Calculus - single variable
Limits and continuity
Motivational applications (estimation)
Finding limits using graphs
Rules of limits - basic
Evaluating limits - factoring
Evaluating limits - rationalizing
Evaluating limits - rational expressions
Evaluating limits - trigonometric
Squeeze theorem
One-sided limits - concept of
Continuity - concept of
Continuity - classifying discontinuities
Continuity - intermediate value theorem
Infinite limits and vertical asymptotes
Limits at infinity, horizontal and oblique asymptotes
Estimating limits numerically
Applications - instantaneous rate of change
Applications - tangent lines and slopes
Applications - finding all asymptotes
Applications - other
Definitions and existence (conceptual)
Differentiation
Definition of the derivative
Conceptual understanding of derivatives
Derivatives of polynomials and power functions
Product rule (without trigonometric functions)
Product rule (with trigonometric functions)
Quotient rule (without trigonometric functions)
Quotient rule (with trigonometric functions)
Derivatives of trigonometric functions
Derivatives of exponential functions
Derivatives of logarithmic functions
Derivatives of inverse functions
Derivatives of inverse trigonometric functions
Hyperbolic functions
Derivatives of hyperbolic functions
Chain rule (without trigonometric functions)
Chain rule (with trigonometric functions)
Higher-order derivatives
Derivatives involving multiple rules (no product, quotient, or chain rule)
Derivatives involving multiple rules (no chain rule)
Derivatives involving multiple rules (all rules)
Logarithmic differentiation
Implicit differentiation
Applications of differentiation
Mean value theorem
Rates of change - general
Rates of change - business and economics
Rates of change - engineering and physics
Rates of change - natural and social sciences
Increasing/decreasing functions and local extrema
Concavity and points of inflection
Global extrema
Summary of curve sketching
Optimization - general
Optimization - business and economics
Optimization - engineering and physics
Optimization - natural and social sciences
Linear approximation and differentials
Related rates
Indeterminate forms and L'Hopital's rule
Newton's method
Elasticity of demand
Integrals
Conceptual understanding of integration
Antiderivatives (without trigonometric functions)
Antiderivatives (with trigonometric functions)
Indefinite integrals (without trigonometric functions)
Indefinite integrals (with trigonometric functions)
Riemann sums
Definite integrals (without trigonometric functions)
Definite integrals (with trigonometric functions)
Fundamental theorem of calculus
Improper integrals
Techniques of integration
Substitution (without trigonometric functions)
Substitution (with trigonometric functions)
Integration by parts (without trigonometric functions)
Integration by parts (with trigonometric functions)
Trigonometric integrals
Hyperbolic functions
Partial fractions
Trigonometric substitution
Tables of integrals
Computer algebra system
Mixed techniques
Challenging integrals
Approximation
Applications of integration
Average value of a function
Areas between curves
Volumes by slices
Volumes by disks
Volumes by washers
Volumes by cylindrical shells
Volumes by multiple methods
Arc length
Surfaces of revolution
Distance, velocity, acceleration
Work
Hydrostatic pressure
Center of gravity
Other physics and engineering applications
Economics
Biology
Probability and statistics
Infinite sequences and series
Limit of a sequence
Series notation
Partial sums
Taylor polynomials
Geometric
Test for divergence
Comparison tests
Integral test
Ratio test
Root test
Alternating series test
Absolute and conditional convergence
Strategy for testing series
Interval of convergence of a power series
Maclaurin series
Taylor series
Power series
Representations of functions as series
Applications of Taylor polynomials
Fourier series
Parametric
Curves
Eliminating the parameter
Tangents, velocity, and speed
Higher order derivatives
Arc length
Area
Volumes of revolution
Surface area of surfaces of revolution
Polar
Similar figures <<< Trigonometry__Geometric and algebraic foundations for trigonometry__Similar figures
Polar and rectangular coordinates <<< Trigonometry__Polar coordinates & vectors__Polar and rectangular coordinates
Curves <<< Trigonometry__Polar coordinates & vectors__Curves
Inequalities <<< Trigonometry__Polar coordinates & vectors__Inequalities
Tangents
Area
Arc length
Other applications
Calculus - multivariable
Vector geometry <<<
Vectors and vector arithmetic <<< Geometry__Vector geometry__Vectors and vector arithmetic
Dot product, length, and unit vectors <<< Geometry__Vector geometry__Dot product, length, and unit vectors
Cross product <<< Geometry__Vector geometry__Cross product
