DELA_InquiryNotes.out

\BOOKMARK [0][-]{chapter.0}{To the Student and the Instructor}{}% 1
\BOOKMARK [1][-]{section.0.1}{An Inquiry Based Approach}{chapter.0}% 2
\BOOKMARK [1][-]{section.0.2}{Online Texts and Other Resources}{chapter.0}% 3
\BOOKMARK [1][-]{section.0.3}{To the Instructor}{chapter.0}% 4
\BOOKMARK [0][-]{chapter.1}{First Order Differential Equations}{}% 5
\BOOKMARK [1][-]{section.1.1}{Modeling and Differential Equations}{chapter.1}% 6
\BOOKMARK [1][-]{section.1.2}{Differential Equation Terminology}{chapter.1}% 7
\BOOKMARK [1][-]{section.1.3}{Solution Technique: Integration}{chapter.1}% 8
\BOOKMARK [1][-]{section.1.4}{Solution Technique: Separation of Variables}{chapter.1}% 9
\BOOKMARK [1][-]{section.1.5}{Solution Technique: Undetermined Coefficients}{chapter.1}% 10
\BOOKMARK [1][-]{section.1.6}{Solution Technique: Integrating Factors}{chapter.1}% 11
\BOOKMARK [1][-]{section.1.7}{Mixing Problems}{chapter.1}% 12
\BOOKMARK [1][-]{section.1.8}{Software for Differential Equations}{chapter.1}% 13
\BOOKMARK [2][-]{subsection.1.8.1}{MATLAB's dsolve command}{section.1.8}% 14
\BOOKMARK [2][-]{subsection.1.8.2}{Slope Fields and Phase Plots}{section.1.8}% 15
\BOOKMARK [2][-]{subsection.1.8.3}{An Exploration with MATLAB}{section.1.8}% 16
\BOOKMARK [1][-]{section.1.9}{Additional Exercises}{chapter.1}% 17
\BOOKMARK [0][-]{chapter.2}{Qualitative and Numerical Methods}{}% 18
\BOOKMARK [1][-]{section.2.1}{Equilibrium Points and Stability}{chapter.2}% 19
\BOOKMARK [2][-]{subsection.2.1.1}{Classifying Equilibrium Points}{section.2.1}% 20
\BOOKMARK [2][-]{subsection.2.1.2}{Phase Line and Slope Field Analysis}{section.2.1}% 21
\BOOKMARK [1][-]{section.2.2}{Numerical Methods}{chapter.2}% 22
\BOOKMARK [2][-]{subsection.2.2.1}{Euler's Method}{section.2.2}% 23
\BOOKMARK [2][-]{subsection.2.2.2}{Runge-Kutta Method}{section.2.2}% 24
\BOOKMARK [2][-]{subsection.2.2.3}{Numerical Explorations}{section.2.2}% 25
\BOOKMARK [1][-]{section.2.3}{Additional Exercises}{chapter.2}% 26
\BOOKMARK [0][-]{chapter.3}{Linear Systems and Matrices}{}% 27
\BOOKMARK [1][-]{section.3.1}{Matrix Operations and Definitions}{chapter.3}% 28
\BOOKMARK [1][-]{section.3.2}{Gaussian Elimination: Reduced Row Echelon Form}{chapter.3}% 29
\BOOKMARK [1][-]{section.3.3}{Linear Combinations}{chapter.3}% 30
\BOOKMARK [1][-]{section.3.4}{Inverses and Determinants}{chapter.3}% 31
\BOOKMARK [2][-]{subsection.3.4.1}{Inverses of Square Matrices}{section.3.4}% 32
\BOOKMARK [2][-]{subsection.3.4.2}{Determinants}{section.3.4}% 33
\BOOKMARK [2][-]{subsection.3.4.3}{The Geometry of Determinants}{section.3.4}% 34
