UG Mathematics Entrance syllabus.

Core proof methods

• Direct Proof and the Structure of an Argument
• Proof by Contradiction
• Proof by Contrapositive
• The Role of Counterexamples

Writing mathematics

• Writing a Complete Olympiad Solution
• Quantifiers and Logical Flow
• Handling Degenerate and Boundary Cases

Choosing a method

• Contradiction versus Contrapositive
• Proving Equivalence in Two Directions
• Constructive and Non-Constructive Existence

Advanced presentation

• Structuring Multi-Part Solutions
• Partial Results and Their Value

Rigour at olympiad depth

• Minimal Counterexample Arguments
• Detecting Hidden Assumptions
• Writing a Solution a Grader Cannot Fault

Induction

• The Principle of Mathematical Induction
• Strong Induction
• Induction in Divisibility and Inequalities

Recursion

• Recursive Definitions
• Computing Terms of a Recursive Sequence

Structural induction

• Double Induction
• Structural Induction on Configurations
• Induction on the Number of Variables

Well-ordering and descent

• The Well-Ordering Principle
• Infinite Descent
• Descent on a Well-Chosen Measure

Olympiad depth

• Descent in Diophantine Problems
• Termination Arguments in Games and Processes
• Constructing the Right Descent Measure

Core identities

• Difference of Squares and Cubes
• Sum and Difference of Powers
• The Identity x^3 + y^3 + z^3 - 3xyz

Manipulation

• Completing the Square
• Substitution and Normalisation
• Telescoping Expressions

Special factorisations

• The Sophie Germain Identity
• Cyclic and Symmetric Factorisations
• Factoring by Grouping and Adding Terms

Technique

• Normalising a Constraint
• Homogeneous Expressions
• Substitutions that Simplify an Expression

Olympiad depth

• Identity Construction in Hard Problems
• Factorisation as a Descent Tool
• Recognising Hidden Factorisation

First moves

• Understanding the Problem and Restating It
• Small Cases and Pattern Finding
• Working Backwards

Exploration

• Drawing the Right Diagram
• Introducing Notation and Auxiliary Elements

Standard strategies

• The Extremal Principle
• Invariants and Monovariants
• Symmetry Reduction
• Generalisation and Specialisation

Organisation

• Problem Decomposition
• Choosing Between Two Routes

Competition execution

• Time Allocation in a Four and a Half Hour Paper
• Post-Mortem Analysis of Failed Attempts
• Building a Personal Error Log

Polynomial Structure and Division

• Division Algorithm
• Remainder Theorem and Factor Theorem
• Identity Theorem for Polynomials
• Root Structure

Roots, Coefficients and Arithmetic Constraints

• Vieta's Relations
• Rational and Integer Roots
• Polynomials with Integer Values
• Root Location
• Symmetric and Power Sum Methods

Discriminants, Resultants and Common Roots

• Discriminant of a Polynomial
• Resultant and Common Roots
• Reduced Polynomial Problems

Irreducibility and Special Families

• Eisenstein Criterion
• Modular Reduction
• Interpolation and Construction

Elementary symmetric functions

• Elementary Symmetric Functions
• Newton's Identities
• Symmetric Polynomials in Three Variables

Roots of unity

• Roots of Unity and Their Properties
• The Roots of Unity Filter

Structure theorems

• The Fundamental Theorem of Symmetric Polynomials
• Cyclotomic Polynomials
• Chebyshev Polynomials

Reductions

• The uvw Reduction
• The pqr Method

Olympiad depth

• Power Sums and Recurrence Relations
• Constructing Polynomials from Symmetric Data
• Symmetric Techniques in Hard Problems

Standard sequences

• Arithmetic and Geometric Progressions
• Means and Their Properties
• Telescoping Sums

Summation technique

• The Method of Differences
• Sigma Notation and Standard Finite Sums

Advanced summation

• Abel Summation by Parts
• Sums of Powers
• Recursive Sequences and Their Limits

Series

• Convergence of Elementary Series
• Continued-Fraction Convergents as Sequences

Olympiad depth

• Summation Identities in Hard Problems
• Estimating Sums and Asymptotic Bounds
• Generating-Function Proofs of Identities