Cross product <<< Geometry__Vector geometry__Lines
Cross product <<< Geometry__Vector geometry__Planes
Cross product <<< Geometry__Vector geometry__Lines with planes
Coordinate systems <<< Geometry__Vector geometry__Coordinate systems
Calculus of vector valued functions
Parameterized curves
Limits and continuity
Derivatives
Integrals
Arc length and curvature
Frames, motions, and other applications
Concepts for multivariable functions
Notation, domain, and range
Surfaces
Quadratic surfaces
Surfaces in other coordinate systems
Parameterized surfaces
Traces, contours, and level sets
Differentiation of multivariable functions
Limits and continuity
Partial derivatives
Chain rule
Differentiability, linearization and tangent planes
Directional derivatives and the gradient
Extreme values and optimization
Lagrange multipliers and constrained optimization
Integration of multivariable functions
Double integrals over rectangles
Iterated integrals and Fubini's theorem
Double integrals over general regions
Double integrals in polar
Triple integrals
Change of variable
Triple integrals in cylindrical and spherical
Applications of double integrals
Applications of triple integrals
Vector fields
Graphs, flows lines, and level surfaces
Identifying extrema from graphs
Vector calculus
Derivatives
Line integrals
Conservative vector fields
Applications of line integrals
Curl and divergence
Surface integrals of scalar fields
Surface integrals of vector fields
Fundamental theorems
Line integrals
Green's theorem
Stokes' theorem
Divergence theorem
Algebra
Algebra of real numbers and simplifying expressions
Properties
Algebraic expressions
Evaluating expressions
Inequalities and intervals
Simplifying expressions
Solving linear equations in one variable
Isolating variables
Scientific notation
Absolute value expressions and functions
Solving equations with absolute values
Graphs with absolute values
Absolute value inequalities
Applications using absolute values
Properties of exponents, rational exponents and radicals
Properties of exponents
Properties of rational exponents and radicals
Cartesian coordinate system
Plotting points
Midpoint and distance formulas
Circles
Graphs of equations
Factoring
Factoring: common factors
Factoring by grouping
Factoring trinomials
Factoring: special forms
Factoring polynomials: general
Functions
Definition, concept
Function notation
Graphs
Domain and range
Piecewise functions
Compositions and combinations of functions
Difference quotient
Interpretation and applications
Transformations of functions and graphs
Shifts: vertical and horizontal
Scale changes: vertical and horizontal
Shift and scale change
Reflect
Reflect and shift
Reflect and scale change
Symmetry: even, odd, neither
Three or more transformations
Vertical shifts
Horizontal shifts
Vertical stretches and compressions
Horizontal stretches and compressions
Reflections and symmetry
Graphs
Linear equations and functions
Finding the slope
Parallel and perpendicular lines
Equations of lines: slope-intercept form
Equations of lines: point-slope form
Equations of lines: standard form
Equations of lines: general
Linear functions
Linear equations
Linear inequalities
Graphs of lines
Applications and models
Quadratic equations and functions
Solve by factoring
Completing the square
Quadratic formula
Complex roots
Solving equations
Forms: vertex, factored, general
Inequalities
Graphs
Applications and models
Operations on polynomial and rational expressions
Polynomials: add, subtract
Polynomials: multiply
Polynomials: divide
Rational expressions: multiply, divide
Rational expressions: add, subtract
Simplify rational expressions
Partial fractions
Polynomial equations and functions
Polynomial equations
Polynomial functions
Inequalities involving polynomials
Remainder and factor theorems
Zeros and multiplicities
Counting zeros
Graphs of polynomials
Applications and models
Complex roots
Variation and power functions
Direct variation
Inverse variation
Mixed variation
Power functions
Applications of power functions
Systems of equations and inequalities
Linear systems
Nonlinear systems
Inequalities
Functions with fractional exponents and radical functions
Functions with fractional exponents
Radical functions
Equations
Applications
Rational equations and functions
Simplifying
Rational equations
Rational functions
Rational inequalities
Graphs of rational functions
Asymptotes
Applications and models
Inverse functions
1-1 functions
Finding the inverse function
Interpreting inverse functions
Exponential and logarithmic expressions and functions
Exponential functions
Properties of logarithms
Logarithmic functions
Exponential and logarithmic equations
Inequalities
Graphs
Applications and models - population growth
Applications and models - radioactive decay
