Mathematical Methods in Engineering and Physics

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Cover Title Page Copyright Contents Preface Chapter 1 Introduction to Ordinary Differential Equations 1.1 Motivating Exercise: The Simple Harmonic Oscillator 1.2 Overview of Differential Equations 1.2.1 Discovery Exercise: Overview of Differential Equations 1.2.2 Explanation: Overview of Differential Equations 1.2.3 Problems: Overview of Differential Equations 1.3 Arbitrary Constants 1.3.1 Discovery Exercise: Arbitrary Constants 1.3.2 Explanation: Arbitrary Constants 1.3.3 Problems: Arbitrary Constants 1.4 Slope Fields and Equilibrium 1.4.1 Discovery Exercise: Slope Fields 1.4.2 Explanation: Slope Fields and Equilibrium 1.4.3 Problems: Slope Fields and Equilibrium 1.5 Separation of Variables 1.5.1 Discovery Exercise: Separation of Variables 1.5.2 Explanation: Separation of Variables 1.5.3 Problems: Separation of Variables 1.6 Guess and Check, and Linear Superposition 1.6.1 Discovery Exercise: Guess and Check, and Linear Superposition 1.6.2 Explanation: Guess and Check, and Linear Superposition 1.6.3 Problems: Guess and Check, and Linear Superposition 1.7 Coupled Equations (see felderbooks.com) 1.8 Differential Equations on a Computer (see felderbooks.com) 1.9 Additional Problems (see felderbooks.com) Chapter 2 Taylor Series and Series Convergence 2.1 Motivating Exercise: Vibrations in a Crystal 2.2 Linear Approximations 2.2.1 Discovery Exercise: Linear Approximations 2.2.2 Explanation: Linear Approximations 2.2.3 Problems: Linear Approximations 2.3 Maclaurin Series 2.3.1 Discovery Exercise: A Polynomial Equivalent of the Sine 2.3.2 Explanation: Maclaurin Series 2.3.3 Problems: Maclaurin Series 2.4 Taylor Series 2.4.1 Explanation: Taylor Series 2.4.2 Problems: Taylor Series 2.5 Finding One Taylor Series from Another 2.5.1 Explanation: Finding One Taylor Series from Another 2.5.2 Problems: Finding One Taylor Series from Another 2.6 Sequences and Series 2.6.1 Discovery Exercise: Geometric Series 2.6.2 Discovery Exercise: Infinite Series 2.6.3 Explanation: Sequences and Series 2.6.4 Problems: Sequences and Series 2.7 Tests for Series Convergence 2.7.1 Explanation: The Ratio Test 2.7.2 Problems: The Ratio Test 2.7.3 Explanation: Other Tests for Series Convergence 2.7.4 Problems: Other Tests for Series Convergence 2.8 Asymptotic Expansions (see felderbooks.com) 2.9 Additional Problems (see felderbooks.com) Chapter 3 Complex Numbers 3.1 Motivating Exercise: The Underdamped Harmonic Oscillator 3.2 Complex Numbers 3.2.1 Discovery Exercise: Imaginary Numbers 3.2.2 Explanation: Complex Numbers 3.2.3 Problems: Complex Numbers 3.3 The Complex Plane 3.3.1 Discovery Exercise: The Complex Plane 3.3.2 Explanation: The Complex Plane 3.4 Euler's Formula I—The Complex Exponential Function 3.4.1 Discovery Exercise: Euler’s Formula I—The Complex Exponential Function 3.4.2 Explanation: Euler’s Formula I—The Complex Exponential Function 3.4.3 Problems: Euler’s Formula I—The Complex Exponential Function 3.5 Euler's Formula II---Modeling Oscillations 3.5.1 Discovery Exercise: Euler’s Formula II—Modeling Oscillations 3.5.2 Explanation: Euler’s Formula II—Modeling Oscillations 3.5.3 Problems: Euler’s Formula II—Modeling