\BOOKMARK [1][-]{section.3.5}{Additional Exercises}{chapter.3}% 35
\BOOKMARK [0][-]{chapter.4}{Vector Spaces}{}% 36
\BOOKMARK [1][-]{section.4.1}{What is a Vector Space}{chapter.4}% 37
\BOOKMARK [1][-]{section.4.2}{Linear Independence and Linear Dependence}{chapter.4}% 38
\BOOKMARK [1][-]{section.4.3}{Span}{chapter.4}% 39
\BOOKMARK [1][-]{section.4.4}{Subspaces}{chapter.4}% 40
\BOOKMARK [1][-]{section.4.5}{Basis}{chapter.4}% 41
\BOOKMARK [1][-]{section.4.6}{Row, Column, and Null Spaces in Euclidean Space}{chapter.4}% 42
\BOOKMARK [1][-]{section.4.7}{The Invertible Matrix Theorem}{chapter.4}% 43
\BOOKMARK [1][-]{section.4.8}{Additional Exercises}{chapter.4}% 44
\BOOKMARK [0][-]{chapter.5}{Geometry in Vector Spaces}{}% 45
\BOOKMARK [1][-]{section.5.1}{The Geometry of Euclidean Space}{chapter.5}% 46
\BOOKMARK [2][-]{subsection.5.1.1}{The Dot Product}{section.5.1}% 47
\BOOKMARK [2][-]{subsection.5.1.2}{Projections}{section.5.1}% 48
\BOOKMARK [2][-]{subsection.5.1.3}{The Gram-Schmidt Process: Making Orthogonal Sets}{section.5.1}% 49
\BOOKMARK [1][-]{section.5.2}{Inner Product Spaces}{chapter.5}% 50
\BOOKMARK [1][-]{section.5.3}{Fourier Series}{chapter.5}% 51
\BOOKMARK [1][-]{section.5.4}{Linear Transformations}{chapter.5}% 52
\BOOKMARK [2][-]{subsection.5.4.1}{Matrix Transformation}{section.5.4}% 53
\BOOKMARK [2][-]{subsection.5.4.2}{Linear Transformations in Abstract Vector Spaces}{section.5.4}% 54
\BOOKMARK [1][-]{section.5.5}{Additional Exercises}{chapter.5}% 55
\BOOKMARK [0][-]{chapter.6}{The Eigenvalue Eigenvector Problem}{}% 56
\BOOKMARK [1][-]{section.6.1}{Introduction To Eigenvalues}{chapter.6}% 57
\BOOKMARK [1][-]{section.6.2}{Diagonlization of Matrices}{chapter.6}% 58
\BOOKMARK [1][-]{section.6.3}{Powers of Matrices}{chapter.6}% 59
\BOOKMARK [1][-]{section.6.4}{The Google Page Rank Algorithm}{chapter.6}% 60
\BOOKMARK [1][-]{section.6.5}{Additional Exercises}{chapter.6}% 61
\BOOKMARK [0][-]{chapter.7}{Second Order Differential Equations}{}% 62
\BOOKMARK [1][-]{section.7.1}{Intro to Second Order Differential Equations}{chapter.7}% 63
\BOOKMARK [1][-]{section.7.2}{Types of Mechanical Vibrations}{chapter.7}% 64
\BOOKMARK [1][-]{section.7.3}{Undetermined Coefficients}{chapter.7}% 65
\BOOKMARK [1][-]{section.7.4}{Resonance and Beats}{chapter.7}% 66
\BOOKMARK [1][-]{section.7.5}{Additional Exercises}{chapter.7}% 67
\BOOKMARK [0][-]{chapter.8}{Linear and Nonlinear Systems of ODEs}{}% 68
\BOOKMARK [1][-]{section.8.1}{From Second Order ODE's to Systems}{chapter.8}% 69
\BOOKMARK [1][-]{section.8.2}{Matrices and Linear Systems}{chapter.8}% 70
