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Mexico Spain Paraninfo Calle/Magallanes, 25 28015 Madrid, Spain Contents PART 1 Ordinary Differential Equations 1 Chapter 1 First-Order Differential Equations 3 1.1 General and Particular Solutions 3 1.2 Implicitly Defined Solutions 4 1.4 The Initial Value Problem 6 1.1 Some Applications of Separable Differential Equations 14 1.3 Linear Differential Equations 22 1.4 Exact Differential Equations 26 1.1 Separable Equations and Integrating Factors 37 1.2 Linear Equations and Integrating Factors 37 1.6 Homogeneous, Bernoulli, and Riccati Equations 38 1.1 Homogeneous Differential Equations 38 1.2 The Bernoulli Equation 42 1.3 The Riccati Equation 43 1.7 Applications to Mechanics, Electrical Circuits, and Orthogonal Trajectories 46 1.8 Existence and Uniqueness for Solutions of Initial Value Problems 58 Chapter 2 Second-Order Differential Equations 61 2.2 Theory of Solutions of y + pxy + qxy = fx 62 2.1 The Homogeneous Equation y + pxy + qx = 0 64 2.2 The Nonhomogeneous Equation y + pxy + qxy = fx 68 2.3 Reduction of Order 69 2.4 The Constant Coefficient Homogeneous Linear Equation 73 2.2 Case 2: A 2 − 4B = 0 74 v vi Contents 2.4 An Alternative General Solution in the Complex Root Case 75 2.6 The Nonhomogeneous Equation y + pxy + qxy = fx 82 2.1 The Method of Variation of Parameters 82 2.2 The Method of Undetermined Coefficients 85 2.3 The Principle of Superposition 91 2.4 Higher-Order Differential Equations 91 2.7 Application of Second-Order Differential Equations to a Mechanical System 93 2.5 Analogy with an Electrical Circuit 103 Chapter 3 The Laplace Transform 107 3.1 Definition and Basic Properties 107 3.2 Solution of Initial Value Problems Using the Laplace Transform 116 3.3 Shifting Theorems and the Heaviside Function 120 3.1 The First Shifting Theorem 120 3.2 The Heaviside Function and Pulses 122 3.3 The Second Shifting Theorem 125 3.4 Analysis of Electrical Circuits 129 3.5 Unit Impulses and the Dirac Delta Function 139 3.6 Laplace Transform Solution of Systems 144 3.7 Differential Equations with Polynomial Coefficients 150 Chapter 4 Series Solutions 155 4.1 Power Series Solutions of Initial Value Problems 156 4.2 Power Series Solutions Using Recurrence Relations 161 4.3 Singular Points and the Method of Frobenius 166 4.4 Second Solutions and Logarithm Factors 173 Chapter 5 Numerical Approximation of Solutions 181 5.1 A Problem in Radioactive Waste Disposal 187 5.2 One-Step Methods 190 5.1 The Second-Order Taylor Method 190 5.2 The Modified Euler Method 193 5.3 Runge-Kutta Methods 195 5.4 Case 4 r = 4 199 Contents vii PART 2 Vectors and Linear Algebra 201 Chapter 6 Vectors and Vector Spaces 203 6.1 The Algebra and Geometry of Vectors 203 6.2 The Dot Product 211 6.3 The Cross Product 217 6.4 The Vector Space R n 223 6.5 Linear Independence, Spanning Sets, and Dimension in R n 228 Chapter 7 Matrices and Systems of Linear Equations 237 7.2 Matrix Notation for Systems of Linear Equations 242 7.3 Some Special Matrices 243 7.4 Another Rationale for the Definition of Matrix Multiplication 246 7.5 Random Walks in Crystals 247 7.2 Elementary Row Operations and Elementary Matrices 251 7.3 The Row Echelon Form of a Matrix 258 7.4 The Row and Column Spaces of a Matrix and Rank of a Matrix 266 