Oscillations 3.6 Special Application: Electric Circuits (see felderbooks.com) 3.7 Additional Problems (see felderbooks.com) Chapter 4 Partial Derivatives 4.1 Motivating Exercise: The Wave Equation 4.2 Partial Derivatives 4.2.1 Discovery Exercise: Partial Derivatives 4.2.2 Explanation: Partial Derivatives 4.2.3 Problems: Partial Derivatives 4.3 The Chain Rule 4.3.1 Discovery Exercise: The Chain Rule 4.3.2 Explanation: The Chain Rule 4.3.3 Problems: The Chain Rule 4.4 Implicit Differentiation 4.4.1 Discovery Exercise: Implicit Differentiation 4.4.2 Explanation: Implicit Differentiation 4.4.3 Problems: Implicit Differentiation 4.5 Directional Derivatives 4.5.1 Discovery Exercise: Directional Derivatives 4.5.2 Explanation: Directional Derivatives 4.5.3 Problems: Directional Derivatives 4.6 The Gradient 4.6.1 Discovery Exercise: The Gradient 4.6.2 Explanation: The Gradient 4.7 Tangent Plane Approximations and Power Series (see felderbooks.com) 4.8 Optimization and the Gradient 4.8.1 Discovery Exercise: Optimization and the Gradient 4.8.2 Explanation: Optimization and the Gradient 4.8.3 Problems: Optimization and the Gradient 4.9 Lagrange Multipliers 4.9.1 Explanation: Lagrange Multipliers 4.9.2 Problems: Lagrange Multipliers 4.10 Special Application: Thermodynamics (see felderbooks.com) 4.11 Additional Problems (see felderbooks.com) Chapter 5 Integrals in Two or More Dimensions 5.1 Motivating Exercise: Newton's Problem (or) The Gravitational Field of a Sphere 5.2 Setting Up Integrals 5.2.1 Discovery Exercise: Adding Up the Pieces 5.2.2 Explanation: Setting Up Integrals in One Dimension 5.2.3 Problems: Setting Up Integrals in One Dimension 5.2.4 Explanation: Single Integrals in Multiple Dimensions 5.2.5 Problems: Single Integrals in Multiple Dimensions 5.3 Cartesian Double Integrals over a Rectangular Region 5.3.1 Discovery Exercise: Cartesian Double Integrals over a Rectangular Region 5.3.2 Explanation: Cartesian Double Integrals over a Rectangular Region 5.3.3 Problems: Cartesian Double Integrals over a Rectangular Region 5.4 Cartesian Double Integrals over a Non-Rectangular Region 5.4.1 Discovery Exercise: Cartesian Double Integrals over a Non-Rectangular Region 5.4.2 Explanation: Cartesian Double Integrals over a Non-Rectangular Region 5.4.3 Problems: Cartesian Double Integrals over a Non-Rectangular Region 5.5 Triple Integrals in Cartesian Coordinates 5.5.1 Explanation: Triple Integrals in Cartesian Coordinates 5.5.2 Problems: Triple Integrals in Cartesian Coordinates 5.6 Double Integrals in Polar Coordinates 5.6.1 Discovery Exercise: Double Integrals in Polar Coordinates 5.6.2 Explanation: Double Integrals in Polar Coordinates 5.6.3 Problems: Double Integrals in Polar Coordinates 5.7 Cylindrical and Spherical Coordinates 5.7.1 Discovery Exercise: Cylindrical and Spherical Coordinates 5.7.2 Explanation: Cylindrical and Spherical Coordinates 5.7.3 Problems: Cylindrical and Spherical Coordinates 5.8 Line Integrals 5.8.1 Discovery Exercise: Line Integrals 5.8.2 Explanation: Line Integrals 5.8.3 Problems: Line Integrals 5.9 Parametrically Expressed Surfaces 5.9.1 Discovery Exercise: Parametrically Expressed Surfaces 5.9.2 Explanation: Parametrically Expressed Surfaces 5.9.3 Problems: Parametrically