Basic methods

• Substitution and Special Values
• Cauchy-Type Equations
• Functional Equations on Integers and Rationals

Standard families

• Equations Involving f(x + y)
• Equations Involving f(xy)

Structural arguments

• Injectivity and Surjectivity Arguments
• Monotonicity and Boundedness Hypotheses
• Polynomial Solutions of Functional Equations

Two-variable equations

• Equations Involving f(f(x))
• Functional Equations in Two Variables

Olympiad depth

• Hard Functional Equations from Shortlists
• The Regularity Trap and Unstated Assumptions
• Functional Inequalities

Algebra of complex numbers

• Complex Numbers and the Argand Plane
• Modulus, Argument and Polar Form
• De Moivre's Theorem

Roots

• Roots of Unity
• Complex Roots of Polynomial Equations

Geometry with complex numbers

• Lines and Circles in Complex Form
• Concyclic, Collinear and Perpendicular Criteria
• Rotations and Reflections as Multiplication

Classical theorems

• The Complex Form of Classical Theorems
• Sums of Roots of Unity

Olympiad depth

• Hard Geometry by Complex Numbers
• The Complex Proof of Ptolemy's Theorem
• The Triangle Inequality for Moduli

Determinants

• Determinants and Their Properties
• Expansion by Minors
• Determinants in Geometry

Matrices

• Matrix Operations and Inverses
• Solving Systems of Linear Equations

Structure

• Rank and Linear Independence
• Vandermonde and Circulant Determinants

Applications

• Matrices and Linear Recurrences
• Linear Algebra in Combinatorial Problems

Olympiad depth

• Rank Arguments in Combinatorics
• Elementary Use of Eigenvalues
• Determinant Identities and the Trace Trick

Groups in number theory

• Orders of Elements
• Fermat's Little Theorem as a Group Statement
• Cyclic Groups Modulo a Prime

Permutations

• Permutation Groups and Cycle Decomposition
• Parity of Permutations

Group actions

• Group Actions and Orbits
• Burnside's Lemma
• Counting Coloured Configurations

Elementary structure

• Cauchy's Theorem and Elements of Prime Order
• Subgroups and Cosets

Olympiad depth

• Counting Up to Symmetry in Hard Problems
• Group Actions on Sets of Configurations
• Combinatorial Existence by Group Theory

Divisibility

• Divisibility and Its Basic Properties
• Division with Remainder
• Divisibility Rules for Small Numbers
• Divisibility of Sums and Products

Greatest common divisor

• Greatest Common Divisor and Least Common Multiple
• The Euclidean Algorithm
• Bezout's Identity

Extended techniques

• The Extended Euclidean Algorithm
• Linear Combinations and Coprimality
• GCD of Several Numbers and of Polynomial Values

Divisibility structures

• The gcd of x^m - 1 and x^n - 1
• Divisibility of a^n - b^n
• Coprime Integer Pairs and Their Consequences

Olympiad depth

• Divisibility Arguments in Hard Problems
• The GCD of a Set of Binomial Coefficients
• Constructing Divisibility Counterexamples

Hard applications

• Estimating GCDs of Polynomial Values
• Divisibility Chains and Iterated GCDs
• Non-Standard Divisibility Problems

Primes and factorisation

• Prime and Composite Numbers
• The Fundamental Theorem of Arithmetic
• Prime Factorisation Techniques
• Counting Divisors from the Factorisation

Infinitude and distribution

• Euclid's Proof of the Infinitude of Primes
• Primes in Intervals and Bertrand's Postulate
• Primes of Special Forms

Prime structure

• Smooth Numbers and Their Uses
• The Largest Prime Factor of an Integer
• Prime Divisors of Sequences

Quoted results

• Zsigmondy's Theorem as a Quoted Result
• Chebyshev-Type Elementary Bounds
• Using a Deep Theorem Inside a Solution

Olympiad depth

• Primitive Prime Divisors in Hard Problems
• Simultaneous Prime Conditions
• Constructing Integers with Prescribed Prime Structure

Hard applications

• Estimating the Number of Primes in a Range
• Prime Divisors of Factorials and Binomials
• Irreducibility by Prime Arguments