\BOOKMARK [1][-]{section.8.3}{The Eigenvalue Method for Linear Systems}{chapter.8}% 71
\BOOKMARK [1][-]{section.8.4}{The Matrix Exponential}{chapter.8}% 72
\BOOKMARK [1][-]{section.8.5}{Complex Eigenvalues for Linear Systems of ODEs}{chapter.8}% 73
\BOOKMARK [1][-]{section.8.6}{Trace-Determinant Plane}{chapter.8}% 74
\BOOKMARK [1][-]{section.8.7}{Building ODE Models for Nonlinear Systems}{chapter.8}% 75
\BOOKMARK [1][-]{section.8.8}{Equilibria and Linearization of Nonlinear Systems}{chapter.8}% 76
\BOOKMARK [1][-]{section.8.9}{Applied Nonlinear Systems}{chapter.8}% 77
\BOOKMARK [1][-]{section.8.10}{Additional Exercises}{chapter.8}% 78
\BOOKMARK [0][-]{chapter.9}{Laplace Transforms}{}% 79
\BOOKMARK [1][-]{section.9.1}{Introduction to Laplace Transforms}{chapter.9}% 80
\BOOKMARK [1][-]{section.9.2}{Basic Laplace Transforms and Basic Properties}{chapter.9}% 81
\BOOKMARK [1][-]{section.9.3}{Important Theorems for Laplace Transforms}{chapter.9}% 82
\BOOKMARK [1][-]{section.9.4}{Common Laplace Transforms}{chapter.9}% 83
\BOOKMARK [1][-]{section.9.5}{Solving Differential Equations with Laplace Transforms}{chapter.9}% 84
\BOOKMARK [1][-]{section.9.6}{The Heaviside Function and Delayed Forcing Terms}{chapter.9}% 85
\BOOKMARK [1][-]{section.9.7}{Impulses and The Delta Function}{chapter.9}% 86
\BOOKMARK [1][-]{section.9.8}{Additional Exercises}{chapter.9}% 87
\BOOKMARK [0][-]{chapter.10}{Power Series Method: The Ultimate Guess}{}% 88
\BOOKMARK [1][-]{section.10.1}{Taylor Series Solutions to Diff. Equations}{chapter.10}% 89
\BOOKMARK [1][-]{section.10.2}{Radius of Convergence for Power Series}{chapter.10}% 90
\BOOKMARK [1][-]{section.10.3}{Power Series Solutions to Diff. Equations}{chapter.10}% 91
\BOOKMARK [1][-]{section.10.4}{Additional Exercises}{chapter.10}% 92
\BOOKMARK [0][-]{chapter.11}{Partial Differential Equations}{}% 93
\BOOKMARK [1][-]{section.11.1}{Some Reminders from Multivariable Calculus}{chapter.11}% 94
\BOOKMARK [1][-]{section.11.2}{Where to PDEs Come From?}{chapter.11}% 95
\BOOKMARK [2][-]{subsection.11.2.1}{Derivation of General Balance Law}{section.11.2}% 96
\BOOKMARK [2][-]{subsection.11.2.2}{Simplifications of the Local Balance Law}{section.11.2}% 97
\BOOKMARK [2][-]{subsection.11.2.3}{Laplace's Equation and Poisson's Equation}{section.11.2}% 98
\BOOKMARK [1][-]{section.11.3}{The 1D Heat Equation}{chapter.11}% 99
\BOOKMARK [2][-]{subsection.11.3.1}{1D Heat Equation with Zero Temperature Ends}{section.11.3}% 100
\BOOKMARK [2][-]{subsection.11.3.2}{1D Heat Equation with Insulated Ends}{section.11.3}% 101