Expressed Surfaces 5.10 Surface Integrals 5.10.1 Discovery Exercise: Surface Integrals 5.10.2 Explanation: Surface Integrals 5.10.3 Problems: Surface Integrals 5.11 Special Application: Gravitational Forces (see felderbooks.com) 5.12 Additional Problems (see felderbooks.com) Chapter 6 Linear Algebra I 6.1 The Motivating Example on which We're Going to Base the Whole Chapter: The Three-Spring Problem 6.1.1 Explanation: All I Really Need to Know about Matrices I Learned from the Three-Spring Problem 6.1.2 Problems: The Three-Spring Problem 6.2 Matrices: The Easy Stuff 6.2.1 Discovery Exercise: The Easy Matrix Stuff 6.2.2 Explanation: The Easy Matrices Stuff 6.2.3 Problems: The Easy Matrix Stuff 6.3 Matrix Times Column 6.3.1 Discovery Exercise: Multiple Value Problems 6.3.2 Explanation: Matrix Times Column 6.3.3 Problems: Matrix Times Column 6.4 Basis Vectors 6.4.1 Discovery Exercise: Basis Vectors 6.4.2 Explanation: Basis Vectors 6.4.3 Problems: Basis Vectors 6.5 Matrix Times Matrix 6.5.1 Discovery Exercise: Matrix Times Matrix 6.5.2 Explanation: Matrix Times Matrix 6.5.3 Problems: Matrix Times Matrix 6.6 The Identity and Inverse Matrices 6.6.1 Discovery Exercise: The Identity and Inverse Matrices 6.6.2 Explanation: The Identity and Inverse Matrices 6.6.3 Problems: The Identity and Inverse Matrices 6.7 Linear Dependence and the Determinant 6.7.1 Discovery Exercise: Linear Dependence and the Determinant 6.7.2 Explanation: Linear Dependence and the Determinant 6.7.3 Problems: Linear Dependence and the Determinant 6.8 Eigenvectors and Eigenvalues 6.8.1 Discovery Exercise: Eigenvectors and Eigenvalues 6.8.2 Explanation: Eigenvectors and Eigenvalues 6.8.3 Problems: Eigenvectors and Eigenvalues 6.9 Putting It Together: Revisiting the Three-Spring Problem 6.9.1 Explanation: Revisiting the Three-Spring Problem 6.9.2 Problems: Revisiting the Three-Spring Problem 6.10 Additional Problems (see felderbooks.com) Chapter 7 Linear Algebra II 7.1 Geometric Transformations 7.1.1 Discovery Exercise: Geometric Transformations 7.1.2 Explanation: Geometric Transformations 7.1.3 Problems: Geometric Transformations 7.2 Tensors 7.2.1 Discovery Exercise: Tensors 7.2.2 Explanation: Tensors 7.2.3 Problems: Tensors 7.3 Vector Spaces and Complex Vectors 7.3.1 Explanation: Vector Spaces and Complex Vectors 7.3.2 Problems: Vector Spaces and Complex Vectors 7.4 Row Reduction (see felderbooks.com) 7.5 Linear Programming and the Simplex Method (see felderbooks.com) 7.6 Additional Problems (see felderbooks.com) Chapter 8 Vector Calculus 8.1 Motivating Exercise: Flowing Fluids 8.2 Scalar and Vector Fields 8.2.1 Discovery Exercise: Scalar and Vector Fields 8.2.2 Explanation: Scalar and Vector Fields 8.2.3 Problems: Scalar and Vector Fields 8.3 Potential in One Dimension 8.3.1 Discovery Exercise: Potential in One Dimension 8.3.2 Explanation: Potential in One Dimension 8.3.3 Problems: Potential in One Dimension 8.4 From Potential to Gradient 8.4.1 Discovery Exercise: From Potential to Gradient 8.4.2 Explanation: From Potential to Gradient 8.4.3 Problems: From Potential to Gradient 8.5 From Gradient to Potential: The Gradient Theorem 8.5.1 Problems: From Gradient to Potential 8.6 Divergence, Curl, and Laplacian 8.6.1 Discovery