7.5 Solution of Homogeneous Systems of Linear Equations 272 7.6 The Solution Space of AX = O 280 7.7 Nonhomogeneous Systems of Linear Equations 283 7.1 The Structure of Solutions of AX = B 284 7.2 Existence and Uniqueness of Solutions of AX = B 285 7.1 A Method for Finding A −1 295 Chapter 8 Determinants 299 8.2 Definition of the Determinant 301 8.3 Properties of Determinants 303 8.4 Evaluation of Determinants by Elementary Row and Column Operations 307 8.6 Determinants of Triangular Matrices 314 8.7 A Determinant Formula for a Matrix Inverse 315 8.9 The Matrix Tree Theorem 320 Chapter 9 Eigenvalues, Diagonalization, and Special Matrices 323 9.1 Eigenvalues and Eigenvectors 324 9.2 Diagonalization of Matrices 330 9.3 Orthogonal and Symmetric Matrices 339 viii Contents 9.5 Unitary, Hermitian, and Skew Hermitian Matrices 352 PART 3 Systems of Differential Equations and Qualitative Methods 359 Chapter 10 Systems of Linear Differential Equations 361 10.1 Theory of Systems of Linear First-Order Differential Equations 361 10.1 Theory of the Homogeneous System X = AX 365 10.2 General Solution of the Nonhomogeneous System X = AX + G 372 10.2 Solution of X = AX when A is Constant 374 10.1 Solution of X = AX when A has Complex Eigenvalues 377 10.2 Solution of X = AX when A does not have n Linearly Independent Eigenvectors 379 10.3 Solution of X = AX by Diagonalizing A 384 10.4 Exponential Matrix Solutions of X = AX 386 10.3 Solution of X = AX + G 394 10.1 Variation of Parameters 394 10.2 Solution of X = AX + G by Diagonalizing A 398 Chapter 11 Qualitative Methods and Systems of Nonlinear Differential Equations 403 11.1 Nonlinear Systems and Existence of Solutions 403 11.2 The Phase Plane, Phase Portraits and Direction Fields 406 11.3 Phase Portraits of Linear Systems 413 11.4 Critical Points and Stability 424 11.5 Almost Linear Systems 431 11.6 Lyapunov’s Stability Criteria 451 11.7 Limit Cycles and Periodic Solutions 461 PART 4 Vector Analysis 473 Chapter 12 Vector Differential Calculus 475 12.1 Vector Functions of One Variable 475 12.2 Velocity, Acceleration, Curvature and Torsion 481 12.1 Tangential and Normal Components of Acceleration 488 12.2 Curvature as a Function of t 491 12.3 The Frenet Formulas 492 12.3 Vector Fields and Streamlines 493 12.4 The Gradient Field and Directional Derivatives 499 12.1 Level Surfaces, Tangent Planes and Normal Lines 503 12.5 Divergence and Curl 510 12.1 A Physical Interpretation of Divergence 512 12.2 A Physical Interpretation of Curl 513 Contents ix Chapter 13 Vector Integral Calculus 517 13.1 Line Integral with Respect to Arc Length 525 13.1 An Extension of Green’s Theorem 532 13.3 Independence of Path and Potential Theory in the Plane 536 13.1 A More Critical Look at Theorem 13.4 Surfaces in 3-Space and Surface Integrals 545 13.1 Normal Vector to a Surface 548 13.2 The Tangent Plane to a Surface 551 13.3 Smooth and Piecewise Smooth Surfaces 552 13.5 Applications of Surface Integrals 557 13.2 Mass and Center of Mass of a Shell 557 13.3 Flux of a Vector Field Across a Surface 560 13.6 Preparation for the Integral Theorems of Gauss and Stokes 562 13.7 The Divergence Theorem of Gauss 564 13.2 The Heat Equation 568 13.3 The Divergence Theorem as a Conservation of Mass Principle 570 13.8 The Integral Theorem of Stokes 572 13.1 An Interpretation of Curl 576 13.2 Potential Theory in 3-Space 576 PART 5 Fourier Analysis, Orthogonal Expansions, and Wavelets 581 Chapter 14 Fourier Series 583 14.1 Why Fourier Series? 