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 byDr. RAMA VERMA

M.Sc. (Gold Medalist), Ph.D.HOD (Mathematics)Mata Sundri College

Delhi University

Secular India Award - 98 for National Integration and Communal Harmonygiven by Prime Minister Shri Atal Behari Vajpayee on 12th June 1999.

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© 1997, H.K. 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. 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, 2014Seventh Revised Edition 2014ISBN : 81-219-1469-8 Code : 16C 224PRINTED IN INDIA

By Rajendra Ravindra Printers Pvt. Ltd., 7361, Ram Nagar, New Delhi -110 055and published by S. Chand & Company Pvt. Ltd., 7361, Ram Nagar, New Delhi -110 055.

PREFACE TO THE SEVENTH REVISED EDITION

The demand of Mathematical Physics by the students and teachers has encouraged me torevise the text book. The entire book is rewritten in such a way that it can cover the syllabus ofB.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 andshort 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 areincluded at the end of the textbook.

This 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 inthem.

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 H.K. DASSNew Delhi-110 058Mob. 9350055078 011-28525078, 32985078e-mail: hk_dass@yahoo.com

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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. 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 whatsoever.S.Chand 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. S. Chand shall notbe 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 purelycoincidental and work of imagination. Thus the same should in no manner be termed as defamatory to any individual.

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 canunderstand how to apply the mathematical operations to the problems of Physics.

All valuable suggestions for the improvement of the book will be highly appreciated andgratefully 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 Delhiwithout whose co-operation, it would not have been possible to put this book in such a fine formatand that too in record time.

D-1/87, Janakpuri H.K. DASSNew Delhi-110 058Tel. 5555078

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UNIT – I1. 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 ofTwo Vectors 3; 1.8 Scalar, or Dot Product 3; 1.9 Useful Results 4; 1.10 Work Done Asa Scalar Product 4; 1.11 Vector Product or Cross Product 4; 1.12 Vector ProductExpressed As a Determinant 4; 1.13 Area of Parallelogram 5; 1.14 Moment of a force5; 1.15 Angular Velocity 5; 1.16 Scalar Triple Product 5; 1.17 Geometrical Interpretation6; 1.18 Coplanarity Questions 8; 1.19 Vector Product of Three Vectors 10; 1.20 ScalarProduct of Four Vectors 12; 1.21 Vector Product of Four Vectors 21.

2. DIFFERENTIATION OF VECTOR (POINT FUNCTION, GRADIENT, DIVERGENCE AND CURL OFA VECTOR AND THEIR PHYSICAL INTERPRETATIONS) 15 – 55

2.1 Vector Function 15; 2.2 Differentiation of Vectors 15; 2.3 Formulae of Differentiation16; 2.4 Scalar and Vector Point Functions 18; 2.5 Gradient of a Scalar Function 18; 2.6Geometrical Meaning of Gradient, Normal 19; 2.7 Normal and Directional Derivative19; 2.8 Divergence of a Vector Function 31; 2.9 Physical Interpretation of Divergence32; 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’sTheorem 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 ofProving Stoke’s theorem 73; 3.8 Gauss’s theorem of Divergence 88; 3.9 Deductionfrom 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 GeometricalSignificance 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 PolarCo-ordinates Into ˆˆ ˆ, ,i j k 117; 4.11 Conversion of Spherical Polar Co-ordinates(r, , ) into ˆˆ ˆ, ,i j k 117; 4.12 Relation Between Cylindrical and Spherical Co-ordinates118.

5. DOUBLE INTEGRALS 122 – 148

5.1 Double Integration 122; 5.2 Evaluation of Double Integral 122; 5.3 Evaluation ofdouble integrals in Polar Co-ordinates 127; 5.4 Change of order of Integration 132;5.5 Change of Variables 142.

C O N T E N T S

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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 PolarCo-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 CartesianCoordinates 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 ofMass 189; 8.8 Centre of Gravity 191; 8.9 Moment of Inertia of a Solid 191; 8.10 Centreof Pressure 195.

9. GAMA, BETA FUNCTIONS 198 – 240

9.1 Gamma Function 198; 9.2 Prove that 200; 9.3 Transformation of Gamma Function201; 9.4 Beta Function 202; 9.5 Evaluation of Beta Function 202; 9.6 A Property of BetaFunction 203; 9.7 Transformation of Beta Function 204; 9.8 Relation Between Betaand Gamma Functions 204; 9.9 Show that 205; 9.10 Duplication formula 211; 9.11 Toshow that 212; 9.12 To show that 213; 9.13 Double Integration 219; 9.14 Dirichlet’sintegral (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 ofDifferentiation Under the Integral Sign When the Limits of Integration are Functionsof 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 ofErrors 242; 10.5 Error due to Approximation of the Function 244; 10.6 Error in aseries Approximation 245; 10.7 Order of Approximation 246; 10.8 Most Probable Valueand 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 AFourier 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 Seriesfor Discontinuous Functions 255; 11.8 Function Defined in Two or MoreSub-ranges 256; 11.9 Discontinuous Functions 257; 11.10 Even Function and OddFunction 261; 11.11 Half-range Series, Period 0 to p 265; 11.12 Change of Interval andFunctions 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 PracticalHarmonic Analysis 285.

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UNIT – II

12. DIFFERENTIAL EQUATIONS OF FIRST ORDER 289 – 325

12.1 Definition 289; 12.2 order and Degree of a Differential Equation 289; 12.3Formation of Differential Equations 289; 12.4 Solution of a Differential Equation292; 12.5 Geometrical Meaning of the Differential Equation of the First order andFirst 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; 12.9Equations Reducible to Homogeneous form 297; 12.10 Linear Differential Equations299; 12.11 Equations Reducible to the Linear form (Bernoulli Equation) 302; 12.12Exact Differential Equation 309; 12.13 Equations Reducible to the Exact Equations313; 12.14 Differential Equations Reducible to Exact form (by Inspection) 317; 12.15Equations 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; 13.6Homogeneous 327; 13.7 Superposition or Linearity Principle 327; 13.8 LinearIndependence and dependence 328; 13.9 Wronskian 328; 13.10 Existence of linearlyIndependence 328; 13.11 Structure Theorem 328; 13.12 Super position Principle 32913.13 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; 13.171 1( ) ( )

ax axe ef D f a

339; 13.18

11 [ ( )] .( )

n nx f D xf D

341; 13.19 2 21 sinsin

( ) ( )axax

f D f a

342;

13.201 1. ( ) . . ( )( ) ( )

ax axe x e xf D f D a

347; 13.21 To find the Value of

1 sin .( )

nx axf D

353;

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'sHomogeneous Differential Equations 364; 14.3 Method of Variation of Parameters367; 14.4 Method Undetermined Coefficients 377.

15. DIFFERENTIAL EQUATION OF OTHER TYPES 382 – 405

15.1 Introduction 382; 15.2 Equation of the Type ( )n

nd y f xdx

382; 15.3 Equation of

the Type ( )n

nd y f ydx

383; 15.4 Equations which do not contain ‘y’ directly 386; 15.5Equations which do not contain ‘x’ directly 385; 15.6 Equations whose solution isknown 389; 15.7 Normal form (Removal of first derivative) 394; 15.8 Method of Solving

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Mathematical Physics

Publisher : SChand Publications ISBN : 9788121914697 Author : H K Dass

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