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Wiki pages by topic
Algebra
Expressions and Variables
Solving Equations
- Setting Up Equations
- Simple Equations
- Solving Equations
- Verifying Solutions
- Multi-step Equations
- Isolating a Variable
- Balance Puzzles
- System of Linear Equations
- How are exponent towers evaluated?
- Does a square root have two values?
- Do Square Roots Always Multiply?
- Is 0.999... = 1?
- What is 0 to the power of 0?
- If AB=AC does B=C?
- Is (a/b)/c = a/(b/c)?
- Does sqrt(x²+y²) = x+y?
- Does cross multiply always work for inequalities?
- How does addition in the denominator work?
Linear Equations
Systems of Linear Equations
Absolute Value
Linear Inequalities
Non-Linear Inequalities
- Polynomial Inequalities
- Classical Inequalities
- Absolute Value Inequalities (1 Quadratic Term)
- Exponential Inequalities
- Logarithmic Inequalities
Quadratic Equations
- Factoring Quadratics
- Quadratic Equations
- Completing The Square
- Applications of Completing the Square
- Quadratic Discriminant
Square Roots (Radicals)
Arithmetic and Geometric Progressions
- Arithmetic Progressions
- Arithmetic Mean
- Harmonic Progression
- Harmonic Mean
- Geometric Progressions
- Geometric Mean
- Arithmetic and Geometric Progressions Problem Solving
- Arithmetic-Geometric Progression
Functions
- Functions
- Evaluating Functions
- Function Composition
- Inverse Functions
- Function Terminology
- Relations
- Graphs of Functions
- Transforming Graphs of Functions
- Functional Equations
Polynomials
- Polynomials
- Multiplying Polynomials
- Polynomial Division
- Synthetic Division
- Solving Identity Equations
- Difference Of Squares
- Applying the Perfect Square Identity
- Applying the Perfect Cube Identity
- Factoring Polynomials
- Factoring by Substitution
- Rational Expressions
- Simplifying Rational Expressions
- Factoring Compound Quadratics
- Factoring Cubic Polynomials
- Descartes' Rule of Signs
- Fundamental Theorem of Algebra
- Method of Undetermined Coefficients
- Remainder Factor Theorem
- Transforming Roots of Polynomials
Rational Functions
- Intercepts of Rational Functions
- Rational Equations
- Asymptotes
- Graphing Rational Equations
- Partial Fractions (Linear Factors)
- Partial Fractions (Cover Up Rule)
- Partial Fractions (Irreducible Quadratics)
- Partial Fractions (Repeated Factors)
- Telescoping Series (Sum)
Piecewise Functions
- Floor Function
- Ceiling Function
- Trailing Number of Zeros
- Fractional Part Function
- Hermite's Identity
Exponents
Exponential Functions
- Rules of Exponents (Algebraic)
- Solving Exponential Equations
- Graphs of Exponential Functions
- Exponential Functions (Problem Solving)
- Interest Rate
- Leonhard Euler
Logarithms
Polar Coordinates
Parametric Equations
Complex Numbers
- Complex Numbers
- Complex Conjugates
- Complex Numbers (Absolute Values)
- Complex Plane
- Complex Numbers in Geometry
- Gaussian Integers
- Discrete Fourier Transform
- Euler's Formula
- De Moivre's Theorem
- Roots of Unity
Advanced Linear Algebra (Matrices)
- Linear Algebra
- Matrices
- Determinants
- Eigenvalues and Eigenvectors
- Kernel (Nullspace)
- Vector Space
- Rank
- Cayley-Hamilton Theorem
- Row And Column Spaces
- Spectral Theorem
- Fundamental Subspaces
- Change of Basis
- Basis
- Rank-Nullity Theorem
- Linear Transformations
- Linear Independence
- Jordan Canonical Form
- Affine Transformations
- Vieta Root Jumping
Advanced Polynomials
- Algebraic Manipulation
- Algebraic Manipulation Identities
- Sum of n/n²/n³
- Telescoping Series (Product)
- Nested Functions
- Cardano's Method
- Cubic Discriminant
- Gauss: The Prince of Mathematics
- Srinivasa Ramanujan
- Binomial Coefficient
- Pascal's Triangle
- Binomial Theorem
- Negative Binomial Theorem
- Fractional Binomial Theorem
- Rational Root Theorem
- Vieta's Formula
- Newton's Identities
- Completing the Square (Multiple Variables)
- Algebraic Identities
- Factorization of Polynomials
- Sophie Germain Identity
- Algebraic Number Theory
- Fermat's Last Theorem
- Eisenstein's Irreducibility Criterion
- Group Theory
- Lagrange's Theorem
- Schwartz-Zippel Lemma
- Polynomial Interpolation by Remainder Factor Theorem
- Lagrange Interpolation
- Method of Differences
- Primitive Roots of Unity
- Symmetric Polynomials
- The uvw Method
- Chebyshev Polynomials (Definition and Properties)
Advanced Inequalities
- Cauchy-Schwarz Inequality
- Titu's Lemma
- Rearrangement Inequality
- Chebyshev's Inequality
- Jensen's Inequality
- Holder's Inequality
- Young's Inequality
- Muirhead Inequality
- Reverse Rearrangement Inequality
- AM-GM Inequality
- Applying the AM-GM Inequality
- Power Mean Inequality (QAGH)
- Quadratic Mean
- Triangle Inequality
- RMO Math Contest Preparation
- INMO Math Contest Preparation
Basic Mathematics
Pattern Recognition
- Identifying Patterns
- Identifying Pattern Relationships
- What Comes Next?