Congruence basics

• Congruences and Their Properties
• Arithmetic with Congruences
• Linear Congruences
• Residue Systems

Core theorems

• Fermat's Little Theorem
• Euler's Theorem
• Wilson's Theorem

Systems and structure

• The Chinese Remainder Theorem
• Non-Coprime Moduli
• The Multiplicative Order of an Element
• Primitive Roots Modulo a Prime

Classical congruences

• Wolstenholme's Theorem
• Quadratic Congruences and Their Solutions
• Congruences Involving Factorials

Olympiad depth

• Order Arguments in Hard Problems
• Constructing Solutions to Congruence Systems
• Congruence Obstructions to Existence

Hard applications

• Exponential Congruences and Index Methods
• Orders Modulo Composite Numbers
• Covering Systems of Congruences

Multiplicative functions

• The Divisor Function tau(n)
• The Sum-of-Divisors Function sigma(n)
• Euler's Totient Function
• Multiplicative and Completely Multiplicative Functions

Elementary results

• The Number of Divisors of a Product
• Sums over Divisors
• Perfect Numbers

Inversion

• The Mobius Function
• Mobius Inversion
• The Dirichlet Convolution

Averages and bounds

• Summatory Estimates for tau and sigma
• Elementary Bounds for phi(n)
• The Average Order of a Divisor Function

Olympiad depth

• Mobius Inversion in Counting Problems
• Multiplicative Functions in Hard Problems
• The Euclid-Euler Characterisation of Perfect Numbers

Hard applications

• Constructing Integers with Prescribed Divisor Data
• Sums Involving phi and mu
• Estimating Divisor Sums Sharply

Linear and quadratic equations

• Linear Diophantine Equations
• The Frobenius Coin Problem
• Pythagorean Triples
• Sums of Two Squares

Method

• Modular Obstructions
• Infinite Descent in Equations
• Bounding and Size Arguments

Classical families

• The Pell Equation
• Vieta Jumping
• The Markov Equation
• Equations Solvable by Factorisation

Deep inputs

• The Sum of Two Squares Theorem
• Gaussian Integers in Diophantine Problems
• Fermat's Last Theorem for Small Exponents

Olympiad depth

• Descent on Quadratic Forms
• Simultaneous Diophantine Conditions
• Exponential Diophantine Equations

Hard applications

• Catalan's Conjecture as a Quoted Result
• Equations with No Small Solutions
• Constructing Infinite Solution Families

Finite continued fractions

• Finite Continued Fractions
• Convergents and Their Recurrence
• Converting a Rational to a Continued Fraction
• The Euclidean Algorithm as a Continued Fraction

Approximation

• Best Rational Approximations
• The Approximation Theorem
• Irrationality Proofs by Approximation

Infinite continued fractions

• Infinite Continued Fractions and Convergence
• Periodic Continued Fractions
• Quadratic Irrationals and Their Periods

Applications

• Solving Pell's Equation by Continued Fractions
• The Hurwitz Approximation Bound
• Approximating Algebraic Numbers

Olympiad depth

• Continued Fractions in Hard Number Theory
• Constructing Good Rational Approximations
• Divisibility Consequences of Approximation

Hard applications

• Simultaneous Approximation Problems
• Continued Fractions and Quadratic Forms
• Estimating the Error of a Convergent

Basic counting

• The Multiplication and Addition Principles
• Permutations and Combinations
• Counting with Repetition
• Arrangements with Restrictions

Structured counting

• Bijections and Double Counting
• Stars and Bars
• Distributions into Boxes

Lattice and arrangement methods

• Lattice Paths
• The Reflection Principle
• Circular and Multiset Arrangements

Special counts

• Derangements
• Counting Onto Functions

Olympiad depth

• Bijection Construction in Hard Problems
• Double Counting Two Ways
• Counting Up to Symmetry

Hard applications

• Extremal Counting Problems
• Counting with Forbidden Patterns

Binomial coefficients

• The Binomial Theorem
• Pascal's Rule
• Properties of Binomial Coefficients
• The Multinomial Theorem

Standard identities

• Vandermonde's Identity
• The Hockey Stick Identity
• Sums of Binomial Coefficients

