# Study Notes

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

- Concept 1.1.1: Propositions, Logical Operators, and Truth Tables
- Concept 1.1.2: Conditionals and Logical Equivalences

### 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