Exercise: Divergence, Curl, and Laplacian 8.6.2 Explanation: Divergence, Curl, and Laplacian 8.6.3 Problems: Divergence, Curl, and Laplacian 8.7 Divergence and Curl II---The Math Behind the Pictures 8.7.1 Explanation: Divergence and Curl II—The Math Behind the Pictures 8.7.2 Problems: Divergence and Curl II—The Math Behind the Pictures 8.8 Vectors in Curvilinear Coordinates 8.8.1 Discovery Exercise: Vectors in Curvilinear Coordinates 8.8.2 Explanation: Vector Derivatives in Curvilinear Coordinates 8.8.3 Problems: Vectors in Curvilinear Coordinates 8.9 The Divergence Theorem 8.9.1 Discovery Exercise: The Divergence Theorem 8.9.2 Explanation: The Divergence Theorem 8.9.3 Problems: The Divergence Theorem 8.10 Stokes' Theorem 8.10.1 Explanation: Stokes’ Theorem 8.10.2 Problems: Stokes’ Theorem 8.11 Conservative Vector Fields 8.11.1 Discovery Exercise: Conservative Vector Fields 8.11.2 Explanation: Conservative Vector Fields 8.11.3 Problems: Conservative Vector Fields 8.12 Additional Problems (see felderbooks.com) Chapter 9 Fourier Series and Transforms 9.1 Motivating Exercise: Discovering Extrasolar Planets 9.2 Introduction to Fourier Series 9.2.1 Discovery Exercise: Introduction to Fourier Series 9.2.2 Explanation: Introduction to Fourier Series 9.2.3 Problems: Introduction to Fourier Series 9.3 Deriving the Formula for a Fourier Series 9.3.1 Explanation: Deriving the Formula for a Fourier Series 9.3.2 Problems: Deriving the Formula for a Fourier Series 9.4 Different Periods and Finite Domains 9.4.1 Discovery Exercise: Different Periods and Finite Domains 9.4.2 Explanation: Different Periods and Finite Domains 9.4.3 Problems: Different Periods and Finite Domains 9.5 Fourier Series with Complex Exponentials 9.5.1 Discovery Exercise: Fourier Series with Complex Exponentials 9.5.2 Explanation: Fourier Series with Complex Exponentials 9.5.3 Problems: Fourier Series with Complex Exponentials 9.6 Fourier Transforms 9.6.1 Discovery Exercise: Fourier Transforms 9.6.2 Explanation: Fourier Transforms 9.7 Discrete Fourier Transforms (see felderbooks.com) 9.8 Multivariate Fourier Series (see felderbooks.com) 9.9 Additional Problems (see felderbooks.com) Chapter 10 Methods of Solving Ordinary Differential Equations 10.1 Motivating Exercise: A Damped, Driven Oscillator 10.2 Guess and Check 10.2.1 Discovery Exercise: Guess and Check 10.2.2 Explanation: Guess and Check 10.2.3 Problems: Guess and Check 10.3 Phase Portraits (see felderbooks.com) 10.4 Linear First-Order Differential Equations (see felderbooks.com) 10.5 Exact Differential Equations (see felderbooks.com) 10.6 Linearly Independent Solutions and the Wronskian (see felderbooks.com) 10.7 Variable Substitution 10.7.1 Discovery Exercise: Variable Substitution 10.7.2 Explanation: Variable Substitution 10.7.3 Problems: Variable Substitution 10.8 Three Special Cases of Variable Substitution 10.8.1 Explanation: Three Special Cases of Variable Substitution 10.8.2 Problems: Three Special Cases of Variable Substitution 10.9 Reduction of Order and Variation of Parameters (see felderbooks.com) 10.10 Heaviside, Dirac, and Laplace 10.10.1 Discovery Exercise: Heaviside and Dirac 10.10.2 Explanation: Heaviside, Dirac, and Laplace 10.10.3 Problems: Heaviside, Dirac, Laplace 