Notes and Documents

Mathematics and Music

  1. Hamiltonian Cycles in Music
  2. Ionian, Dorian and Phrygian Modes
  3. Lydian,Mixolydian and Aeolian Modes
  4. D Minor Inversions
  5. Modes

Elementary Math for Teachers

  1. Course Introduction
  2. Math 3041 Homeworks 1-13
  3. Math 3041 Practice Problems
  4. Relations
  5. Errata
  6. Section 1.1 Counting
  7. Section 1.2 Place Value
  8. Exercise on the Addition Table
  9. Section 1.4 Subtraction
  10. Section 1.5 Multiplication
  11. Exercise on the Multiplication Table
  12. Section 1.6 Division
  13. Section 3.3 Algorithms
  14. Section 3.4 A Division Algorithm
  15. Section 3.5 Estimation
  16. Section 4.1 Prealgebra
  17. Section 4.3 Divisibility
  18. Section 5.3 Factors and Prime Numbers
  19. Exercises on Factors and Primes
  20. Exercises on prime numbers and the greatest common factor
  21. Section 6.1 Fractions
  22. Section 6.1   Partitive and Measure Division
  23. Notes on the conversion of units
  24. Solving Algebra and Other Story Problems by Sybilla Beckmann
  25. Tree of Porphyry
  26. Some Educational References

Precalculus/Foundations of Mathematics

  1. On Definition (Definition according to Aristotle)
  2. On Truth
  3. On implication
  4. On Sets
  5. Relations and Functions
  6. Fractions
  7. Properties of the real numbers
  8. Order and Inequalities
  9. Absolute Value
  10. Straight Lines
  11. The Plane
  12. Some Standard Forms of the Straight Line
  13. Graphing examples
  14. Image of a function
  15. The Circle
  16. The Parabola
  17. Exponential functions–review
  18. Sequences
  19. Limits of sequences
  20. Complex Numbers
  21. Notes on Trigonometry

Calculus

  1. The Limit of a Sequence
  2. The epsilon-delta definition of a limit -From Brilliant.org
  3. Limits using Epsilon and Delta – Example.
  4. The Chain Rule
  5. Points of Inflexion
  6. Rates of Change
  7. Rectilinear Motion
  8. Mean Value Theorem
  9. Optimization (Application to Optics)
  10. Antiderivative Formulae
  11. Antiderivatives Involving the Secant Function
  12. The Definite Integral
  13. The Fundamental Theorem
  14. Applications of the Definite Integral
  15. Work
  16. Harder Substitution Examples
  17. The Irrationality of e
  18. Nintendo Calculus ☺
  19. Some Ways to Define Functions
  20. Remarks on Parity
  21. Hyperbolic Sectors
  22. Hyperbolic Functions
  23. Inverse Trigonometric and Hyperbolic Functions
  24. Miscellaneous Trigonometric and Hyperbolic Integrals
  25. Hyperbolic Functions – Examples
  26. Euler’s Formula – Useful Corollaries
  27. Terminal Velocity
  28. The Substitution $t = \tan (x/2)$
  29. Rational Functions
  30. Integration of Rational Functions
  31. Improper Integrals
  32. Convergence of Series
  33. Absolute Convergence
  34. Ratio Test
  35. Cauchy Condensation
  36. Alternating Series
  37. Limit Comparison Test
  38. Power Series-I
  39. Power Series-II
  40. Computing Taylor Series
  41. Exponential Series
  42. The Irrationality of e
  43. Equality up to Order $n$
  44. Limit Superior and Limit Inferior
  45. The Cycloid
  46. Curvature
  47. Complex Numbers – Miscellaneous Computations

Calculus of Several Variables

  1. Determinants and Orientation in R^2
  2. The Vector Product
  3. Matrix Multiplication -Brief Review
  4. Reformulation of the usual Derivative
  5. The Derivative of a Scalar Function
  6. The Derivative of a Vector Valued Function
  7. The Chain Rule for Curves
  8. The Chain Rule for Vector Valued Functions
  9. Unit Speed Reparametrization
  10. Line Integrals of Scalar Functions
  11. Line Integrals of Vector Fields
  12. Vector Fields and Potential Functions
  13. Optimization – Summary
  14. Optimization with Constraints
  15. Double Integrals
  16. Formulae for Integration

Differential Equations

  1. Introduction to ODE’s
  2. Properties of the Direction Field
  3. Separable ODE’s
  4. Integrating Factors
  5. Second Order Linear ODE’s
  6. Second Order Linear ODE’s with Constant Coefficients
  7. Method of Variation of Parameters
  8. Forced Oscillator – Steady State Solution by Fourier Transfor
  9. Convolution and Time Invariant Linear Systems
  10. Forced Damped Oscillatory Systems
  11. Scaling of First and Second Order Linear Equations (nondimensionalization)

