Notes

This is the place where I'll put any interesting course notes I might have, once I get around to retyping them in LaTeX or markdown.

Calculus I (AP Calculus AB + University of Waterloo Math 117)

This is the first calculus course most students take, and it is similar, if somewhat more advanced, in comparison to most high school courses I've seen.

Covered topics

  1. Limits
    • Intuitive Definition of a Limit
    • Delta-Epsilon Definition of a Limit
    • Techniques for computing Limits and Limit Laws
  2. Derivatives
    • Limit Definition of the Derivative
    • Common Derivatives
    • Differentiation Rules
    • Derivatives of Trigonometric Functions
    • L'Hopital's Rule
  3. Integrals
    • Riemann Sums
    • Definition of the Integral
    • The Fundamental Theorem of Calculus
    • Properties of the Integral
    • Antiderivatives
    • The Substitution Method
    • Integration by Parts
    • Trigonometric Substitution
    • Partial Fraction Decomposition
    • Approximation Methods for Integrals

Notes

Currently, I haven't posted my notes yet. Sorry! I'll try to get some kind of online storage working so that I can post my typed LaTeX notes. Soon(tm).

Linear Algebra (University of Waterloo Math 115)

This is a course in linear algebra that covers the basics needed for a 1A Engineering student at the University of Waterloo.

Covered topics

  1. Vectors and Vector Geometry
    • Definition of Vectors
    • Linear Combinations of Vectors
    • Norm/Magnitude and the Dot Product
    • Vector Equations of Lines and Planes
    • The Cross Product in 3-D space
    • Scalar Equtions of Planes
    • Projections and Applications
  2. Systems of Linear Equations
    • Augmented Matrices
    • Solving Linear Equations through Gaussian Elimination
    • Ranks and Pivots
    • Homogenous Systems
  3. Matrices
    • Matrix Algebra
    • The Matrix-Vector Product
    • Matrix Equations
    • The Matrix-Matrix Product
    • Matrix Inversion
  4. Subspaces
    • Spanning Sets and Geometry
    • Linear Independence
    • Subspaces
    • Bases and Dimensions of Subspaces
    • Fundamental Subspaces of a Matrix
  5. Linear Transformations
    • Matrix Transofmrations and Linear Transformations
    • Algebra of Linear Transformations
    • Linear Transformation Inversion
    • Kernel and Range of Linear Transformations
  6. Determinants
    • Invertibility
    • Elementary Operations on Determinants
    • Properties of Determinants
  7. Complex Numbers
    • Complex Number Algebra
    • Conjugate and Modulus of Complex Numbers
    • Polar and Exponential Forms
    • Complex Polynomials
    • Complex Roots
  8. Eigenvalues and Eigenvectors
    • Definition of Eigenvalues and Eigenvectors
    • Computing Eigenvalues and Eigenvectors
    • Eigenspaces
    • Diagonalization
    • Matrix Exponentiation

Notes

Currently, I haven't posted my notes yet. Sorry! I'll try to get some kind of online storage working so that I can post my typed LaTeX notes. Soon(tm).

Introduction to Programming (University of Waterloo CS 137)

This is an introductory course to programming in C for University of Waterloo Software Engineering students. The course teaches basic operations in C and design patterns, culminating in basic algorithm analysis and the linked list data structure.

Covered topics

  1. Introduction to C
    • Variables
    • Arithmetic Operators
    • Relational Operators
    • Bit Manipulation
  2. Decision-Making
    • Conditional Statements
    • Switch Statements
  3. Loops
    • While Loops
    • Do-While Loops
    • For Loops
    • Early Escaping
  4. Functions and Modules
    • Utilizing Functions
    • Creating Function Declarations
    • Variable Scoping
  5. Recursion
    • Recursive Functions
    • Tail and Accumulative Recursion
  6. Arrays
    • Size of Arrays
    • Arrays in Functions
    • Copying Arrays
    • Variable-Length Arrays
    • Multidimensional Arrays
  7. Floating Point Numbers and Mathematics
    • Floating Point Standard
    • Floating Point Errors
    • C Standard Math Library
  8. Structures in C
    • Creating and Using Structures
    • Structure Arrays
    • Nested Structures
  9. Pointers and Memory Management
    • Pointers
    • Pointer Arithmetic
    • Dynamic Memory Model
    • The Struct Hack
    • Vectors
  10. Characters and Strings
    • ASCII and C characters
    • Strings
    • C Standard String Library
    • String IO
  11. Big-O Notation and Runtime Complexity
    • Definition of Big-O
    • Theorems for Determining Big-O
  12. Searching and Sorting Algorithms
    • Linear Search
    • Binary Search
    • Selection Sort
    • Insertion Sort
    • Merge Sort
    • Quick Sort
  13. Linked Lists
    • Structure
    • Generalization
    • Copying

Notes

Currently, I haven't posted my notes yet. Sorry! I'll try to get some kind of online storage working so that I can post my typed LaTeX notes. Soon(tm).

