Studies in theoretical physics. Volume 1, Fundamental mathematical methods. / Daniel Erenso and Victor Montemayor.

Format:

Book

Author/Creator:

Erenso, Daniel, author.

Montemayor, Victor, author.

Series:

IOP Ebooks Series

Language:

English

Subjects (All):

Mathematical physics.

Physical Description:

1 online resource (629 pages)

Edition:

First edition.

Place of Publication:

Bristol, England : IOP Publishing, [2022]

Summary:

Studies in Theoretical Physics, Volume 1: Fundamental mathematical methods provides a modern and integrated way to teach the mathematical methods needed in theoretical physics and engineering courses. It introduces analytical and computer problem-solving techniques using Mathematica.

Contents:

Intro

Preface

Acknowledgement

Author biographies

Daniel Erenso

Victor Montemayor

Chapter 1 Series and convergence

1.1 Sequence and series

1.2 Testing series for convergence

1.3 Series representations of real functions

1.4 Sequence, series and Mathematica

1.5 Homework assignment

Chapter 2 Complex numbers, functions, and series

2.1 Complex numbers

2.2 Complex infinite series

Definition and convergence

Test of convergence

2.3 Powers and roots of complex numbers

2.4 Algebraic versus transcendental functions

Euler's formula, trigonometric, and hyperbolic functions

Logarithmic functions

Orthogonal and orthonormal set of functions

2.5 Complex numbers, functions and Mathematica

2.6 Homework assignment

Chapter 3 Vectors

3.1 Vector fundamentals

3.2 Vector addition

3.3 Vector multiplication

3.4 Vectors and equations of a line and a plane

Equation of a line

Equation of a plane

3.5 Vectors and Mathematica

Orthogonal vector operations

3.6 Homework assignment

Chapter 4 Matrices and determinants

4.1 Important terminologies

A matrix

4.2 Matrix arithmetic and manipulation

Addition and subtraction

Multiplication by a scalar

Multiplication of matrices

The commutator

4.3 Matrix representation of a set of linear equations

4.4 Solving a set of linear equations using matrices

Gaussian Elimination method and Row Echelon Form

Rank of a Matrix

4.5 Determinant of a square matrix

Properties of determinants

4.6 Cramer's rule

4.7 The adjoint and inverse of a matrix

The cofactor and adjoint matrices

Inverse of a square matrix

4.8 Orthogonal matrices and the rotation matrix

Orthogonal matrices

Linear operators

4.9 Linear dependence and independence

Set of linear functions

Basis vectors.

4.10 Gram-Schmidt orthogonalization

4.11 Matrices and Mathematica

4.12 Homework assignment

Chapter 5 Introduction to differential calculus I

5.1 Partial differentiation

5.2 Total differential

5.3 The multivariable form of the chain rule

5.4 Extremum (max/min) problems

5.5 The method of Lagrangian multipliers

5.6 Change of variables

Function of single variable differential equation

Function of multivariable differential equation

5.7 Partial differentiation and Mathematica

5.8 Homework assignments

Chapter 6 Introduction to differential calculus II

6.1 First-order ordinary DE

6.2 The first-order ODE and exact total differential

6.3 First-order ODE and non-exact total differential

6.4 Higher-order ODE

Indicial equation with degenerate roots

B. NHODE with constant coefficients

6.5 The particular solution and the method of superposition

6.6 The method of successive integration

6.7 Introduction to partial differential equations

6.8 Linear differential equations and Mathematica

6.9 Homework assignment

Chapter 7 Integral calculus-scalar functions

7.1 Integration in Cartesian coordinates

7.2 Physical applications

7.3 1-D and 2-D curvilinear coordinates

7.4 3-D curvilinear coordinates: cylindrical

7.5 3-D curvilinear coordinate: spherical

7.6 Scalar integrals and Mathematica

7.7 Homework assignment

Chapter 8 Vector calculus

8.1 Review of vector products

8.2 Vectors product physical applications

8.3 Vectors derivatives

8.4 The gradient operator and directional derivative

8.5 The divergence, the curl, and the Laplacian

8.6 Line vector integrals

8.7 Conservative vectors and exact differentials

8.8 Double integral and Green's theorem

8.9 The Stokes' theorem

8.10 The divergence theorem

8.11 Vector calculus and Mathematica.

8.12 Homework assignment

Chapter 9 Introduction to the calculus of variations

9.1 Stationary points and geodesic

9.2 The general problem of the calculus of variations

9.3 The Brachistochrone problem

9.4 The Euler-Lagrange equation in classical mechanics

9.5 The calculus of variations and Mathematica

9.6 Homework assignment

Chapter 10 Introduction to the eigenvalue problem

10.1 Eigenvalue problem in physics

10.2 Matrix review

10.3 Orthogonal transformations and Dirac's notation

10.4 Eigenvalues and eigenvectors

10.5 Eigenvalue equation and Hermitian matrices

10.6 The similarity transformation

10.7 Eigenvalue equation and Mathematica

10.8 Homework assignment

Chapter 11 Special functions

11.1 The factorial, the gamma function, and Stirling's formula

11.2 The beta function

11.3 The error function

11.4 Elliptic integrals

11.5 The Dirac delta function

11.6 Mathematica and special functions

11.7 Homework assignments

Chapter 12 Power series and differential equations

12.1 Power series substitution

12.2 Orthonormal set of vectors and functions

12.3 Complete set of functions

12.4 The Legendre differential equation

12.5 The Legendre polynomials

12.6 The generating function for the Legendre polynomials

12.7 Legendre series

12.8 The associated Legendre differential equation

12.9 Spherical harmonics and the addition theorem

12.10 The method of Frobenius and the Bessel equation

The zeroes of the Bessel function

12.11 The orthogonality of the Bessel functions

Limiting (asymptotic) forms for the Bessel functions

12.12 Fuch's theorem

12.13 Mathematica and serious substitution method

12.14 Homework assignments

Chapter 13 Partial differential equation

13.1 PDE in physics

13.2 Laplace's equation in Cartesian coordinates.

13.3 Laplace's equation in spherical coordinates

13.4 Laplace's equation in cylindrical coordinates

13.5 Poisson's equation

13.6 Homework assignment

Chapter 14 Functions of complex variables

14.1 Review of complex numbers

14.2 Analytic functions

14.3 Essential terminologies

14.4 Contour integration and Cauchy's theorem

14.5 Cauchy's integral formula

14.6 Laurent's theorem

14.7 The residue theorem

14.8 Methods of finding residues

14.9 Applications of the residue theorem

14.10 The modified residue theorem

14.11 Mathematica and complex functions

14.12 Homework assignment

Chapter 15 Laplace transform

15.1 Integral transform

15.2 The Laplace transform

15.3 Inverse Laplace transform

15.4 Applications of Laplace transforms

15.5 Mathematica and Laplace transform

15.6 Homework assignment

Chapter 16 Fourier series and transform

16.1 Average and root-mean-sqaure values

16.2 The Fourier series

16.3 Dirichlet conditions

16.4 Fourier series with spatial and temporal arguments

16.5 The Fourier transform and inverse transform

16.6 The Dirac delta function and the Fourier inverse transform

16.7 Applications of the Fourier transform

16.8 Fourier transform and convolution

16.9 Mathematica, Fourier series, transform, and inverse transform

Bibliography.

Notes:

Description based on publisher supplied metadata and other sources.

Description based on print version record.

Includes bibliographical references.

ISBN:

9780750344272

075034427X

OCLC:

1429724070