Proof techniques

• Combinatorial Proof by Double Counting
• Generating-Function Proofs of Identities
• The Roots of Unity Filter

Arithmetic of binomial coefficients

• Lucas's Theorem
• Kummer's Theorem

Olympiad depth

• Hard Binomial Identities from Shortlists
• Alternating Sums and Weighted Identities
• The Chu-Vandermonde Identity

Hard applications

• Binomial Coefficients in Number Theory
• Bounding Binomial Coefficients Sharply

Inclusion-exclusion

• The Inclusion-Exclusion Formula
• Inclusion-Exclusion for Two and Three Sets
• The Complement Method
• Derangements by Inclusion-Exclusion

Counting applications

• Counting Onto Functions
• Counting with Forbidden Positions
• Counting Coprime Integers

Sieve technique

• Legendre's Formula for v_p(n!)
• The Sieve of Eratosthenes as a Counting Device
• Bonferroni-Type Bounds

Weighted forms

• Inclusion-Exclusion with Weights
• Mobius Inversion as a Sieve

Olympiad depth

• Sieve Arguments in Hard Problems
• Partial Sieves with Controlled Error
• Simultaneous Restrictions

Hard applications

• Counting with Dependence Between Properties
• Sharp Bonferroni Constructions

Pigeonhole

• The Pigeonhole Principle
• The Generalised Pigeonhole Principle
• Choosing the Right Boxes
• Pigeonhole Arguments in Geometry

Extremal principle

• The Extremal Principle
• Choosing the Largest or Smallest Object
• Minimal Counterexample Arguments

Structural existence

• Ramsey-Type Existence by Pigeonhole
• The Erdos-Szekeres Monotone Subsequence Theorem
• Lattice Point Arguments

Extremal technique

• Double Counting an Extremal Object
• Degree Arguments in Graphs

Olympiad depth

• Hard Existence Problems
• Forcing Structure by Counting
• Extremal Arguments with Equality Cases

Hard applications

• Dilworth-Type Chain and Antichain Results
• Constructing Counterexamples to Existence Claims

Recurrences

• Setting Up a Recurrence
• Linear Recurrences with Constant Coefficients
• The Characteristic Equation
• The Fibonacci Recurrence

Solving recurrences

• Non-Homogeneous Recurrences
• The Method of Undetermined Coefficients
• Recurrences from Counting Problems

Generating functions

• Ordinary Generating Functions
• Operations on Generating Functions
• The Catalan Generating Function

Applications

• The Transfer Matrix Method
• Counting by Coefficient Extraction

Olympiad depth

• Generating Functions in Hard Counting Problems
• Lagrange Inversion in Elementary Form
• Counting with Obstacles

Hard applications

• Recurrences with Variable Coefficients
• Asymptotics of a Counting Sequence

Graph basics

• Graphs, Vertices and Edges
• Degree and the Handshake Lemma
• Paths, Cycles and Connectivity
• Isomorphism and Subgraphs

Trees and traversals

• Trees and Their Properties
• Spanning Trees
• Eulerian and Hamiltonian Cycles

Structure

• Bipartite Graphs
• Planar Graphs and Euler's Formula
• Graph Colouring

Matchings and trees

• Matchings and Hall's Theorem
• Cayley's Formula for Labelled Trees

Olympiad depth

• Hall's Theorem in Hard Problems
• Menger's Theorem and Connectivity
• Tournament and Orientation Arguments

Hard applications

• Colouring Arguments in Existence Problems
• Building a Graph Model of a Problem

Extremal graph results

• Turan's Theorem
• Triangle-Free Graphs
• Edges Forcing a Complete Subgraph
• Extremal Constructions

Ramsey theory

• Ramsey Numbers
• Ramsey's Theorem for Pairs
• Monochromatic Structures

Further extremal results

• The Kovari-Sos-Turan Theorem
• Zarankiewicz-Type Bounds
• Erdos-Stone as Context

Extremal set theory

• Sperner's Theorem
• The Erdos-Ko-Rado Theorem

Olympiad depth

• Extremal Constructions in Shortlists
• Proving Sharpness of an Extremal Bound
• Ramsey Arguments in Hard Problems

