Basic Mathematics and Physics to be a Great Amateur Theoretical Physicist

You will want to study basic mathematics before you get into mechanics.

For what follows these are good free resources:

If you wish to buy a book, I recommend these:

You should master these topics/skills before you move on to classical mechanics:

This requires either Mathematica 8 or later, or the free Mathematica CDF Viewer, though the viewer cannot run the programs, (you can find that here). You will also need to download the MAST Writing Style into the folder SystemFiles/Front End/Stylesheets. You can download that here. Once you load this file into the folder rename it MAST Writing Style 3. Reload Mathematica and it will be there.

The topics below form several logical units.

Unit 1: Sets of Numbers and Their Generalization

This constitutes a modern introductory course in algebra. The key reason to study this, even if you altready have studied elementary algebra, is that the language of modern algebraic structures are introdude early and used throughout. This is the language upon which all of modern mathematics is based.

  1. Numbers and Sets
  2. Logic:
  3. Relations
  4. Binary Operations, Addition, Summation, Multiplication, Products, and Exponentiation
  5. Semigroups and Solving Equations by Subtraction
  6. Integers
  7. Monoids and Groups
  8. Rings and Integral Domains
  9. Solving Equations by Division
  10. Primes, Factoring, and Division
  11. The Fundamental Theorem of Arithmetic
  12. Rational Numbers
  13. Fields
  14. Decimal Expansions
  15. Scientific Notation
  16. Other Simple Algebraic Structures
  17. Modular Arithmetic
  18. Finite Arithmetic
  19. Diophantine Equations
  20. Rules for Exponents
  21. Roots and Logarithms
  22. Real Numbers
  23. Dedekind Cuts and Ordered Fields
  24. Imaginary Numbers
  25. Complex Numbers
  26. Quaternions and Gaussian Integers
  27. Scientific and Mathematical Writing
  28. Algebraic Expressions
  29. Polynomials
  30. Partial Fractions
  31. The Binomial Theorem
  32. Combinatorics
  33. Probability
  34. Measurement and Error
  35. Equations
  36. Functions, Maps, and Morphisms
  37. Infinite Sets
  38. Algebraic Functions
  39. The Theory of Equations
  40. Exponential and Logarithmic Functions

Unit 2: Sets of Points

This is a modern course in introductory geometry where the concepts of geometric symmetry using the language of algebraic structures is used extensively. This merges the concepts of traditional geometry, algebraic structures, trigonometry, analytic geometry, and linear algebra.

  1. Geometry
  2. Segments, Rays, and Lines
  3. Distance and Time
  4. Angles and Triangles
  5. Polygons
  6. Circles and Arcs
  7. Perimeter and Area
  8. Constructions
  9. Congruence and Similarity
  10. Ratio and Proportion
  11. Scaling
  12. Lines and Planes in Space
  13. Polyhedra
  14. Round Figures in Space
  15. Surface Area and Volume
  16. Estimation
  17. The Real Line
  18. Coordinate Systems
  19. Plotting Data
  20. Operations on Points
  21. Plotting Functions
  22. Transformations
  23. Right Angle Trigonometry
  24. General Trigonometry
  25. Trigonomettric Functions and Periodicity
  26. Trigonometric Identities
  27. Matrices, Transformations, and Isometries
  28. Complex Geometry and de Moivre's Theorem
  29. Simple Groups
  30. Linear Systems of Equations
  31. Determinants
  32. Dimensional Analysis
  33. Lines in Coordinate Systems
  34. Vector Algebra and Geometry
  35. Scalar and Vector Products
  36. Conic Sections
  37. Parametric Representation of Curves
  38. Quadric Surfaces
  39. Vector Spaces and Linear Mappings
  40. Quadratic Forms

Unit 3: Sets of Functions

This consistutes a course in differential and integral calculus.

  1. Sequences
  2. Dynamical Systems
  3. Limits of Sequences
  4. Limits of Functions
  5. The Formal Definition of Limits
  6. Infinity and Limits
  7. Continuous Functions
  8. Uniform Continuity
  9. The Derivative
  10. Differentiation Rules
  11. Implicit Differentiation
  12. Critical Points
  13. Optimization
  14. Curve Sketching
  15. Important Theorems of Differentiation
  16. Approximations and Differentials
  17. Related Rates
  18. Motion in One Dimension
  19. Riemann Sums
  20. The Definite Integral
  21. The Indefinite Integral
  22. The Fundamental Theorem of Calculus
  23. Hyperbolic Functions
  24. Arc Length
  25. Mean Values
  26. Areas
  27. Volumes
  28. Integrals in Polar Coordinates
  29. Curvature
  30. An Introduction to Differential Equations
  31. The Properties of Integrals
  32. Integration by Substitution
  33. Integration by Partial Fractions
  34. Integration by Parts
  35. Trigonometric Integrands and Substitutions
  36. Approximate Integration
  37. Improper Integrals
  38. Other Methods of Integration
  39. Separable and Homogeneous Differential Equations
  40. First-Order Linear Differential Equations

Unit 4: Modeling, More About Differential Equations, and Series

This extends the ideas of basic calculus.

  1. Models
  2. Electric Circuits
  3. Second-Order Differential Equations
  4. Electric Circuits II
  5. Difference Equations
  6. The Laplace Transform
  7. Solving Differential Equations by Laplace Transforms
  8. Continuous Dynamical Systems
  9. Newton's Equation in One Dimension
  10. Falling Bodies
  11. Parachuting
  12. Probability Distributions
  13. Random Experiments
  14. Taylor Polynomials
  15. Taylor's Theorem
  16. Infinite Series
  17. Transformaing Series
  18. The Idea of Covnergence and The Preliminary Test
  19. The Comparison Test
  20. The Root Test
  21. The Ratio Test
  22. The Ratio Comparison Test
  23. The Integral Test
  24. Gauss's Test
  25. Alternating Series
  26. Absolute and Conditional Convergence
  27. Operations on Series
  28. Kummer's Method for Improving Convergence
  29. Uniform Convergence
  30. The Weierstrass M Test and Abel's Test
  31. Properties of Uniformly Convergent Series
  32. Taylor's Expansion
  33. Power Series
  34. Convergence of Power Series
  35. Operations with Power Series
  36. Indeterminate Forms
  37. Inversion of a Power Series
  38. Tricks for Series Expansions
  39. Important Series
  40. Power Series Solution of Differential Equations

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