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Mathematics of Physics and Engineering

(coco) #1



    1. Euclidean Geometry and Vectors Preface vii

      • 1.1 Euclidean Geometry

        • 1.1.1 The Postulates of Euclid

        • 1.1.2 Relative Position and Position Vectors

        • 1.1.3 Euclidean Space as a Linear Space



      • 1.2 Vector Operations

        • 1.2.1 Inner Product

        • 1.2.2 Cross Product

        • 1.2.3 Scalar Triple Product







    • 1.3 Curves in Space

      • 1.3.1 Vector-Valued Functions of a Scalar Variable

      • 1.3.2 The Tangent Vector and Arc Length

      • 1.3.3 Frenet's Formulas

      • 1.3.4 Velocity and Acceleration







    1. Vector Analysis and Classical and Relativistic Mechanics

      • 2.1 Kinematics and Dynamics of a Point Mass

        • 2.1.1 Newton's Laws of Motion and Gravitation

        • 2.1.2 Parallel Translation of Frames

        • 2.1.3 Uniform Rotation of Frames

        • 2.1.4 General Accelerating Frames



      • 2.2 Systems of Point Masses

        • 2.2.1 Non-Rigid Systems of Points

        • 2.2.2 Rigid Systems of Points



      • 2.2.3 Rigid Bodies xii Mathematics of Physics and Engineering





    • 2.3 The Lagrange-Hamilton Method

      • 2.3.1 Lagrange's Equations

      • 2.3.2 An Example of Lagrange's Method

      • 2.3.3 Hamilton's Equations



    • 2.4 Elements of the Theory of Relativity

      • 2.4.1 Historical Background

      • 2.4.2 The Lorentz Transformation and Special Relativity

      • 2.4.3 Einstein's Field Equations and General Relativity







    1. Vector Analysis and Classical Electromagnetic Theory



    • 3.1 Functions of Several Variables

      • 3.1.1 Functions, Sets, and the Gradient

      • 3.1.2 Integration and Differentiation

      • 3.1.3 Curvilinear Coordinate Systems



    • 3.2 The Three Integral Theorems of Vector Analysis

      • 3.2.1 Green's Theorem

      • 3.2.2 The Divergence Theorem of Gauss

      • 3.2.3 Stokes's Theorem

      • 3.2.4 Laplace's and Poisson's Equations



    • 3.3 Maxwell's Equations and Electromagnetic Theory

      • 3.3.1 Maxwell's Equations in Vacuum

      • 3.3.2 The Electric and Magnetic Dipoles

      • 3.3.3 Maxwell's Equations in Material Media







    1. Elements of Complex Analysis



    • 4.1 The Algebra of Complex Numbers

      • 4.1.1 Basic Definitions

      • 4.1.2 The Complex Plane

      • 4.1.3 Applications to Analysis of AC Circuits



    • 4.2 Functions of a Complex Variable

      • 4.2.1 Continuity and Differentiability

      • 4.2.2 Cauchy-Riemann Equations

      • 4.2.3 The Integral Theorem and Formula of Cauchy

      • 4.2.4 Conformal Mappings



    • 4.3 Power Series and Analytic Functions

      • 4.3.1 Series of Complex Numbers

        • 4.3.2 Convergence of Power Series



      • 4.3.3 The Exponential Function Contents xiii



    • 4.4 Singularities of Complex Functions

      • 4.4.1 Laurent Series

      • 4.4.2 Residue Integration

      • 4.4.3 Power Series and Ordinary Differential Equations







    1. Elements of Fourier Analysis



    • 5.1 Fourier Series

      • 5.1.1 Fourier Coefficients

      • 5.1.2 Point-wise and Uniform Convergence

      • 5.1.3 Computing the Fourier Series



    • 5.2 Fourier Transform

      • 5.2.1 From Sums to Integrals

      • 5.2.2 Properties of the Fourier Transform

      • 5.2.3 Computing the Fourier Transform



    • 5.3 Discrete Fourier Transform

      • 5.3.1 Discrete Functions

      • 5.3.2 Fast Fourier Transform (FFT)



    • 5.4 Laplace Transform

      • 5.4.1 Definition and Properties

      • 5.4.2 Applications to System Theory







    1. Partial Differential Equations of Mathematical Physics



    • 6.1 Basic Equations and Solution Methods

      • 6.1.1 Transport Equation

      • 6.1.2 Heat Equation

      • 6.1.3 Wave Equation in One Dimension



    • 6.2 Elements of the General Theory of PDEs

      • 6.2.1 Classification of Equations and Characteristics

      • 6.2.2 Variation of Parameters

      • 6.2.3 Separation of Variables



    • 6.3 Some Classical Partial Differential Equations

      • 6.3.1 Telegraph Equation

      • 6.3.2 Helmholtz's Equation

        • 6.3.3 Wave Equation in Two and Three Dimensions

        • 6.3.4 Maxwell's Equations

        • 6.3.5 Equations of Fluid Mechanics





    • 6.4 Equations of Quantum Mechanics

      • 6.4.1 Schrodinger's Equation xiv Mathematics of Physics and Engineering

        • 6.4.2 Dirac's Equation of Relativistic Quantum Mechanics

        • 6.4.3 Introduction to Quantum Computing



      • 6.5 Numerical Solution of Partial Differential Equations

        • 6.5.1 General Concepts in Numerical Methods

          • 6.5.2 One-Dimensional Heat Equation

          • 6.5.3 One-Dimensional Wave Equation

          • 6.5.4 The Poisson Equation in a Rectangle

          • 6.5.5 The Finite Element Method











    1. Further Developments and Special Topics

      • 7.1 Geometry and Vectors

      • 7.2 Kinematics and Dynamics

      • 7.3 Special Relativity

        • 7.4 Vector Calculus



      • 7.5 Complex Analysis

        • 7.6 Fourier Analysis

        • 7.7 Partial Differential Equations









      1. Appendix



      • 8.1 Linear Algebra and Matrices

      • 8.2 Ordinary Differential Equations

      • 8.3 Tensors

        • 8.4 Lumped Electric Circuits

        • 8.5 Physical Units and Constants





    • Bibliography

    • List of Notations

    • Index



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