Hard applications

• Extremal Problems on Hypergraphs
• Stability of an Extremal Configuration

Elementary probability

• Sample Spaces and Events
• Conditional Probability
• Independence
• The Law of Total Probability

Expectation

• Random Variables and Expectation
• Linearity of Expectation
• Variance and Covariance

Probabilistic method

• The Probabilistic Method
• Existence by Positive Probability
• Markov's Inequality

Second moment

• Chebyshev's Inequality
• The Second Moment Method

Olympiad depth

• Hard Probabilistic Existence Problems
• The Lovasz Local Lemma in Simple Form
• Expected Value Arguments in Combinatorics

Hard applications

• Alteration Arguments
• Randomised Constructions with Constraints

Centres of a triangle

• The Circumcentre and the Circumcircle
• The Incentre and the Incircle
• The Orthocentre
• The Centroid

Elementary triangle results

• The Angle Bisector Theorem
• The Sine and Cosine Rules in Geometry
• The Area Formulae
• Congruence and Similarity Criteria

Classical lines and circles

• The Euler Line
• The Nine-Point Circle
• The Feuerbach Theorem
• The Excentres and Excircles

Metric relations

• Ceva's Theorem
• Menelaus's Theorem
• Stewart's Theorem
• Apollonius's Theorem

Advanced triangle geometry

• Symmedians and the Lemoine Point
• The Gergonne and Nagel Points
• The Trigonometric Form of Ceva and Menelaus
• Coordinates of the Classical Centres

Olympiad configurations

• Hard Triangle Problems from Shortlists
• The Orthic Triangle and Its Properties
• The Intouch Triangle

Coordinate methods

• Cartesian Coordinates in Geometry
• The Distance and Area Formulae in Use
• Lines and Circles in Coordinates
• Solving a Configuration by Coordinates

Vector methods

• Vectors in the Plane
• The Dot Product and Perpendicularity
• The Cross Product and Area
• Vector Proofs of Classical Results

Barycentric and trilinear coordinates

• Barycentric Coordinates
• Trilinear Coordinates
• Coordinates of the Classical Centres
• The Equation of a Line in Barycentrics

Complex methods

• The Complex Plane and the Unit Circle
• Complex Criteria for Collinearity and Concyclicity
• Rotations as Complex Multiplication
• The Complex Form of Classical Theorems

Advanced analytic geometry

• Hard Geometry by Complex Numbers
• The Complex Proof of Ptolemy's Theorem
• Computing with Named Triangle Centres
• Choosing the Right Coordinate System

Olympiad configurations

• Proving a Concyclicity by Computation
• Algebraic Proofs of Collinearity
• Hybrid Synthetic and Analytic Solutions

Trigonometric fundamentals

• Radian Measure and the Unit Circle
• The Sine and Cosine Rules
• Trigonometric Identities for Geometry
• The Area Formulae in Trigonometric Form

Triangle trigonometry

• The Half-Angle Formulae
• The Inradius and Circumradius Formulae
• Solving a Triangle
• Heights and Distances

Trigonometric technique in geometry

• The Tangent Half-Angle Substitution
• The Trigonometric Form of Ceva and Menelaus
• The Auxiliary Angle Method
• Trigonometric Substitutions in Optimisation

Trigonometric sums

• Sums of Sines and Cosines in Arithmetic Progression
• The Roots of Unity Filter in Trigonometric Form
• The Lagrange Trigonometric Identity
• Chebyshev Polynomials in Trigonometric Form

Advanced trigonometric methods

• Hard Trigonometric Problems from Shortlists
• Trigonometric Inequalities
• Bounds for Trigonometric Sums
• Trigonometric Parametrisation of a Configuration

Olympiad configurations

• Angle Chasing with Trigonometric Identities
• Proving an Identity in a Triangle
• Trigonometric Solutions of Optimisation Problems

Solids and their measures

• The Tetrahedron and Its Volumes
• The Sphere, Cone and Cylinder
• Surface Area and Volume Formulae
• Sections of a Solid

Three-dimensional configurations

• Angles Between Lines and Planes
• The Dihedral Angle
• The Distance Between Skew Lines
• Perpendicularity in Space

