Below, you can find the course syllabus for Study Notes and the respective articles for each section. The syllabus is designed so that it can be read in sequence to understand everything, but can also be read in buffet-style. It deviates from normal curricula, covering each topic from high school to university levels, all in one go. Alternatively, you can read a specific article, and its linked prerequisite concepts and further readings.

Mathematics

The mathematics course covers each broad topic of mathematics from precalc to university. It gives you a rigorous foundation of mathematics from the get-go, allowing you to form a deeper understanding of concepts normally taught with some hand-waving in high school mathematics courses. Each unit spans from simple to complex. You may skip around as you please to get the desired level of difficulty. For instance, those interested in the physics course may only study units 2 and 3, and read parts of unti 4, 5, and 6 as needed.

Unit 1: Logic, Proofs, and Discrete Structures

Section 1.1: Propositional Logic

Section 1.2: Predicate Logic

  • Concept 1.2.1: Predicates
  • Concept 1.2.2: Quantifiers
  • Concept 1.2.3: Nested Quantifiers

Section 1.3: Proof Techniques

  • Concept 1.3.1: Rules of Inference
  • Concept 1.3.2: Direct Proofs
  • Concept 1.3.3: Proof by Contraposition
  • Concept 1.3.4: Proof by Contradiction
  • Concept 1.3.5: Proof by Exhaustion or Cases
  • Concept 1.3.6: Proof for existence, uniqueness, or nonexistence

Section 1.4: Discrete Structures

  • Concept 1.4.1: Sets
  • Concept 1.4.2: Functions
  • Concept 1.4.3: Series and its Sums

Section 1.5: Mathematical Induction

  • Concept 1.5.1: Mathematical Induction
  • Concept 1.5.2: Strong Induction
  • Concept 1.5.3: Structural Induction

Unit 2: Real-Valued Functions

Section 2.1: Functions

  • Concept 2.2.1: Polynomial Functions and the Fundamental Theorem of Algebra
  • Concept 2.2.2: Rational Functions
  • Concept 2.2.3: Transformation of Functions
  • Concept 2.2.4: Inverse Functions

Section 2.2: Exponential and Logarithmmic Functions

  • Concept 2.2.1: Laws of exponents
  • Concept 2.2.2: Laws of logarithms
  • Concept 2.2.3: Exponential and logarithmic functions

Section 2.3: Trigonometric Functions

  • Concept 2.3.1: Unit Circle and Radians
  • Concept 2.3.2: Trigonometric Ratios
  • Concept 2.3.3: Trigonometric Functions
  • Concept 2.3.4: Trigonometric Identities
  • Concept 2.3.5: Inverse Trigonometric Functions

Unit 3: Single-Variable Calculus

Section 3.1: Limits and Differentiation

  • Concept 3.1.1: Limits and Continuity
  • Concept 3.1.2: Difference Quotient and Derivatives
  • Concept 3.1.3: Derivatives of Common Functions
  • Concept 3.1.4: Product and Quotient Rules
  • Concept 3.1.5: Chain Rule and Implicit Differentiation
  • Concept 3.1.6: Hyperbolic Trigonometric Functions
  • Concept 3.1.7: Approximations
  • Concept 3.1.8: Optimizations
  • Concept 3.1.9: Mean Value Theorem
  • Concept 3.1.10: L’Hopital’s Rule

Section 3.2: Antiderivatives and Definite Integrals

  • Concept 3.2.1: Antiderivatives and Riemann Sums
  • Concept 3.2.2: Fundamental Theorem of Calculus
  • Concept 3.2.3: Second Fundamental Theorem of Calculus
  • Concept 3.2.4: Trig Substitution
  • Concept 3.2.5: Integration by Parts
  • Concept 3.2.6: Surface Areas, Volumes, and Arc Lengths
  • Concept 3.2.7: Parametric Equations and Polar Coordinates
  • Concept 3.2.8: Improper Integrals
  • Concept 3.2.9: Taylor Series

Unit 4: Multivariable Calculus

Section 4.1: Vectors

Section 4.2: Partial Derivatives

Section 4.3: Double Integrals

Section 4.4: Triple Integrals

Unit 5: Differential Equations

Unit 6: Linear Algebra

Unit 7: Probability and Statistics

Unit 8: Complex Analysis

unti 9: Discrete Mathematics

Unit 10: Number Theory and Cryptography

Physics

Unit 1: Mechanics

Unit 2: Thermodynamics

Unit 3: Electricity and Magnetism

Unit 4: Circular Motion, Waves, and Light

Unit 5: Particle Physics

Unit 6: Fluid Dynamics

Unit 7: Relativity

Computer Science

Unit 1: Computer Organization

Unit 2: Computer Engineering

Unit 3: Data Structures and Algorithms

Unit 4: Theory of Computation

Unit 5: Artificial Intelligence

Unit 6: Programming Language Design

Unit 7: Human-Computer Interaction

Unit 8: Computer Networking

Unit 9: Cybersecurity

Unit 10: Computer Graphics

Unit 11: Databases

Unit 12: Societal Impacts of Computing