Mathematical Physics , livre ebook

M ATHEMATICAL P HYSICS
LICENSE, DISCLAIMER OF LIABILITY, AND LIMITED WARRANTY
By purchasing or using this book (the Work ), you agree that this license grants permission to use the contents contained herein, but does not give you the right of ownership to any of the textual content in the book or ownership to any of the information or products contained in it. This license does not permit uploading of the Work onto the Internet or on a network ( of any kind ) without the written consent of the Publisher . Duplication or dissemination of any text, code, simulations, images, etc. contained herein is limited to and subject to licensing terms for the respective products, and permission must be obtained from the Publisher or the owner of the content, etc., in order to reproduce or network any portion of the textual material (in any media) that is contained in the Work.
M ERCURY L EARNING AND I NFORMATION ( MLI or the Publisher ) and anyone involved in the creation, writing, or production of the companion disc, accompanying algorithms, code, or computer programs ( the software ), and any accompanying Web site or software of the Work, cannot and do not warrant the performance or results that might be obtained by using the contents of the Work. The author, developers, and the Publisher have used their best efforts to insure the accuracy and functionality of the textual material and/or programs contained in this package; we, however, make no warranty of any kind, express or implied, regarding the performance of these contents or programs. The Work is sold as is without warranty (except for defective materials used in manufacturing the book or due to faulty workmanship).
The author, developers, and the publisher of any accompanying content, and anyone involved in the composition, production, and manufacturing of this work will not be liable for damages of any kind arising out of the use of (or the inability to use) the algorithms, source code, computer programs, or textual material contained in this publication. This includes, but is not limited to, loss of revenue or profit, or other incidental, physical, or consequential damages arising out of the use of this Work.
The sole remedy in the event of a claim of any kind is expressly limited to replacement of the book, and only at the discretion of the Publisher. The use of implied warranty and certain exclusions vary from state to state, and might not apply to the purchaser of this product.
M ATHEMATICAL P HYSICS
An Introduction
Derek Raine, PhD et al.

M ERCURY L EARNING AND I NFORMATION
Dulles, Virginia
Boston, Massachusetts
New Delhi
Copyright 2019 by M ERCURY L EARNING AND I NFORMATION LLC. All rights reserved.
Original title and copyright: Mathematical Methods for Physical Science by Derek Raine et al. Copyright 2018 Pantaneto Press. All rights reserved.
This publication, portions of it, or any accompanying software may not bereproduced in any way, stored in a retrieval system of any type, or transmittedby any means, media, electronic display or mechanical display, including, but notlimited to, photocopy, recording, Internet postings, or scanning, without priorpermission in writing from the publisher.
Publisher: David Pallai
M ERCURY L EARNING AND I NFORMATION
22841 Quicksilver Drive
Dulles, VA 20166
info@merclearning.com
www.merclearning.com
1-(800)-232-0223
D. Raine et al. Mathematical Physics .
ISBN: 9781683922056
The publisher recognizes and respects all marks used by companies, manufacturers, and developers as a means to distinguish their products. All brand names andproduct names mentioned in this book are trademarks or service marks of theirrespective companies. Any omission or misuse (of any kind) of service marks ortrademarks, etc. is not an attempt to infringe on the property of others.
Library of Congress Control Number: 2018949983
181920321 This book is printed on acid-free paper in the United States of America.
Our titles are available for adoption, license, or bulk purchase by institutions, corporations, etc. For additional information, please contact the Customer
Service Dept. at (800) 232-0223 (toll free).
All of our titles are available in digital format at authorcloudware.com and otherdigital vendors. The sole obligation of M ERCURY L EARNING AND I NFORMATION tothe purchaser is to replace the book, based on defective materials or faultyworkmanship, but not based on the operation or functionality of the product.
