Mathematical Methods in Engineering and Physics

Gary N. Felder and Kenny M. Felder are the authors of Mathematical Methods in Engineering and Physics, published by Wiley.

• Preface xi1 Introduction to Ordinary Differential Equations 11.1 Motivating Exercise: The Simple Harmonic Oscillator 21.2 Overview of Differential Equations 31.3 Arbitrary Constants 151.4 Slope Fields and Equilibrium 251.5 Separation of Variables 341.6 Guess and Check, and Linear Superposition 391.7 Coupled Equations (see felderbooks.com)1.8 Differential Equations on a Computer (see felderbooks.com)1.9 Additional Problems (see felderbooks.com)2 Taylor Series and Series Convergence 502.1 Motivating Exercise: Vibrations in a Crystal 512.2 Linear Approximations 522.3 Maclaurin Series 602.4 Taylor Series 702.5 Finding One Taylor Series from Another 762.6 Sequences and Series 802.7 Tests for Series Convergence 922.8 Asymptotic Expansions (see felderbooks.com)2.9 Additional Problems (see felderbooks.com)3 Complex Numbers 1043.1 Motivating Exercise: The Underdamped Harmonic Oscillator 1043.2 Complex Numbers 1053.3 The Complex Plane 1133.4 Euler’s Formula I—The Complex Exponential Function 1173.5 Euler’s Formula II—Modeling Oscillations 1263.6 Special Application: Electric Circuits (see felderbooks.com)3.7 Additional Problems (see felderbooks.com)4 Partial Derivatives 1364.1 Motivating Exercise: The Wave Equation 1364.2 Partial Derivatives 1374.3 The Chain Rule 1454.4 Implicit Differentiation 1534.5 Directional Derivatives 1584.6 The Gradient 1634.7 Tangent Plane Approximations and Power Series (see felderbooks.com)4.8 Optimization and the Gradient 1724.9 Lagrange Multipliers 1814.10 Special Application: Thermodynamics (see felderbooks.com)4.11 Additional Problems (see felderbooks.com)5 Integrals in Two or More Dimensions 1885.1 Motivating Exercise: Newton’s Problem (or) The Gravitational Field of a Sphere 1885.2 Setting Up Integrals 1895.3 Cartesian Double Integrals over a Rectangular Region 2045.4 Cartesian Double Integrals over a Non-Rectangular Region 2115.5 Triple Integrals in Cartesian Coordinates 2165.6 Double Integrals in Polar Coordinates 2215.7 Cylindrical and Spherical Coordinates 2295.8 Line Integrals 2405.9 Parametrically Expressed Surfaces 2495.10 Surface Integrals 2535.11 Special Application: Gravitational Forces (see felderbooks.com)5.12 Additional Problems (see felderbooks.com)6 Linear Algebra I 2666.1 The Motivating Example on which We’re Going to Base the Whole Chapter: The Three-Spring Problem 2666.2 Matrices: The Easy Stuff 2766.3 Matrix Times Column 2806.4 Basis Vectors 2866.5 Matrix Times Matrix 2946.6 The Identity and Inverse Matrices 3036.7 Linear Dependence and the Determinant 3126.8 Eigenvectors and Eigenvalues 3256.9 Putting It Together: Revisiting the Three-Spring Problem 3366.10 Additional Problems (see felderbooks.com)7 Linear Algebra II 3467.1 Geometric Transformations 3477.2 Tensors 3587.3 Vector Spaces and Complex Vectors 3697.4 Row Reduction (see felderbooks.com)7.5 Linear Programming and the Simplex Method (see felderbooks.com)7.6 Additional Problems (see felderbooks.com)8 Vector Calculus 3788.1 Motivating Exercise: Flowing Fluids 3788.2 Scalar and Vector Fields 3798.3 Potential in One Dimension 3878.4 From Potential to Gradient 3968.5 From Gradient to Potential: The Gradient Theorem 4028.6 Divergence, Curl, and Laplacian 4078.7 Divergence and Curl II—The Math Behind the Pictures 4168.8 Vectors in Curvilinear Coordinates 4198.9 The Divergence Theorem 4268.10 Stokes’ Theorem 4328.11 Conservative Vector Fields 4378.12 Additional Problems (see