583 14.2 The Fourier Series of a Function 586 14.1 Even and Odd Functions 589 14.3 Convergence of Fourier Series 593 14.1 Convergence at the End Points 599 14.2 A Second Convergence Theorem 601 14.3 Partial Sums of Fourier Series 604 14.4 The Gibbs Phenomenon 606 14.4 Fourier Cosine and Sine Series 609 14.1 The Fourier Cosine Series of a Function 610 14.2 The Fourier Sine Series of a Function 612 14.5 Integration and Differentiation of Fourier Series 614 14.6 The Phase Angle Form of a Fourier Series 623 14.7 Complex Fourier Series and the Frequency Spectrum 630 14.1 Review of Complex Numbers 630 14.2 Complex Fourier Series 631 x Contents Chapter 15 The Fourier Integral and Fourier Transforms 637 15.1 The Fourier Integral 637 15.2 Fourier Cosine and Sine Integrals 640 15.3 The Complex Fourier Integral and the Fourier Transform 642 15.4 Additional Properties and Applications of the Fourier Transform 652 15.1 The Fourier Transform of a Derivative 652 15.3 The Fourier Transform of an Integral 656 15.5 Filtering and the Dirac Delta Function 660 15.6 The Windowed Fourier Transform 661 15.7 The Shannon Sampling Theorem 665 15.8 Lowpass and Bandpass Filters 667 15.5 The Fourier Cosine and Sine Transforms 670 15.6 The Finite Fourier Cosine and Sine Transforms 673 15.7 The Discrete Fourier Transform 675 15.1 Linearity and Periodicity 678 15.2 The Inverse N -Point DFT 678 15.3 DFT Approximation of Fourier Coefficients 679 15.8 Sampled Fourier Series 681 15.1 Approximation of a Fourier Transform by an N -Point DFT 685 15.9 The Fast Fourier Transform 694 15.1 Use of the FFT in Analyzing Power Spectral Densities of Signals 695 15.2 Filtering Noise From a Signal 696 15.3 Analysis of the Tides in Morro Bay 697 Chapter 16 Special Functions, Orthogonal Expansions, and Wavelets 701 16.1 A Generating Function for the Legendre Polynomials 704 16.2 A Recurrence Relation for the Legendre Polynomials 706 16.3 Orthogonality of the Legendre Polynomials 708 16.4 Fourier–Legendre Series 709 16.5 Computation of Fourier–Legendre Coefficients 711 16.6 Zeros of the Legendre Polynomials 713 16.7 Derivative and Integral Formulas for Pn x 715 16.1 The Gamma Function 719 16.2 Bessel Functions of the First Kind and Solutions of Bessel’s Equation 721 16.3 Bessel Functions of the Second Kind 722 16.4 Modified Bessel Functions 725 16.5 Some Applications of Bessel Functions 727 16.6 A Generating Function for Jn x 732 16.7 An Integral Formula for Jn x 733 16.8 A Recurrence Relation for Jv x 735 16.9 Zeros of Jv x 737 Contents xi 16.10 Fourier–Bessel Expansions 739 16.11 Fourier–Bessel Coefficients 741 16.3 Sturm–Liouville Theory and Eigenfunction Expansions 745 16.1 The Sturm–Liouville Problem 745 16.2 The Sturm–Liouville Theorem 752 16.4 Approximation in the Mean and Bessel’s Inequality 759 16.5 Convergence in the Mean and Parseval’s Theorem 762 16.6 Completeness of the Eigenfunctions 763 16.1 The Idea Behind Wavelets 765 16.2 The Haar Wavelets 767 16.4 Multiresolution Analysis with Haar Wavelets 774 16.5 General Construction of Wavelets and Multiresolution Analysis 775 16.
