CORE 003: Linear Algebra and Ordinary Differential Equations
Syllabus

The topic list for this project is: vector algebra and vector calculus of a single variable, matrix algebra, linear algebra, first-order differential equations, higher order differential equations, systems of differential equations, integral transform methods, power series methods, and eigenfunction methods.

Instructor: George E. Hrabovsky, george@madscitech.org, 608-276-6832.

Prerequisite: It is assumed that you have completed CORE 002 or the equivalent. You must be comfortable with the principles of single-variable calculus such as CORE 001, with matrices and determinants such as MATH R003, and with vectors such as in MATH R004.

Task #1: Start and keep a notebook for your study. This should be bound and have at least 300 sheets. You may need more than one notebook of this size. Smaller notebooks than 300-sheets can be used, but the total number of sheets should be at least 300. Each set of 300 pages started and completed is worth a point towards your final total of 4. To begin your notebook you will need a list of topics. The one listed below is only one possible choice. This choice is the default. Any choice other than this one must be approved by your instructor.

Procedure for the Course

If a topic from the list below is underscored that means there is some resource material for it. If there is no resource material for it then you must develop that for yourself.

It is expected that you will develop one or more questions for each topic. Questions can be of the form who, what, when, where, why, and how.

Once you have written down a set of questions for a topic, you either answer each of these qurestions or you explain how you attempted to answer the question and failed. Don't be alarmed; even some elementary questions resist answering. You can learn a lot just by making the effort.

The next step is to ask a set of new questions based on your previous attempts at answering your first set of questions (this can include those questions you were unable to answer before). Answer each of those questions as best you can and create another set of questions for each answer. Answer each of those to the best of your ability and ask another set of questios for each, but do not answer them right away. If you are really interested in one or more of these questions attempt to answer them in a, "topic of personal interest," session; or you may answer them in a personal research project.

Wherever possible give at least three examples of any definition, principle, or procedure.

This course requires three pages of notes per topic to fill a 300 page notebook.

1. The nature of linear algebra and differential equations
2. Pure linear algebra and differential equations
3. Applied linear algebra and differential equations
4. Vector algebra
5. Vector calculus of a single variable
6. Vector algebra and vector calculus of a single variable in Mathematica
7. Topic of Personal Interest (including, but not limited to, 5 practice problems, linear dependence and independence of vectors, euclidean space, transformation equations in euclidean space, the inner product, the Kronecker delta, the Levi-Civita symbol, the cross product, space curves, rotating coordinate systems)
8. Topic of Personal Interest
9. Topic of Personal Interest
10. Review of topics to date
11. Matrix algebra
12. Special types of square matrices
13. Linear systems of equations
14. Matrix representation of linear systems
15. Gaussian elimination
16. Elementary row and column operations
17. Determinants
18. Minors and cofactors
19. Cramer's rule
20. Eigenvalues and eigenvectors
21. Matrix algebra in Mathematica
22. Topic of Personal Interest (including, but not but not limited to, 5 practice problems, vectors as matrices, block matrices, complex matrices, LU decomposition, the volume of a parallelepipied, the characteristic polynomial of a matrix, the minimal polynomial of a matrix, matrix diagonalization)
23. Topic of Personal Interest
24. Topic of Personal Interest
25. Review of topics 11-24
26. Review of topics to date
27. Vector spaces
28. Linear mappings
29. Linear operators
30. Matrix representation of a linear operator
31. Inner product spaces
32. Gram-Schmidt orthogonalization
33. Eigenvalues and eigenvectors of linear operators
34. Linear algebra in Mathematica
35. Topic of Personal Interest (including, but not but not limited to, 5 practice problems, linear dependence and independence of vector spaces, bases and dimension, coordinate systems, kernel and image, the characteristic polynomial of a linear operator, the minimal polynomial of a linear operator)
36. Topic of Personal Interest
37. Topic of Personal Interest
38. Review of topics 27-37
39. Review of topics to date
40. Separable equations
41. Linear first-order equations
42. Exact equations
43. Integrating factors
44. Nonlinear equations
45. First-order differential equations in Mathematica
46. Topic of Personal Interest (including, but not but not limited to, 5 practice problems, orthogonal trajectories, direction fields and integral curves, Euler's method for first-order differential equations, an improved Euler's method, Bernoulli's equation, existence and uniqueness, Bernoulli's equation, Riccatti's equation)
47. Topic of Personal Interest
48. Topic of Personal Interest
49. Review of topics 40-48
50. Review of topics to date
51. Second-order equations with constant coefficients
52. Higher-order equations with constant coefficients
53. Higher-order equations with variable coefficients
54. Higher order differential equations in Mathematica
55. Topic of Personal Interest (including, but not but not limited to, 5 practice problems, the Runge-Kutta method, existence and uniqueness, equations with the dependent variable missing, equations with the independent variable missing, linear dependence and the Wronskian)
56. Topic of Personal Interest
57. Topic of Personal Interest
58. Review of topics 51-57
59. Review of topics to date
60. Systems of first-order linear differential equations
61. Systems of differential equations with constant coefficients
62. Nonlinear systems
63. Stability
64. Systems of differential equations in Mathematica
65. Topic of Personal Interest (including, but not but not limited to, 5 practice problems, existence and uniqueness, reduction of order, the phase plane, limit cycles and periodic solutions, perturbation theory, dynamical systems, chaos, variation of parameters, complex and repeated roots, the fundamental matrix)
66. Topic of Personal Interest
67. Topic of Personal Interest
68. Review of topics 60-67
69. Review of topics to date
70. Laplace transform methods
71. Fourier transform methods
72. Integral transform methods in Mathematica
73. Topic of Personal Interest (including, but not but not limited to, 5 practice problems, initial value problems, step functions, impulse functions, solving systems by integral transforms, convolutions, other integral transforms)
74. Topic of Personal Interest
75. Topic of Personal Interest
76. Review of topics 70-75
77. Review of topics to date
78. Power series methods and ordinary points
79. Regular singular points
80. The gamma and beta functions
81. The hypergeometric functions
82. Bessel functions
83. Legendre polynomials
84. Confluent hypergeometric function
85. Mathieu functions
86. Elliptic functions
87. Power series methods in Mathematica
88. Topic of Personal Interest (including, but not but not limited to, 5 practice problems, the method of Frobenius, Euler equations, analytic functions)
89. Topic of Personal Interest
90. Topic of Personal Interest
91. Review of topics 78-90
92. Review of topics to date
93. Boundary value problems
94. Sturm-Liouville problems
95. Eigenfunction methods in Mathematica
96. Topic of Personal Interest (including, but not but not limited to, 5 practice problems, adjoint operators, Green's functions, eigenfunction expansions)
97. Topic of Personal Interest
98. Topic of Personal Interest
99. Review of topics 93-98
100. Review of topics to date

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