- Finding Terms in a Sequence
- Recognizing Visual Patterns
Arithmetic
- Estimation
- Irrelevant Information
- Trial and Error
- Checking Cases
- Counter-Examples
- SAT Logic
- SAT Reasoning Perfect Score
- Mental Math Tricks
- Subtracting Integers
- Division
- Remainder
- Exponents
- Rational Exponents
- Order of Operations
- Representation on the Real Line
- Natural Numbers
- Fractions
- Complex Fractions
- Componendo and Dividendo
- Continued Fractions
- Decimals
- Converting Decimals and Fractions
- Converting Fractions into Decimals
- Place Value
- Scientific Notation
- Percentages
- Converting Decimals and Percentages
- Converting Percentages and Fractions
- Ratio and Proportion
- Direct Variation
- Inverse Variation
- Speed Distance and Time
Geometry
- Angles
- Parallel Lines
- Perimeter
- Area of a Triangle
- Area of a Rectangle
- Circles
- Composite Figures
- Length and Area Problem Solving
Calculus
Sequences and Limits
- Sequences
- Series
- Arithmetic Progressions
- Geometric Progressions
- Arithmetic-Geometric Progression
- Telescoping Series (Sum)
- Telescoping Series (Product)
- Convergence Tests
- Harmonic Number
- Absolutely Convergent
- Sums Of Divergent Series
- Limits of Sequences
- Infimum/Supremum
- Nested Functions
- Dedekind Cuts
- Limits of Functions
- Limits by Substitution
- Limits by Factoring
- Limits by Rationalization
- When Does A Limit Exist?
- Asymptotes
- Continuous Functions
- Epsilon-Delta Definition of a Limit
- Squeeze Theorem
- Extreme Value Theorem
- Intermediate Value Theorem
- Infinity
- Is infinity at the end of the real number line?
- If F(x) is the antiderivative...
- Do local extrema occur if and only if f'(x) = 0?
- Is Infinity / Infinity = 1?
- Is infinity times zero = zero?
- What is 1 divided by 0?
- If the limit of a sequence is 0 does the series converge?
