Calculus
Set Theory
- Logic and Set Notation
- Introduction to Sets
- Set Operations and Venn Diagrams
- Set Identities
- Cartesian Product of Sets
- Binary Relations
- Properties of Relations
- Operations on Relations
- Composition of Relations
- Closures of Relations
- Equivalence Relations
- Equivalence Classes and Partitions
- Partial Orders
- Total Orders
- Hasse Diagrams
- Special Elements of Partially Ordered Sets
- Well Orders
- Lexicographic Orders
- Lattices
- Topological Sorting
- Well Ordering Principle
- Counting Relations
- Relational Databases
- Functions as Relations
- Injection, Surjection, Bijection
- Inverse Functions
- Composition of Functions
- Floor and Ceiling Functions
- Counting Functions
- Pigeonhole Principle
- Cardinality of a Set
- Countable and Uncountable Sets
- Comparing Cardinalities
- Cantor-Schröder-Bernstein Theorem
- Cantor’s Theorem
- Ordinal Numbers
- Cardinal Numbers
- Paradoxes of Set Theory
- Axiomatic Set Theory
Limits and Continuity of Functions
- Definition of Limit of a Function
- Properties of Limits
- Trigonometric Limits
- The Number e
- Natural Logarithms
- Indeterminate Forms
- Use of Infinitesimals
- L’Hopital’s Rule
- Continuity of Functions
- Discontinuous Functions
Differentiation of Functions
- Definition of the Derivative
- Basic Differentiation Rules
- Derivatives of Power Functions
- Product Rule
- Quotient Rule
- The Chain Rule
- Derivatives of Exponential Functions
- Derivatives of Inverse Functions
- Derivatives of Logarithmic Functions
- Derivatives of Trigonometric Functions
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Hyperbolic Functions
- Logarithmic Differentiation
- Implicit Differentiation
- Derivatives of Parametric Functions
- Derivatives of Polar Functions
- Derivatives of Vector-Valued Functions
- Second Derivatives
- Higher-Order Derivatives
- Leibniz Formula
- Table of Derivatives
- Differential of a Function
- Higher-Order Differentials
Applications of the Derivative
- Tangent and Normal Lines
- Rolle’s Theorem
- Lagrange’s Mean Value Theorem
- Cauchy’s Mean Value Theorem
- Newton’s Method
- Related Rates
- Linear Approximation
- Rectilinear Motion
- Planar Motion
- Critical Points
- Increasing and Decreasing Functions
- Local Extrema of Functions
- Global Extrema of Functions
- Convex Functions
- Inflection Points
- Asymptotes
- Curve Sketching
- Classical Inequalities
- Proving of Inequalities
- Curvature and Radius of Curvature
- Evolute and Involute
- Envelope of a Family of Curves
- Osculating Curves
- Optimization Problems in 2D Geometry
- Optimization Problems in 3D Geometry
- Optimization Problems Involving Numbers
- Optimization Problems in Physics
- Optimization Problems in Economics
- Van der Waals Equation
Integration of Functions
- Antiderivatives and Initial Value Problems
- The Indefinite Integral and Basic Rules of Integration
- Integration by Substitution
- Integration by Parts
- Integration by Completing the Square
- Partial Fraction Decomposition
- Integration of Rational Functions
- Integration of Irrational Functions
- Weierstrass Substitution
- Trigonometric Integrals
- Integration of Hyperbolic Functions
- Integrals of Vector-Valued Functions
- Trigonometric and Hyperbolic Substitutions
- Riemann Sums and the Definite Integral
- The Fundamental Theorem of Calculus
- Trapezoidal Rule
- Simpson’s Rule
- Improper Integrals
Applications of Integrals
- Area of a Region Bounded by Curves
- Volume of a Solid with a Known Cross Section
- Volume of a Solid of Revolution: Disks and Washers
- Volume of a Solid of Revolution: Cylindrical Shells
- Area of a Surface of Revolution
- Average Value of a Function
- Probability Density Function
- Arc Length
- Net Change Theorem
- Distance, Velocity and Acceleration
- Force, Work and Energy
- Fluid Pressure
- Mass and Density
- Center of Mass and Moments
- Moment of Inertia
- Pappus’s Theorem
- Integrals in Electric Circuits
- Applications of Integrals in Economics
Infinite Sequences and Series
- Infinite Sequences
- Geometric Series
- Infinite Series
- Comparison Tests
- The Integral Test
- The Ratio and Root Tests
- Alternating Series
- Power Series
- Differentiation and Integration of Power Series
- Taylor and Maclaurin Series
Double Integrals
- Definition and Properties of Double Integrals
- Iterated Integrals
- Double Integrals over Rectangular Regions
- Double Integrals over General Regions
- Change of Variables in Double Integrals
- Double Integrals in Polar Coordinates
- Geometric Applications of Double Integrals
- Physical Applications of Double Integrals
Triple Integrals
- Definition and Properties of Triple Integrals
- Triple Integrals in Cartesian Coordinates
- Change of Variables in Triple Integrals
- Triple Integrals in Cylindrical Coordinates
- Triple Integrals in Spherical Coordinates
- Calculation of Volumes Using Triple Integrals
- Physical Applications of Triple Integrals
Line Integrals
- Line Integrals of Scalar Functions
- Line Integrals of Vector Fields
- Green’s Theorem
- Path Independence of Line Integrals
- Geometric Applications of Line Integrals
- Physical Applications of Line Integrals
Surface Integrals
- Surface Integrals of Scalar Functions
- Surface Integrals of Vector Fields
- The Divergence Theorem
- Stoke’s Theorem
- Geometric Applications of Surface Integrals
- Physical Applications of Surface Integrals
Fourier Series
- Definition of Fourier Series and Typical Examples
- Fourier Series of Functions with an Arbitrary Period
- Even and Odd Extensions
- Complex Form of Fourier Series
- Convergence of Fourier Series
- Bessel’s Inequality and Parseval’s Theorem
- Differentiation and Integration of Fourier Series
- Applications of Fourier Series to Differential Equations
- Orthogonal Polynomials and Generalized Fourier Series