Structure of polyhedra

• Euler's Polyhedron Formula
• The Platonic Solids
• Polyhedra and Planar Graphs
• Spheres Inscribed in and Circumscribed about a Polyhedron

Technique

• Volumes by Decomposition
• Cavalieri's Principle
• Vectors and Coordinates in Space
• The Method of Projection onto a Plane

Advanced solid geometry

• Hard Three-Dimensional Problems from Shortlists
• The Geometry of the Regular Tetrahedron
• Spherical Sections and Great Circles
• Optimisation Problems in Space

Olympiad configurations

• The Sphere Through Four Points
• Three-Dimensional Lattice Problems
• Reducing a Space Problem to a Plane

Function concepts and graphs

• Domain, Range and Function Notation
• Graphs and Transformations
• Composition and Inverse Functions
• Modulus and Piecewise Functions

Elementary function families

• Polynomial and Rational Functions
• Exponential and Logarithmic Functions
• Trigonometric and Inverse Trigonometric Functions

Function structure at entrance depth

• Monotonicity and Periodicity
• Inequalities for Elementary Functions
• Equations Mixing Elementary Functions

Limits from first principles

• The Intuitive and Formal Definition of a Limit
• One-Sided Limits
• The Algebra of Limits
• The Squeeze Theorem

Evaluation techniques

• Limits by Rationalisation and Factorisation
• Standard Limits
• Indeterminate Forms

Limits at entrance depth

• Limits at Infinity and Orders of Growth
• Limits Involving the Greatest Integer and Fractional Part
• Hard Indeterminate Forms by Series Expansion

Continuity

• Continuity at a Point
• Continuity on an Interval
• The Algebra of Continuous Functions
• Continuity of Composite and Piecewise Functions

Discontinuity and existence theorems

• Classification of Discontinuities
• The Intermediate Value Theorem
• The Extreme Value Theorem

Continuity at entrance depth

• Epsilon-Delta Proofs of Continuity
• Root Existence and Fixed Points
• Hard Continuity Problems and Counterexamples

Sequences and series

• Arithmetic and Geometric Progressions
• Limits of Sequences
• Convergence of Elementary Series
• Telescoping and the Method of Differences

Summation and recursive sequences

• Sigma Notation and Standard Sums
• Abel Summation by Parts
• Recursive Sequences and Their Limits

Sequences at entrance depth

• Monotone Convergence in Problems
• Estimating Sums and Remainders
• Hard Sequence and Series Problems

Differentiation from the definition

• The Difference Quotient and the Derivative
• First Principles Differentiation
• Geometric Meaning and the Tangent Line
• Differentiability and Continuity

Rules and standard derivatives

• The Product, Quotient and Chain Rules
• Derivatives of the Standard Functions
• Implicit and Logarithmic Differentiation

Differentiation at entrance depth

• Second Derivatives in Implicit and Parametric Form
• Differentiability of Piecewise and Modulus Functions
• Hard Differentiation Problems

Higher derivatives

• Notation and Computation of Higher Derivatives
• The nth Derivative of Standard Functions
• Leibniz's Rule for a Product
• Concavity and Points of Inflection

Mean value theorems

• Rolle's Theorem
• The Lagrange Mean Value Theorem
• The Cauchy Mean Value Theorem

Mean value theorems at entrance depth

• Proofs of the Mean Value Theorems
• The Constancy Criterion and Its Uses
• Hard Problems Using the Mean Value Theorem

Monotonicity and extrema

• Increasing and Decreasing Functions
• Critical Points
• The First Derivative Test
• The Second Derivative Test

Curve analysis

• Curve Sketching and Asymptotes
• Global Extrema on an Interval
• The Number of Real Roots of an Equation

Extrema at entrance depth

• Optimisation with Constraints
• Inequalities Proved by Monotonicity
• Hard Extrema and Curve Problems

Approximation

• Differentials and Linear Approximation
• Percentage and Relative Error
• Small-Angle and Small-Parameter Approximations
• Newton's Method for Roots

Series expansions

• Taylor and Maclaurin Polynomials
• Taylor's Theorem with the Lagrange Remainder
• Standard Maclaurin Expansions