felderbooks.com)9 Fourier Series and Transforms 4459.1 Motivating Exercise: Discovering Extrasolar Planets 4459.2 Introduction to Fourier Series 4479.3 Deriving the Formula for a Fourier Series 4579.4 Different Periods and Finite Domains 4599.5 Fourier Series with Complex Exponentials 4679.6 Fourier Transforms 4729.7 Discrete Fourier Transforms (see felderbooks.com)9.8 Multivariate Fourier Series (see felderbooks.com)9.9 Additional Problems (see felderbooks.com)10 Methods of Solving Ordinary Differential Equations 48410.1 Motivating Exercise: A Damped, Driven Oscillator 48510.2 Guess and Check 48510.3 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 49410.8 Three Special Cases of Variable Substitution 50510.9 Reduction of Order and Variation of Parameters (see felderbooks.com)10.10 Heaviside, Dirac, and Laplace 51210.11 Using Laplace Transforms to Solve Differential Equations 52210.12 Green’s Functions 53110.13 Additional Problems (see felderbooks.com)11 Partial Differential Equations 54111.1 Motivating Exercise: The Heat Equation 54211.2 Overview of Partial Differential Equations 54411.3 Normal Modes 55511.4 Separation of Variables—The Basic Method 56711.5 Separation of Variables—More than Two Variables 58011.6 Separation of Variables—Polar Coordinates and Bessel Functions 58911.7 Separation of Variables—Spherical Coordinates and Legendre Polynomials 60711.8 Inhomogeneous Boundary Conditions 61611.9 The Method of Eigenfunction Expansion 62311.10 The Method of Fourier Transforms 63611.11 The Method of Laplace Transforms 64611.12 Additional Problems (see felderbooks.com)12 Special Functions and ODE Series Solutions 65212.1 Motivating Exercise: The Circular Drum 65212.2 Some Handy Summation Tricks 65412.3 A Few Special Functions 65812.4 Solving Differential Equations with Power Series 66612.5 Legendre Polynomials 67312.6 The Method of Frobenius 68212.7 Bessel Functions 68812.8 Sturm-Liouville Theory and Series Expansions 69712.9 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)13 Calculus with Complex Numbers 70813.1 Motivating Exercise: Laplace’s Equation 70913.2 Functions of Complex Numbers 71013.3 Derivatives, Analytic Functions, and Laplace’s Equation 71613.4 Contour Integration 72613.5 Some Uses of Contour Integration 73313.6 Integrating Along Branch Cuts and Through Poles (see felderbooks.com)13.7 Complex Power Series 74213.8 Mapping Curves and Regions 74713.9 Conformal Mapping and Laplace’s Equation 75413.10 Special Application: Fluid Flow (see felderbooks.com)13.11 Additional Problems (see felderbooks.com)Appendix A Different Types of Differential Equations 765Appendix B Taylor Series 768Appendix C Summary of Tests for Series Convergence 770Appendix D Curvilinear Coordinates 772Appendix E Matrices 774Appendix F Vector Calculus 777Appendix G Fourier Series and Transforms 779Appendix H Laplace Transforms 782Appendix I Summary: Which PDE Technique Do I Use? 787Appendix J Some Common Differential Equations and Their Solutions 790Appendix K Special Functions 798Appendix L Answers to “Check Yourself” in Exercises 801Appendix M Answers to Odd-Numbered Problems (see felderbooks.com)Index 805

"[Mathematical Methods in Engineering and Physics] is my book of choice for teaching undergraduates...I honestly never thought that I could be so enchanted by the heat equation before seeing how Felder and Felder effectively have students derive it as part of honing their intuition for how to think about partial differential equations." - Christine Aidala, PhD, Associate Professor of Physics at University of Michigan for the American Journal of Physics