Differentiation
- Average and Instantaneous Rate of Change
- Tangent Line to a Curve
- Derivative by First Principle
- Derivatives of Polynomials
- Derivatives of Rational Functions
- Derivatives of Exponential Functions
- Derivatives of Logarithmic Functions
- Partial Derivatives
- Applying Differentiation Rules To Logarithmic Functions
- Power Rule
- Product Rule
- Quotient Rule
- Chain Rule
- Differentiation of Inverse Functions
- Applying Differentiation Rules to Trigonometric Functions
- Calculus With Inverse Trigonometric Functions
- Differentiation Rules
- Higher-order Derivatives
- Increasing / Decreasing Functions
- Inflection Points
- Implicit Differentiation
- Differentiable Function
- Mean Value Theorem
- Rolle's Theorem
Applications of Differentiation
- Indeterminate Forms
- L'Hopital's Rule
- Related Rates of Change
- Extrema (Local and Absolute)
- Critical Points
- Second Derivative Test
- Optimization
- Lagrange Multipliers
- Vertical Asymptotes
- Average Velocity
- Instantaneous Velocity
- Taylor Series
- Maclaurin Series
- Taylor Series Approximation
- Taylor Series Manipulation
- Interval and Radius of Convergence
- Taylor Series (Error Bounds)
- Power Series
- Small-Angle Approximation
- Fourier Series
- Taylor's Theorem (with Lagrange Remainder)
- Analytic Continuation
Integration
- Line Integral
- Integration
- Integration of Algebraic Functions
- Integration of Exponential Functions
- Integration of Trigonometric Functions
- Integration of Rational Functions
- Integration of Logarithmic Functions
- Integration of Radical Functions
- Definite Integrals
- Riemann Sums
- Fundamental Theorem of Calculus
- Improper Integrals
- Multiple Integral
- u-Substitution
- Trigonometric Substitution in Integration
- Integration by Parts
- Differentiation Under the Integral Sign
- Integration Tricks
- Lebesgue Integration
- Stokes' Theorem
- Green's Theorem
- Area between curves
- Gamma Function
- Beta Function
- Digamma Function
- Riemann Zeta Function
- Cauchy Integral Formula
- Isolated Singularities and Residue Theorem
Applications of Integration
Parametric Equations Calculus
- Parametric Equations
- Polar Coordinates
- Converting Polar Coordinates to Cartesian
- Polar Curves
- Parametric Derivative
- Parametric Equations (Velocity and Acceleration)
- Polar Equations (Area)
Differential Equations
- Differential Equations
- Separable Differential Equations
- Homogeneous Linear Differential Equations
- Euler's Method
- Euler's Method (Small Step Size)
- Logistic Differential Equations
- Bernoulli Equation
- Systems of Linear Differential Equations
- Chaos Theory
Numerical Methods
- Root Approximation (Bisection)
- Newton Raphson Method
- Integral Approximation (Trapezium Rule)
- Integral Approximation (Simpson's Rule)
- Chebyshev's Formula
Chemistry
Fundamentals
- Chemistry in action
- Atoms/Molecules/Elements/Compounds
- Mixtures
- Dalton's Atomic Model
- Bohr's Model
- Periodic Table
- Atomic/Molecular Weights
- Mole Concept
- Concentration Terminology
- Nuclear Decay
- Balancing Chemical Reactions
- Limiting Reagents
- Laws of Chemical Combination
- Try It At Home Chemistry Experiments
- Chemistry Of Drugs
Chemical Bonding
- Ionic Compounds
- Octet Rule
- Lewis Structure
- Hybridisation
- Valence Shell Electron Pair Repulsion
- Sigma and Pi Bonds
- Molecular Orbital Theory
- Coordinate Bonds
- Polarity
- Van der Waals Force
- Dipole Interactions
- Hydrogen Bonds
- Surface Tension
- Strength of Intermolecular Forces
- Chemistry of Diamonds
Reaction Mechanics
- Chemical Equilibrium
- Basics of Statistical Mechanics
- Le Chatelier's Principle
- Acids and Bases
- Acid/Base Equilibrium
- Thermal Expansion
- Rate of Chemical Reactions
- Order of Chemical Reactions
- Activation Energy
- Chemistry of Fireworks
- Thermometry
- Electrolytic Cells
- Galvanic Cells
- Buffers
Organic Chemistry
- Structural Representations
- IUPAC Nomenclature
- Structural Isomerism
- Geometric Isomerism
- Optical Isomerism
- Electrophiles and Nucleophiles
- Inductive/Electromeric/Resonance Effects
- Reactive Intermediates
- Common Types of Organic Reactions
- Grignard Reagent
- Alkanes/Alkenes/Alkynes
- Do objects gain protons to become positive?