\BOOKMARK [2][-]{subsection.11.3.3}{Heat Equation on a Thin Ring}{section.11.3}% 102
\BOOKMARK [1][-]{section.11.4}{Additional Exercises}{chapter.11}% 103
\BOOKMARK [0][-]{section*.31}{Appendices}{}% 104
\BOOKMARK [0][-]{Appendix.a.A}{Partial Fractions}{}% 105
\BOOKMARK [0][-]{Appendix.a.B}{MATLAB Basics}{}% 106
\BOOKMARK [1][-]{section.a.B.1}{Vectors and Matrices}{Appendix.a.B}% 107
\BOOKMARK [1][-]{section.a.B.2}{Looping}{Appendix.a.B}% 108
\BOOKMARK [2][-]{subsection.a.B.2.1}{For Loops}{section.a.B.2}% 109
\BOOKMARK [2][-]{subsection.a.B.2.2}{The While Loop}{section.a.B.2}% 110
\BOOKMARK [1][-]{section.a.B.3}{Conditional Statements}{Appendix.a.B}% 111
\BOOKMARK [2][-]{subsection.a.B.3.1}{If Statements}{section.a.B.3}% 112
\BOOKMARK [2][-]{subsection.a.B.3.2}{Case-Switch Statements}{section.a.B.3}% 113
\BOOKMARK [1][-]{section.a.B.4}{Functions}{Appendix.a.B}% 114
\BOOKMARK [1][-]{section.a.B.5}{Plotting}{Appendix.a.B}% 115
\BOOKMARK [1][-]{section.a.B.6}{Animations}{Appendix.a.B}% 116
\BOOKMARK [0][-]{Appendix.a.C}{LaTeX}{}% 117
\BOOKMARK [1][-]{section.a.C.1}{Equation Environments and Cross Referencing}{Appendix.a.C}% 118
\BOOKMARK [1][-]{section.a.C.2}{Tables, Tabular, Figures, Shortcuts, and Other Environments}{Appendix.a.C}% 119
\BOOKMARK [2][-]{subsection.a.C.2.1}{Tables and Tabular Environments}{section.a.C.2}% 120
\BOOKMARK [2][-]{subsection.a.C.2.2}{Excel To LaTeX}{section.a.C.2}% 121
\BOOKMARK [2][-]{subsection.a.C.2.3}{Figures}{section.a.C.2}% 122
\BOOKMARK [2][-]{subsection.a.C.2.4}{New Commands: Shortcuts are AWESOME!}{section.a.C.2}% 123
\BOOKMARK [1][-]{section.a.C.3}{Graphics in LaTeX}{Appendix.a.C}% 124
\BOOKMARK [2][-]{subsection.a.C.3.1}{The Tikz and PGFPlots Packages}{section.a.C.3}% 125
\BOOKMARK [1][-]{section.a.C.4}{Bibliography Management}{Appendix.a.C}% 126
\BOOKMARK [2][-]{subsection.a.C.4.1}{Embedded Bibliography}{section.a.C.4}% 127
\BOOKMARK [2][-]{subsection.a.C.4.2}{Bibliography Database: BibTeX}{section.a.C.4}% 128
\BOOKMARK [0][-]{Appendix.a.D}{Miscellaneous Topics}{}% 129
\BOOKMARK [1][-]{section.a.D.1}{Existence and Uniqueness of Solutions to First Order ODEs}{Appendix.a.D}% 130
\BOOKMARK [1][-]{section.a.D.2}{Bifurcations in First Order Differential Equations}{Appendix.a.D}% 131
\BOOKMARK [1][-]{section.a.D.3}{Isomorphic Vector Spaces}{Appendix.a.D}% 132
\BOOKMARK [1][-]{section.a.D.4}{Higher Dimensions are Geometrically Weird}{Appendix.a.D}% 133
\BOOKMARK [1][-]{section.a.D.5}{Where Laplace Transforms Come From}{Appendix.a.D}% 134
\BOOKMARK [1][-]{section.a.D.6}{Convolutions with Laplace Transforms}{Appendix.a.D}% 135