar Product 4; Vector Product or Cross Product 4; Vector ProductExpressed As a Determinant 4; Area of Parallelogram 5; Moment of a force5; Angular Velocity 5; Scalar Triple Product 5; Geometrical Interpretation6; Coplanarity Questions 8; Vector Product of Three Vectors 10; ScalarProduct of Four Vectors 12.

10 Vector Product of Four Vectors DIFFERENTIATION OF VECTOR (POINT FUNCTION, GRADIENT, DIVERGENCE AND CURL OFA VECTOR AND THEIR PHYSICAL INTERPRETATIONS)15 Vector Function 15; Differentiation of Vectors 15; Formulae of Differentiation16; Scalar and Vector Point Functions 18; Gradient of a Scalar Function 18; Meaning of Gradient, Normal 19; Normal and Directional Derivative19; Divergence of a Vector Function 31; Physical Interpretation of Divergence32; Curl 36; Physical Meaning of Curl INTEGRATION OF VECTORS56 Line Integral 56; Surface Integral 64; Volume Integral 66; Green sTheorem 67; Area of the Plane Region by Green s theorem 70; Stoke s theorem(Relation Between Line Integral and Surface Integral) 72; Another Method ofProving Stoke s theorem 73; Gauss s theorem of Divergence 88; Deductionfrom Gauss Diversion Theorem ORTHOGONAL CURVILINEAR COORDINATES108 Curvilinear Coordinates 108; Differential of an Arc Length 109; GeometricalSignificance of h1, h2, h3 109; Differential Operator 109; Divergence 110; Curl 111; Laplacian Operator 2?