- Chemistry Of Poisons
- Chemistry Of Nutrition
Biology
- DNA
- Reverse Transcription
- Mutation and DNA repair
- CRISPR
- DNA Bricks
- Hemoglobin
- Viral Replication
- Zika Virus
Computer Science
Types and Data Structures
- Abstract Data Types
- Lists
- Array (ADT)
- Queues
- Double Ended Queues
- Stacks
- Associative Arrays
- Sets (ADT)
- Priority Queues
- Array (Data Structure)
- Disjoint-set Data Structure
- Dynamic Array
- Linked List
- Unrolled Linked List
- Skip List
- Hash Tables
- Bloom Filter
- Cuckoo Filter
- Merkle Tree
- Trees
- Recursive Backtracking
- Traversals
- Tries
- Treaps
- Fenwick Tree
- Binary Search Trees
- Red-Black Tree
- AVL Tree
- Scapegoat Tree
- Splay Tree
- Heaps
- Binary Heap
- Binomial Heap
- Fibonacci Heap
- Pairing Heap
- Graphs
- Graph implementation
- Adjacency Matrix
- Spanning Trees
- Social Networks
- Kruskal's Algorithm
- Strings
- Regular Expressions
Algorithms
- Algorithms
- Divide and Conquer
- Greedy Algorithms
- Randomized Algorithms
- Complexity Theory
- Big O Notation
- Master Theorem
- Amortized Analysis
- Complexity Classes
- P versus NP
- Dynamic Programming
- Backpack Problem
- Egg Dropping
- Fast Fibonacci Transform
- Karatsuba Algorithm
- Sorting Algorithms
- Insertion Sort
- Bubble Sort
- Merge Sort
- Quick Sort
- Heap Sort
- Radix Sort
- Counting Sort
- Median-finding Algorithm
- Binary Search
- Depth-First Search
- Breadth-First Search
- A* Search
- Shortest Path Algorithms
- Dijkstra's
- Bellman-Ford
- Floyd-Warshall
- Johnson's Algorithm
- Matching (Graph Theory)
- Matching Algorithms
- Flow Network
- Max-flow Min-cut
- Ford-Fulkerson
- Edmonds-Karp
- Shunting Yard Algorithm
- Rabin-Karp
- Knuth-Morris-Pratt
- Basic Shapes/Polygons/Trigonometry
- Convex Hull
- Grids
- Finite State Machines
- Turing Machines
- Halting Problem
- Kolmogorov Complexity
- Traveling Salesperson Problem
- Pushdown Automata
- Regular Languages
- Context Free Grammars
- Context Free Languages
- Signals and Systems
- Linear Time Invariant Systems
- Predicting System Behavior
Programming Languages
- Subroutines
- List comprehension
- While
- For
- Primality Testing
- Pattern matching
- Logic Gates
- Control Flow Statements
- Object-Oriented Programming
- Objects
- Classes
- Methods
- Map
- Foldlist
- Filter
Cryptography and Simulations
- Caesar Cipher
- Vigenere Cipher
- RSA Encryption
- Enigma Machine
- Diffie-Hellman
- Knapsack Cryptosystem
- Secure Hash Algorithms
- Entropy (Information Theory)
- Huffman Code
- Error correcting codes
- Symmetric Ciphers
- Inverse Transform Sampling
- Monte-Carlo Simulation
- Annealing
- Genetic Algorithms
- Programming Blackjack
Machine Learning
- Machine Learning
- Supervised Learning
- Unsupervised Learning
- Feature Vector
- Naive Bayes Classifier
- K-nearest Neighbors
- Perceptron
- Support Vector Machines
- Principal Component Analysis
- Ridge Regression
- k-Means Clustering
- Markov Chains
- Hidden Markov Models
- Gaussian Mixture Model
- Collaborative Filtering
- Artificial Neural Network
- Feedforward Neural Networks
- Backpropagation
- Recurrent Neural Network
Discrete Mathematics
Counting
- Rule of Sum
- Counting Integers in a Range
- Rule of Product
- Rule of Sum and Product Problem Solving
- Counting Shapes
- Discrete Mathematics
- Set Complement
- Permutations
- Permutations with Repetition
- Permutations With Restriction
- Combinatorics
- Combinations
- Rectangular Grid Walk
Sets and Partitions
- Set Theory
- Set
- Sets (Elements)
- Sets (Subsets)
- Classical Sets
- Cardinality
- Partially Ordered Sets
- ZFC
- Ordinal Numbers
- Venn Diagram
- Union and Intersection
- Principle of Inclusion and Exclusion
- Double Counting
- Derangements
- Bijection Injection and Surjection
- Bijective Functions
- Burnside's Lemma
- Pigeonhole Principle
- Ramsey Theory
- Dilworth's Theorem
- Distinct Objects into Distinct Bins