10.11 Using Laplace Transforms to Solve Differential Equations 10.11.1 Explanation: Using Laplace Transforms to Solve Differential Equations 10.11.2 Problems: Using Laplace Transforms to Solve Differential Equations 10.12 Green's Functions 10.12.1 Discovery Exercise: Green’s Functions 10.12.2 Explanation: Green’s Functions 10.12.3 Problems: Green’s Functions 10.13 Additional Problems (see felderbooks.com) Chapter 11 Partial Differential Equations 11.1 Motivating Exercise: The Heat Equation 11.2 Overview of Partial Differential Equations 11.2.1 Discovery Exercise: Overview of Partial Differential Equations 11.2.2 Explanation: Overview of Partial Differential Equations 11.2.3 Problems: Overview of Partial Differential Equations 11.3 Normal Modes 11.3.1 Discovery Exercise: Normal Modes 11.3.2 Explanation: Normal Modes 11.3.3 Problems: Normal Modes 11.4 Separation of Variables—The Basic Method 11.4.1 Discovery Exercise: Separation of Variables—The Basic Method 11.4.2 Explanation: Separation of Variables—The Basic Method 11.4.3 Problems: Separation of Variables—The Basic Method 11.5 Separation of Variables—More than Two Variables 11.5.1 Discovery Exercise: Separation of Variables—More than Two Variables 11.5.2 Explanation: Separation of Variables—More than Two Variables 11.5.3 Problems: Separation of Variables—More than Two Variables 11.6 Separation of Variables—Polar Coordinates and Bessel Functions 11.6.1 Explanation: Bessel Functions—The Unjustified Essentials 11.6.2 Discovery Exercise: Separation of Variables—Polar Coordinates and Bessel Functions 11.6.3 Explanation: Separation of Variables—Polar Coordinates and Bessel Functions 11.6.4 Problems: Separation of Variables—Polar Coordinates and Bessel Functions 11.7 Separation of Variables—Spherical Coordinates and Legendre Polynomials 11.7.1 Explanation: Separation of Variables—Spherical Coordinates and Legendre Polynomials 11.7.2 Explanation: Spherical Harmonics 11.7.3 Stepping Back: An Overview of Separation of Variables 11.7.4 Problems: Separation of Variables—Spherical Coordinates and Legendre Polynomials 11.8 Inhomogeneous Boundary Conditions 11.8.1 Discovery Exercise: Inhomogeneous Boundary Conditions 11.8.2 Explanation: Inhomogeneous Boundary Conditions 11.8.3 Problems: Inhomogeneous Boundary Conditions 11.9 The Method of Eigenfunction Expansion 11.9.1 Discovery Exercise: The Method of Eigenfunction Expansion 11.9.2 Explanation: The Method of Eigenfunction Expansions 11.9.3 Problems: The Method of Eigenfunction Expansion 11.10 The Method of Fourier Transforms 11.10.1 Discovery Exercise: The Method of Fourier Transforms 11.10.2 Explanation: The Method of Fourier Transforms 11.10.3 Problems: The Method of Fourier Transforms 11.11 The Method of Laplace Transforms 11.11.1 Explanation: The Method of Laplace Transforms 11.11.2 Problems: The Method of Laplace Transforms 11.12 Additional Problems (see felderbooks.com) Chapter 12 Special Functions and ODE Series Solutions 12.1 Motivating Exercise: The Circular Drum 12.2 Some Handy Summation Tricks 12.2.1 Explanation: Some Handy Summation Tricks 12.2.2 Problems: Some Handy Summation Tricks 12.3 A Few Special Functions 12.3.1 Explanation: A Few Special Functions 12.3.2 Problems: A Few