Expansions at entrance depth

• Evaluating Hard Limits by Series
• L'Hopital's Rule and When It Fails
• Error Bounds from the Remainder

Standard integration

• Antiderivatives and the Constant of Integration
• Standard Integrals
• Integration by Substitution
• Integration by Parts

Systematic techniques

• Partial Fractions
• Trigonometric Integrals and Substitutions
• The Tangent Half-Angle Substitution

Integration at entrance depth

• Reduction Formulae
• Integrals Involving Roots and Rationalisation
• Hard Indefinite Integrals

The definite integral

• The Definite Integral as Signed Area
• Properties of Definite Integrals
• The Fundamental Theorem of Calculus
• Evaluating Definite Integrals by Substitution

Technique for definite integrals

• The King Property and Symmetry
• Even, Odd and Periodic Integrands
• Differentiation Under the Integral Sign

Definite integrals at entrance depth

• Definite Integrals as Limits of Sums
• Inequalities and Estimates for Integrals
• Hard Definite Integral Problems

Geometric applications

• Area Between Curves
• Volumes of Solids of Revolution
• Volumes by Cross-Section and Cavalieri's Principle
• Arc Length

Further applications

• Surface Area of a Solid of Revolution
• Arc Length and Area in Parametric Form
• Area in Polar Coordinates

Applications at entrance depth

• Centroids and the Theorems of Pappus
• The Average Value of a Function
• Hard Application Problems

Improper integrals

• Improper Integrals of the First Kind
• Improper Integrals of the Second Kind
• The p-Test
• Convergence by Comparison

Special integrals

• The Gamma Function
• The Beta Function
• The Cauchy Principal Value

Improper integrals at entrance depth

• Reduction and Wallis Formulae
• Absolute and Conditional Convergence of Integrals
• Hard Improper Integral Problems

Functions of several variables

• Domains and Level Curves
• Limits and Continuity in Several Variables
• Partial Derivatives
• The Chain Rule and the Total Differential

Extrema and multiple integrals

• Critical Points and the Hessian Test
• Lagrange Multipliers
• Double Integrals and Fubini's Theorem

Vector calculus

• Change of Variables and the Jacobian
• The Gradient, Divergence and Curl
• Green's, Gauss's and Stokes' Theorems

First order equations

• Order and Degree of a Differential Equation
• Separable Equations
• Linear First-Order Equations and the Integrating Factor
• Homogeneous Equations

Second order and special equations

• Second-Order Linear Equations with Constant Coefficients
• The Method of Undetermined Coefficients
• Bernoulli and Exact Equations

Applications of differential equations

• Variation of Parameters and the Wronskian
• Growth, Decay and Newton's Law of Cooling
• Simple Harmonic Motion and the Logistic Equation

Order and completeness

• Field and Order Axioms for the Real Numbers
• Supremum and Infimum
• The Least Upper Bound Property
• The Archimedean Property and the Density of the Rationals

Topology of the real line

• Open and Closed Sets
• Limit Points, Interior and Closure
• Compactness and the Heine-Borel Theorem

Consequences at entrance depth

• Connectedness and Intervals
• The Nested Interval Theorem
• The Cantor Set and Applications of Completeness

Rigorous sequences

• The Epsilon-N Definition of Convergence
• The Algebra of Limits with Proofs
• Monotone Convergence and Bolzano-Weierstrass
• Cauchy Sequences and Completeness

Series and differentiation theory

• Convergence Tests with Proofs
• Absolute and Conditional Convergence
• The Mean Value Theorems with Proof

Integration theory

• Darboux Sums and the Integrability Criterion
• The Fundamental Theorem of Calculus with Proof
• Improper Integrals Rigorously

Convergence of function sequences

• Pointwise Convergence
• Uniform Convergence
• The Supremum Metric
• Uniform Convergence and Continuity

Power series

• The Radius of Convergence
• The Cauchy-Hadamard Formula
• Term-by-Term Differentiation and Integration

Function series at entrance depth

• Taylor Series and Analytic Functions
• The Weierstrass M-Test
• Interchange of Limit, Derivative and Integral