- Distinct Objects into Identical Bins
- Identical Objects into Distinct Bins
- Identical Objects into Identical Bins
- Partition of an Integer
Data
- Mean
- Interquartile Range
- Understanding Data (Problem Solving)
- Data Presentation (Tables)
- Data Presentation (Bar Charts)
- Data Presentation (Pie Charts)
- Data Presentation (Dot Diagram)
- Data Presentation (Histogram)
- Data Presentation (Stem-leaf plot)
Probability
- Probability
- Uniform Probability
- Probability (Rule of Sum)
- Probability (Rule of Product)
- Probability (By Complement)
- Probability (Problem Solving)
- Math of Poker (Basics)
- Conditional Probability
- Probability (Independent events)
- Probabilistic PIE
- Bayes' Theorem
- Conditional Probability Distribution
- Monty Hall Problem
- Bayesian Theory
- Poker Strategy
- Expected Value
- Linearity of Expectation
- Law of Iterated Expectation
- Coupon Collector Problem
- Variance
- Standard Deviation
- Covariance
- Geometric Probability
Random Variables
- Discrete Random Variables - Definition
- Discrete Random Variables - Probability Density Function (PDF)
- Discrete Random Variables - Cumulative Distribution Function
- Discrete Random Variables - Joint Probability Distribution
- Discrete Random Variables - Indicator Variables
- Continuous Random Variables - Definition
- Continuous Random Variables - Probability Density Function (PDF)
- Continuous Random Variables - Cumulative Distribution Function
- Continuous Random Variables - Joint Probability Distribution
- Bernoulli Distribution
- Binomial Distribution
- Geometric Distribution
- Poisson Distribution
- Hypergeometric Distribution
- Uniform Distribution (Continuous)
- Normal Distribution
- Exponential Distribution
- Log-normal Distribution
- Multivariate Normal Distribution
- Statistics
- Sampling
- Chi-Squared Test
- Maximum Likelihood Estimation
- Inverse Transform Sampling
- Statistical Significance
- Markov Chains
- Stationary Distributions
- Transience and Recurrence
- Ergodic Markov Chains
- Absorbing Markov Chains
Binomial Theorem
- Binomial Coefficient
- Pascal's Triangle
- Binomial Theorem
- Negative Binomial Theorem
- Fractional Binomial Theorem
- Multinomial Coefficients
- Multinomial Theorem
Recurrence Relations
- Recursion
- Recurrence Relations
- Fractals
- Minkowski Dimension
- Fibonacci Sequence
- Linear Recurrence Relations
Advanced Topics
- Combinatorial Games (Definition)
- Combinatorial Games (Winning Positions)
- Sprague Grundy Theorem
- Nim
- Dots and Boxes
- Fair Division
- Axiom of Determinacy
- Casework
- Conjectures
- Construction
- Extremal Principle
- Invariant Principle
- Vandermonde's Identity
- Hockey Stick Identity
- Graph Theory
- Eulerian Path
- Hamiltonian Path
- Four Color Theorem
- Graph Coloring and Chromatic Numbers
- Hall's Marriage Theorem
- Applications of Hall's Marriage Theorem
- Art Gallery Problem
- Combinatorial Optimization
- Group actions
- Generating Functions
- Manipulating Generating Functions
- Coloring (Parity Arguments)
- Knots
Electricity and Magnetism
Electrostatics
- Charge and Electric Fields
- Standard Model
- Spin
- Superposition of Electric Fields
- Electrostatics
- Electric Potential
- Electric Dipoles
- Energy of an electric field
- Electric Flux
- Gauss's law
- Conductors
- Poynting Vector
Circuits
- Simple Circuits
- Heat Dissipated by Resistors
- Transformation of Resistances
- Ohm's Law
- Capacitors
- Energy Stored in a Capacitor
- Using Gauss' law to find E-field and capacitance
- Transformers
- Series and Parallel Capacitors
- RC Circuits
- RLC Circuits
- Impedance
- Phasor Diagrams
- Resonant circuits
Magnetism
- Magnetic Field Lines
- Lorentz Force Law
- Biot-Savart Law
- Energy of a magnetic field
- Magnetic vector potential
- Ampere's Law
- Magnetic Flux and Faraday's Law
- Lenz's Law
- Magnetic Moment
- Magnetic Effects of Current
- Is a perfect transparent material black?