CONTENTS
Preface
1 Derivatives and Integrals
1.1 Definition of a derivative
1.2 Some basic derivatives
1.3 Function of a function (chain rule)
1.4 Product (and quotient) rule
1.5 Implicit differentiation
1.6 Piecewise differentiable functions
1.7 Higher order derivatives
1.8 Stationary points
1.9 Integrals of elementary functions
1.10 Integrals of combinations of functions
1.11 Integration by substitution
1.12 Integration by parts
1.13 Integration of rational functions
1.14 Definite integrals
1.15 Area under a graph
1.16 Continuous and discontinuous functions
1.17 Estimates of integrals
1.18 Derivatives of integrals
1.19 Reduction formulae
1.20 Exercises
1.21 Problems
2 Elementary Functions
2.1 Binomial expansion
2.2 Maclaurin series
2.3 Taylor series
2.4 Equilibrium points
2.5 Definition of e (the exponential function)
2.6 The inverse function of e x
2.7 Derivative of the exponential
2.8 Integration of exponentials
2.9 Hyperbolic functions
2.10 Inverse functions
2.11 Inverse hyperbolic functions
2.12 Exercises
2.13 Problems
3 Functions, Limits, and Series
3.1 Curve sketching: Quadratics
3.2 Curve sketching: General polynomials
3.3 Curve sketching: Rational functions
3.4 Graphical solution of inequalities
3.5 The symmetry of functions
3.6 Limits
3.7 Indeterminate forms
3.8 l H pital s rule
3.9 Limits of integrals
3.10 Approximation of functions
3.11 Sequences and series
3.12 Infinite series
3.13 Exercises
3.14 Problems
4 Vectors
4.1 Basic properties
4.2 Linear dependence; basis vectors
4.3 The scalar (dot) product
4.4 The vector (cross) product
4.5 Multiple products
4.6 Equation of a line
4.7 Equation of a plane
4.8 Components of vectors
4.9 Scalar (dot) product (component form)
4.10 Vector (cross) product (component form)
4.11 Scalar triple product
4.12 Vector triple product
4.13 Vector identities
4.14 Examples from physical science
4.15 Extension: Index notation
4.16 Exercises
4.17 Problems
5 Matrices
5.1 Matrix representation
5.2 Solution of systems of equations
5.3 Products
5.4 The identity matrix
5.5 Symmetric and antisymmetric matrices
5.6 Inverses
5.7 The inverse of a 2 2 matrix
5.8 Determinants
5.9 Properties of determinants
5.10 Solution of 2 2 linear systems
5.11 Solution of 3 3 systems
5.12 Homogeneous systems
5.13 A formula for the inverse matrix
5.14 Eigenvalues and eigenvectors
5.15 Matrices as transformations
5.16 Extension: Transformation groups
5.17 Exercises
5.18 Problems
6 Differential Equations
6.1 What are differential equations?
6.2 Solving differential equations
6.3 First order separable equations
6.4 Linearity versus non-linearity
6.5 Homogeneous versus inhomogeneous
6.6 Finding solutions
6.7 Homogeneous linear equations
6.8 Auxiliary equations with repeated roots
6.9 Particular integrals
6.10 Inhomogeneous linear equations
6.11 Integrating factor method
6.12 Extension: Special functions
7 Complex Numbers
7.1 Introduction of complex numbers
7.2 Operations on complex numbers
7.3 Quadratic equations
7.4 The Argand diagram
7.5 Modulus and argument of a complex number
7.6 The complex exponential
7.7 De Moivre s theorem
7.8 The roots of unity
7.9 Roots of real polynomials
7.10 Roots of complex polynomials
7.11 Extension: Complex variable
7.12 Exercises
7.13 Problems
8 Differential Equations 2
8.1 Boundary and initial conditions
8.2 Auxiliary equations with complex roots
8.3 Equations with complex coefficients
8.4 Complex inhomogeneous term
8.5 Boundary (or initial) conditions
8.6 Systems of first order equations
8.7 Complex impedance
8.8 Some examples from physics
8.9 Extension: Green functions
8.10 Exercises
8.11 Problems
9 Multiple Integrals
9.1 Repeated integrals
9.2 Integrals over rectangles in the plane
9.3 Multiple integrals over irregular regions
9.4 Change of order of integration
9.5 Polar coordinates in two dimensions
9.6 Integrals over regions in the plane
9.7 An important definite integral
9.8 Cylindrical polar coordinates
9.9 Volume integrals (cylindrical polar)
9.10 Integrals over cylindrical surfaces
9.11 Spherical polar coordinates
9.12 Volume integrals (spherical polar)
9.13 Integrals over spherical surfaces
9.14 Solid angle
9.15 Sketching surfaces
9.16 Exercises
9.17 Problems
10 Partial Derivatives
10.1 Functions of two variables Functions of two variables
10.2 Partial derivatives
10.3 Directional derivatives
10.4 Functions of many variables
10.5 Higher derivatives
10.6 Function of a function
10.7 Implicit differentiation
10.8 Derivative with respect to a parameter
10.9 Taylor expansion about the origin
10.10 Expansion about an arbitrary point
10.11 Function of more than two variables
10.12 Stationary points
10.13 Definition of the total differential
10.14 Exact differentials
10.15 The chain rule
10.16 Consistency with the chain rule
10.17 Exercises
10.18 Problems
11 Partial Differential Equations
11.1 Examples from physics
11.2 General solution of the wave equation
11.3 Derivation of the general solution
11.4 A string initially at rest
11.5 A string given an initial velocity
11.6 A formula for general initial conditions
11.7 Semi-infinite string
11.8 Simple harmonic waves
11.9 Beats
11.10 Group velocity
11.11 Separation of var