Special Functions 12.4 Solving Differential Equations with Power Series 12.4.1 Discovery Exercise: Solving Differential Equations with Power Series 12.4.2 Explanation: Solving Differential Equations with Power Series 12.4.3 Problems: Solving Differential Equations with Power Series 12.5 Legendre Polynomials 12.5.1 Discovery Exercise: Legendre Polynomials 12.5.2 Explanation: Legendre Polynomials 12.5.3 Problems: Legendre Polynomials 12.6 The Method of Frobenius 12.6.1 Discovery Exercise: The Method of Frobenius 12.6.2 Explanation: The Method of Frobenius 12.6.3 Problems: The Method of Frobenius 12.7 Bessel Functions 12.7.1 Discovery Exercise: Bessel Functions 12.7.2 Explanation: Bessel Functions 12.7.3 Problems: Bessel Functions 12.8 Sturm-Liouville Theory and Series Expansions 12.8.1 Discovery Exercise: Sturm-Liouville Theory and Series Expansions 12.8.2 Explanation: Sturm-Liouville Theory and Series Expansions 12.8.3 Problems: Sturm-Liouville Theory and Series Expansions 12.9 Proof of the Orthgonality of Sturm-Liouville Eigenfunctions (see felderbooks.com) 12.10 Special Application: The Quantum Harmonic Oscillator and Ladder Operators (see felderbooks.com) 12.11 Additional Problems (see felderbooks.com) Chapter 13 Calculus with Complex Numbers 13.1 Motivating Exercise: Laplace's Equation 13.2 Functions of Complex Numbers 13.2.1 Discovery Exercise: Functions of Complex Numbers 13.2.2 Explanation: Functions of Complex Numbers 13.2.3 Problems: Functions of Complex Numbers 13.3 Derivatives, Analytic Functions, and Laplace's Equation 13.3.1 Discovery Exercise: Derivatives and Analytic Functions 13.3.2 Explanation: Derivatives and Analytic Functions 13.3.3 Explanation: Solving Laplace’s Equation with Analytic Functions 13.3.4 Problems: Derivatives, Analytic Functions, and Laplace’s Equation 13.4 Contour Integration 13.4.1 Explanation: Contour Integration 13.4.2 Problems: Contour Integration 13.5 Some Uses of Contour Integration 13.5.1 Explanation: Some Uses of Contour Integration 13.5.2 Explanation: Deriving the Formula for an Inverse Laplace Transform 13.5.3 Problems: Some Uses of Contour Integration 13.6 Integrating Along Branch Cuts and Through Poles (see felderbooks.com) 13.7 Complex Power Series 13.7.1 Explanation: Complex Power Series 13.7.2 Problems: Complex Power Series 13.8 Mapping Curves and Regions 13.8.1 Discovery Exercise: Mapping Curves and Regions 13.8.2 Explanation: Mapping Curves and Regions 13.8.3 Problems: Mapping Curves and Regions 13.9 Conformal Mapping and Laplace's Equation 13.9.1 Explanation: Conformal Mapping and Laplace’s Equation 13.9.2 Problems: Conformal Mapping and Laplace’s Equation 13.10 Special Application: Fluid Flow (see felderbooks.com) 13.11 Additional Problems (see felderbooks.com) Appendix A Different Types of Differential Equations Appendix B Taylor Series Appendix C Summary of Tests for Series Convergence Appendix D Curvilinear Coordinates Appendix E Matrices Appendix F Vector Calculus Appendix G Fourier Series and Transforms Appendix H Laplace Transforms Appendix I Summary: Which PDE Technique Do I Use? Appendix J Some Common Differential Equations and Their Solutions Appendix K Special Functions Appendix L Answers to “Check Yourself” in Exercises Index EULA