- Semiconductors
Electrodynamics
Geometry
Geometric Measurement
- Angles
- Parallel Lines
- Area of a Triangle
- Area of a Rectangle
- Circles
- Length and Area Problem Solving
- Composite Figures
- Volume of a Cuboid
- Volume of a Cylinder
- Volume of a Sphere
- Volume of a Pyramid
- Volume Problem Solving
- Calculating the Size of Earth
- Sphere Packing
- Surface Area
- Surface Area of a Cuboid
- Surface Area of a Sphere
- Surface Area of a Cylinder
Triangles
- Triangles
- Pythagorean Theorem
- Triangle Inequality
- Symmedian
- Classification of Triangles
- Properties of Equilateral Triangles
- Properties of Isosceles Triangles
- Congruent and Similar Triangles
- Heron's Formula
- Pascal's Theorem
- Centroid of a Triangle
- Circumcenter
- Incenter
- Orthocenter
- Euler Line
- Ceva's Theorem
- Apollonius's Theorem
Polygons
- Quadrilateral Classification
- Properties of Squares
- Properties of Rectangles
- Properties of Parallelograms
- Properties of Trapezoids
- Similar Figures
- Regular Polygons
- Perimeter
- Basic Composite Figures
- Pick's Theorem
- Irregular Polygons
- Area of a Polygon
Circles
- Pi
- Tangent (Perpendicular to Radius)
- Alternate Segment Theorem
- Two Secants
- Tangent-Secant
- Incircle of Triangle
- Incircles and Excircles
- Cyclic Quadrilaterals
- Steiner Ellipse
Coordinate Geometry
- Linear Equations
- Equations of Parallel and Perpendicular Lines
- Midpoint of a Line Segment
- Distance Formula
- Distance between Point and Line
- Section Formula
- Equation of Locus
- Determining Coordinates
- Coordinate Geometry (Angle Bisector)
- Conic Sections
- Equation of a Circle
- Parabola
- Ellipse
- Hyperbola
- Discriminant of a Conic Section
- 3D Coordinate Geometry (Distance)
- 3D Coordinate Geometry (Equation of a Line)
- 3D Coordinate Geometry (Skew Lines)
- 3D Coordinate Geometry (Equation of a Plane)
- 3D Coordinate Geometry (Parallel Planes)
- 3D Coordinate Geometry (Perpendicular Planes)
- 3D Coordinate Geometry (Intersection of Planes)
Trigonometry
- Trigonometry
- Converting Degrees and Radians
- Basic Trigonometric Functions
- Reciprocal Trigonometric Functions
- Inverse Trigonometric Functions
- Bearing (Word Problems)
- Hyperbolic Trigonometric Functions
- Local Linearization
- Sine and Cosine Graphs
- Tangent and Cotangent Graphs
- Cosecant and Secant Graphs
- Inverse Trigonometric Graphs
- Symmetry in Trigonometric Graphs
- Trigonometric Graphs (Amplitude and Periodicity)
- Graphical Transformation of Trigonometric Functions
- Graphs of Trigonometric Functions (Problem Solving)
- Trigonometric Equations
- Trigonometric Equations (Double Angle Formula)
- Lengths in Right Triangles
- Sine Rule (Law of Sines)
- Cosine Rule (Law of Cosines)
- Solving Triangles
Trigonometric Identities
- Pythagorean Identities
- Ratio of Trigonometric Functions
- Trigonometric Even-odd Functions
- Trigonometric Periodicity Identities
- Trigonometric Co-function Identities
- Trigonometric R method
- Fundamental Trigonometric Identities
- Double Angle Identities
- Sum and Difference Formulas
- Triple Angle Identities
- Expansions of sin(nx) and cos(nx)
- Half Angle Tangent Substitution
- Trigonometric Power Reduction Identities
- Product-to-Sum Trigonometric Formulas
- Sum to Product Trigonometric Identities
- Proving Trigonometric Identities
- Chebyshev Polynomials (Definition and Properties)
Vectors
3D Geometry
Euclidean Geometry
- Isosceles Triangle Theorem
- Angle Bisector Theorem
- Menelaus' Theorem
- Stewart's Theorem
- Orthic Triangle
- Nine-point Circle
- Routh's Theorem
- Pedal Triangle
- Projective Geometry
- Brahmagupta's Formula
- Power of a Point
- Radical Axis of 2 Circles
- Ptolemy's Theorem
- Pitot's Theorem
- Descartes' Circle Theorem
Geometric Transformations
- Symmetry
- Reflection
- Affine Transformations
- Does 1 square meter equal 100 square centimeters?
- Can a triangle have an angle of 0 degrees?