Logic

  1. Some Axioms for Set Theory
  2. Propositional Calculus – Syntax
  3. Propositional Calculus – Theorems
  4. Propositional Calculus – Proofs
  5. Propositional Calculus – Semantics
  6. Functions, Relations and Structures
  7. First Order Logic – Syntax
  8. First Order Logic – Axioms
  9. First Order Logic – Semantics
  10. Logical circuit symbols and connectives
  11. Turing Machines
  12. First Order Logic – Substitution

Geometry

  1. Unit Speed Reparametrization
  2. Winding Number and Signed Curvature
  3. Regular Surfaces I
  4. Regular Surfaces II
  5. Regular Surfaces III
  6. Jacobian of the Differential
  7. Some Regular Surfaces
  8. Surfaces of Revolution
  9. The Second Fundamental Form
  10. Gauss’s Theorema Egregium
  11. Minimal Surfaces
  12. Hyperbolic Paraboloid Exercise
  13. Self Adjointness
  14. Umbilics
  15. Isometries
  16. Connection Coefficients

Combinatorics/Discrete Mathematics

  1. Inverse Functions
  2. Digraph Taxonomy
  3. Directed Graphs
  4. Equivalence Relations
  5. Simple Graphs
  6. Edge unlabelled Multigraphs
  7. Elementary Combinatorial Formulae
  8. The Symmetric Difference
  9. Balls in Boxes
  10. Binomial-Multinomial Theorems
  11. Multisets
  12. Partitions
  13. Inclusion Exclusion Principle
  14. On the Number of Surjections
  15. Function Digraphs
  16. Group Actions
  17. Orbit Stabilizer Theorem
  18. Cycle Structure
  19. Chemical Isomers and Coloring
  20. Closure
  21. Lattice Types
  22. Max–min inequality
  23. Summary of Lecture on the GCD
  24. Whitney’s Theorem

Algebra

  1. Permutations and Transpositions
  2. Group axioms
  3. The Euclidean Group E(3)
  4. Subgroups
  5. Group homomorphisms
  6. Isomorphisms
  7. Cyclic Groups
  8. Lagrange’s Theorem
  9. Normal Subgroups
  10. First Isomorphism Theorem
  11. Greatest Common Divisors
  12. Direct Products of Modules
  13. Direct Sums of Modules
  14. Retracts (Summands)
  15. Products of Groups I
  16. Products of Groups II

Linear Algebra

  1. Matrix Multiplication
  2. Elimination and Back Substitution.
  3. The Inverse of a Matrix.
  4. LU Factorization.
  5. Permutations and Transpositions
  6. Reduction to the Identity
  7. Linear Spaces
  8. Linear Independence
  9. Spanning Sets and Independent Sets
  10. Exam I -Practise Questions
  11. Euclidean Space
  12. Lines in Euclidean Space
  13. Gram Schmidt Algorithm
  14. Orientation in $R^2$
  15. The Vector Product
  16. Planes
  17. Linear Equations
  18. Linear Equations – Simple Examples
  19. Subspaces (updated April 27 2019)
  20. Applications of the Hermite Normal Form (Updated April 27, 2019)
  21. The Hermite Normal Form
  22. Matrix of a Linear Map
  23. Dimensions of Kernel and Image 
  24. The Null Space of a Matrix
  25. Finding the Left Nullspace
  26. Determinants.
  27. Determinant of a Direct Product.
  28. Graph Matrices
  29. Circuits
  30. Closure
  31. Affine Spaces in Physics
  32. Distance to an Affine Subspace
  33. Convex Sets
  34. The Centroid
  35. Projection and Least Squares

Statistics

  1. The Distribution of the Mean.
  2. Projection and Least Squares. .
  3. Linear Correlation..
  4. Some Important Density Functions.
  5. Selected Formulae.
  6. Elementary Combinatorial Formulae.
  7. Normal CDF Table.

Physics and Chemistry

  1. Scientific Units
  2. Formulae for Integration
  3. Isomers
  4. Physical Applications of Stokes’ Theorem
  5. Hamilton’s Equations
  6. Special Relativity
  7. Waves – Terminology
  8. Table of Ph Values
  9. Concerning the Measurability
    of a Global Mean Surface Temperature

Exams for the B.Sc in Physics

  1. Paper I
  2. Paper II
  3. Paper III
  4. Paper IV
  5. Paper V.

US Citizenship Naturalization