Honours Algebra (University of Waterloo Math 135)

This is a math course covering first order logic, elementary number theory, and complex numbers.

Covered topics

  1. Introduction to Mathematical Language
    • Sets
    • Quantified Statements
    • Nested Quantifiers
  2. Logical Analysis of Mathematical Statements
    • Truth Tables
    • Negation, Conjunction, and Disjunction
    • Logical Operators and Algebra
    • Implications
    • Converses and Contrapositives
    • If and Only If Statements
  3. Proving Mathematical Statements
    • Proving Universal Quantifiers
    • Proving Existential Quantifiers
    • Proving Implications
    • Divisibility of Integers
    • Proof by Contrapositive
    • Proof by Contradiction
    • Proving Uniqueness
    • Proving Equivalence
  4. Mathematical Induction
    • Summations, Products, and Recurrences
    • Mathematical Induction
    • Strong Induction
  5. Sets
    • Set Builders
    • Set Operations
    • Subsets
    • Set Equality and Implications
  6. The GCD
    • The Division Algorithm
    • The Greatest Common Divisor
    • The Certificate of Correctness and Bezout's Lemma
    • The Extended Euclidean Algorithm
    • Properties of the GCD
    • Prime Numbers
    • The Unique Factorization Theorem
    • Prime Factorization and the GCD
  7. Linear Diophantine Equations
    • Determining the Existence of Solutions
    • Determining all Solutions
  8. Congruence and Modular Arithmetic
    • Congruence
    • Properties of Congruences
    • Congruences and Remainders
    • Linear Congruences
    • Non-Linear Congruences
    • Congruence Classes and Modular Arithmetic
    • Fermat's Little Theorem
    • The Chinese Remainder Theorem
    • Splitting Congruences
  9. The RSA Encryption Scheme
    • Public-Key Cryptography
    • The RSA Scheme
    • Proof of the RSA Scheme
  10. Complex Numbers
    • Standard Form
    • Conjugates and Moduli/Magnitudes
    • Polar Form and the Complex Plane
    • De Moivre's Theorem
    • Complex nth Roots
    • Square Roots and the Quadratic Formula
    • Arithmetic of Real and Complex Polynomials
    • Roots of Polynomials and Factoring

Notes

Currently, I haven't posted my notes yet. Sorry! I'll try to get some kind of online storage working so that I can post my typed LaTeX notes. Soon(tm).

Chemistry (University of Waterloo Chem 135)

This is a basic course covering fundamental first-year topics in Chemistry like equillibrium, gases, and reaction rates.

Covered topics

  1. Stoichiometry
    • Unit Conversion
    • Atomic Theory
    • Chemical Reactions
    • The Mole
    • Compositions of Compounds and Mixtures
    • Stoichiometric Calculations
  2. States of Matter
    • Molecular Behavior and Gas Pressure
    • The Ideal Gas Law and Applications
    • Gas Mixtures
    • Intermolecular Forces and Real Gases
    • Condensed States and Temperature
  3. Rates of Reaction
    • Collision Theory
    • Measuring Reaction Rates
    • Temperature and Concentration
  4. Phase Equillibrium
    • Phase Transitions
    • Phase Equillibrium
    • Vapor Pressure
    • Phase Diagrams
    • Henry's Law and Raoult's Law
    • Colligative Properties
  5. Chemical Equillibrium
    • Chemical Equillibrium
    • Reaction Constants
    • Net Chemical Change
  6. Electrochemistry
    • Oxidation States
    • Redox Reactions
    • Galvanic Cells and Faraday's Law
    • Standard Cell Potentials
    • The Nernst Equation
    • Electrolytic Cells

Notes

Currently, I haven't posted my notes yet. Sorry! I'll try to get some kind of online storage working so that I can post my typed LaTeX notes. Soon(tm).