- Mobius Strips
- Klein Bottle
- Spherical Geometry
- Hyperbolic Geometry
Topology
- Topology
- Metric Space
- Homeomorphism
- Dense Set
- Cantor Set
- Stone-Weierstrass Theorem
- Brouwer Fixed Point Theorem
- Ham Sandwich Theorem
- Hilbert Space
- Banach Tarski Paradox
- Axiom Of Choice
- Converse Of Intermediate Value Theorem
- Connected Space
- Open Sets
- Closed Sets
- Compact Space
- Subsequences
Mechanics
Dynamics
- Newton's Laws of Motion
- Identifying Action-Reaction Forces
- Lagrangian Mechanics
- Fluid Mechanics
- Continuity Equation
- Bernoulli's Principle
- Pressure
- Density
- Free Body Diagrams
- Solving for Equilibrium
- Friction
- 1D Dynamics Problem Solving
- Motion Along Inclined Planes
- Centripetal acceleration
- Calculating impulse
- Conservation of Momentum
- Analyzing inelastic collisions
- Is force a necessary condition to persist a motion?
- Do heavier objects fall faster than lighter objects?
- If an object is moving then a net force must be acting on it
- Since every action has an equal and opposite reaction nothing will be moved
- Is acceleration the rate of change of speed?
- Is the acceleration of an object at rest zero?
- Does a rolling ball follow Newton's first law of motion?
- Is there gravity in the Space Station?
- Is heat the same as temperature?
- Are Photons Massless?
- Does an object accelerate under uniform circular motion?
Kinematics
- Introduction to Kinematics
- Motion Under Constant Velocity
- Motion under Constant Acceleration
- Equations Of Motion
- 1D Kinematics Problem Solving
- Hyperloop
- Different Representations of Vectors
- Adding velocity vectors
- Physics of Traffic Jams
- Projectile Motion
- Position-Time Graph
- Uniform Circular Motion
Conservation Laws
- Conservation of Energy
- Kinetic energy
- Determining Kinetic Energy Lost in Inelastic Collisions
- Work
- Rocket Physics
- Conservative vs. Non-conservative Forces
- Gravitational Potential Energy
- Analyzing Elastic Collisions
- Conservative Forces
Units and Measurements
- SI Multiplier Prefixes
- SI Divisor Prefixes
- SI Length
- SI Fluid Measure
- Converting Units
- Dimensional Analysis
- Fermi Estimate
Rotational Motion
- Converting Degrees and Radians
- Angular Kinematics
- Calculating Torque as a Cross Product
- Torque (Equilibrium)
- Torque (Dynamical Behavior)
- Rotational Form of Newton's Second Law
- Moment of Inertia
- Rotational Kinetic Energy (Translational)
- Rotational Kinetic Energy (Work-Kinetic Theorem)
- Rotational Kinetic Energy (Conservation of Energy)
- Rotational Kinetic Energy (Problem Solving)
Oscillation and Waves
- Periodic Motion
- Hooke's Law
- Understanding Basic Traveling Waves
- Waves
- Amplitude/Frequency/Wave Number/Phase Shift
- Doppler Effect
- Snell's Law
- Power of a Wave
- Applying Boundary Conditions to Standing Waves
- Wave Equation
- Sound
- Deriving Exponential Decay from Damping Forces
- Using Exponential Decay to Explain Amplitude Decreases
- Damped Harmonic Oscillators
- Damped Oscillators (Problem Solving)
- Geometrical Optics
- Lenses
- Mirrors
- Dispersion and Scattering
Gravity and Space
- Black Holes
- Dark Matter
- Isaac Newton
- Deriving Kepler's Laws
- Applying Kepler's Laws
- Characteristics of Circular Orbits
Modern Physics
- Entropy
- Intensive and Extensive Properties
- Microstates and macrostates
- Configurational entropy (qualitative)
- Configurational entropy (quantitative)
- Zeroth law of thermodynamics
- First Law of Thermodynamics
- Second Law of Thermodynamics
- Equilibrium
- Ideal Gas Law
- Specific Heat
- Phase Changes
- Quantum Mechanics
- Wave-Particle Duality
- Schrodinger Equation
- Wavefunctions and Measurement
- Heisenberg Uncertainty Principle
- Quantum Harmonic Oscillator
- Quantum Tunneling
- Double-slit Experiment
- Schrodinger's Cat
- Hydrogen Atom
- Blackbody Radiation
- Quantum Entanglement
- Bell's Theorem
- Quantum Teleportation
- Quantum Computing
- Spacetime
- Special Relativity
- General Relativity
- Gravitational Waves
- Cosmology
- Albert Einstein
- Quantum Mechanical Model
- Energy Level and Transition of Electrons
- Orbitals and Quantum Numbers
- Electron Configurations
- Spherical Harmonics
- Green's Functions
- String Theory
Number Theory
Integers
- Even and Odd Numbers
- Divisibility Rules
- Remainder
- Prime Numbers
- Infinitely Many Primes
- Distribution of Primes
- Mersenne Prime
- Prime Factorization
- Perfect Squares/Cubes/Powers
- Factors
- Perfect Numbers
- Fundamental Theorem of Arithmetic
- Greatest Common Divisor
- Lowest Common Multiple
- Division Algorithm
- Euclidean Algorithm
- Bezout's Identity
- Extended Euclidean Algorithm
- Decimal Expansion
- Binary Numbers
- Hexadecimal Numbers
- Number Base
- Fractional and Non-Integer Number Bases
- Negative Integer Number Base
- Factorials
- Stirling's Formula
- Double Factorials and Multifactorials
- Trailing Number of Zeros
- Rightmost Non-zero Digit of n!
- Wilson's Theorem
- Fibonacci Sequence
- Tribonacci Sequence
- Catalan Numbers
- Is 0 even odd or neither?
- Is 0 a prime number?
- Is 1 prime?
- Is 2 prime?
- Modular Arithmetic Misconceptions
- Converting Decimals and Fractions
- Converting Fractions into Decimals
- Converting Repeating Decimals into Fractions
- Rational Numbers
- Irrational Numbers
- History of Irrational Numbers
- Real Numbers
- Transcendental Numbers
Modular Arithmetic
- Modular Arithmetic
- Parity of Integers
- Number Theory
- Chinese Remainder Theorem
- Fermat's Little Theorem
- Lucas' Theorem
- Sum of Squares Theorems
- Euler's Totient Function
- Euler's Theorem
- Finding the Last Digit of a Power
- Order of an Element
- Primitive Roots
- Lifting The Exponent
- Carmichael's Lambda Function
- Arithmetic Function
- Multiplicative Function
- Mobius Function
- Dirichlet Convolution
- Dirichlet Series
- Riemann Zeta Function
- Prime Zeta Function
- Cyclotomic Polynomials
- Quadratic Residues
- Legendre Symbol
- Jacobi Symbol
- Law of Quadratic Reciprocity
- Euler's Criterion
- Public-Key Cryptography
- Diffie-Hellman
- RSA Encryption
Diophantine Equations
- Linear Diophantine Equations
- System of Linear Diophantine Equations
- Postage Stamp Problem / Chicken McNugget Theorem
- Integer Equations (Stars and Bars)
- Finding The Number of Digits
- Diophantine Equations (Solve by Factoring)
- Diophantine Equations (Modular Arithmetic Considerations)
- Fermat's Method of Infinite Descent
- Pell's Equation
- Hensel's Lemma
- Vieta Root Jumping
- General Diophantine Equations (Problem Solving)
Quantitative Finance
Fixed Income
- Types of Fixed Income Securities
- Time value of money
- Bond Valuation
- Bond Yield
- Macaulay Duration
- Modified Duration
- Fixed Income Glossary
- Bond issuers (Treasury)
- Bond issuers (Municipals)
- Interest Rate
Derivatives
- Forward contract
- Futures contract
- Pricing Forward and Futures
- Call and Put Options
- American and European Options
- Option value
- Put Call Parity
- Efficient Market Hypothesis
- Option Greeks (Delta)
- Option Greeks (Gamma)
- Option Greeks (Vega)
- Straddle/Strangle
- Straddle Approximation Formula
- Call and Put Spreads
- Butterfly/Condor
Mathematics Prerequisites
- Rule of Sum and Product Problem Solving
- Permutations
- Combinations
- Principle of Inclusion and Exclusion
- Understanding Data (Problem Solving)
- Data Presentation (Tables)
- Data Presentation (Bar Charts)
- Data Presentation (Pie Charts)
- Data Presentation (Dot Diagram)
- Data Presentation (Histogram)
- Data Presentation (Stem-leaf plot)
- Probability (Problem Solving)
- Expected Value
- Central Limit Theorem
- Markov Chains
- Limits of Sequences
- Limits of Functions
- Differentiation Rules
- Implicit Differentiation
- Integration
- u-Substitution
- Integration by Parts
- Eigenvalues and Eigenvectors
- Matrix Exponentiation
- Subspace
- Linear Regression
- Hypothesis Testing
- Confidence Intervals
- Root Approximation (Bisection)
- Newton Raphson Method
- Fermi Estimate
Computer Science Concepts
- For
- Subroutines
- Pattern matching
- Big O Notation
- Stacks
- Graphs
- Backpack Problem
- Recursive Backtracking
- Monte-Carlo Simulation
Logical Reasoning