Applications and models - general
Finite sequences and series
Notation
Arithmetic
Binomial theorem
Summation formulas
Conic sections
Parabolas
Circles
Ellipses
Hyperbolas
Intersections of conics
Polar or parametric form
Trigonometry
Geometric and algebraic foundations for trigonometry
Similar figures
The Pythagorean theorem & its converse
Radians, converting radians & degrees
Arc length, sector area, angular and linear velocity
Reference angles (using coterminal angles)
Trigonometric functions
Unit circle
Sine & cosine functions - definitions, graphs, & properties
Tangent & cotangent functions - definitions, graphs, & properties
Secant & cosecant functions - definitions, graphs, & properties
Inverse trigonometric functions - definitions, graphs, & properties
Combinations and compositions of functions
Trigonometric functions of special angles
Trigonometric functions of non-special angles
Modeling with trigonometric functions
Triangle trigonometry
Sine, cosine, and tangent of an angle in a right triangle
Applications of special triangles & right triangles
Law of sines (angle-side-angle)
Law of cosines (side-angle-side, side-side-side)
Law of sines or law of cosines (side-side-angle)
Applications of law of sines & law of cosines
Area of a triangle
Analytic trigonometry
Double-angle & half-angle formulas
Addition & subtraction formulas
Using and proving basic identities
Using and proving general identities
Solving trigonometric equations exactly
Solving trigonometric equations numerically
Solving trigonometric inequalities exactly
Solving trigonometric inequalities numerically
Product-to-sum & sum-to-product formulas
Polar coordinates & vectors
Polar and rectangular coordinates
Curves
Inequalities
Differential equations
Introductory concepts
Verification of solutions
Classifications of differential equations
First order differential equations
Linear
Exact
Separable
Substitutions
Equilibrium points and phase lines
Applications - exponential growth & decay
Applications - logistic
Applications - mixing problems
Applications - circuits
Applications - Newton's law of cooling
Applications - other
Direction fields
Integrating factor
Bifurcations
Existence and uniqueness
Higher order differential equations
Reduction of order
Applications
Linear, constant coefficients, homogeneous
Linear, constant coefficients, homogeneous (distinct real roots)
Linear, constant coefficients, homogeneous (repeated roots)
Linear, constant coefficients, homogeneous (complex roots)
Euler equations
Undetermined coefficients
Variation of parameters
Boundary value problems
Linear independence
Differential operators
Existence and uniqueness
Laplace transforms
Applications and solving differential equations
Basic transformations
Inverse transformations
Convolutions
Impulse functions
Step functions
Shift functions
Systems of differential equations
Matrix notation for systems
Verification of solutions
Applications
Distinct real eigenvalues
Complex eigenvalues
Repeated eigenvalues
Reduction to first order systems
Nonhomogeneous systems
Phase planes
Nonlinear systems
Numerical methods
Euler
Runge-Kutta
Systems
Series solutions
Ordinary point
Singular point
Bessel functions
Partial differential equations
Classification
Verification of solutions
Heat equation
Wave equation
Laplace's equation
Other separable equations
Fourier series
Inhomogenous equations
Linear algebra
Systems of linear equations
Systems with 2 variables
Systems with 3 variables
Systems with 4 or more variables
Matrix-vector forms
Vector equations
Augmented matrices
Applications
Matrices
Matrix algebra
Row operations
Echelon form
Rank
Transpose and trace
Inverses
Elementary matrices
Complex entries
Markov chains
Matrix factorizations
Diagonalization
LU factorization
QR factorization
Singular value decomposition
Jordan form
Euclidean spaces
Vectors
Linear combinations
Span
Linear independence
Subspaces
Basis and dimension
Row, column, and null spaces
Coordinate vectors and change of basis
Vector geometry <<<
Vectors and vector arithmetic <<< Geometry__Vector geometry__Vectors and vector arithmetic
Dot product, length, and unit vectors <<< Geometry__Vector geometry__Dot product, length, and unit vectors
Cross product <<< Geometry__Vector geometry__Cross product
Lines <<< Geometry__Vector geometry__Lines
Planes <<< Geometry__Vector geometry__Planes
Lines with planes <<< Geometry__Vector geometry__Lines with planes
Coordinate systems <<< Geometry__Vector geometry__Coordinate systems
Abstract vector spaces
Definition and properties
Linear combinations
Span
Linear independence
Subspaces
Basis and dimension
Coordinate vectors and change of basis
Examples
Eigenvalues and eigenvectors
Computing eigenvalues and eigenvectors
Properties
Complex eigenvalues and eigenvectors
Quadratic forms
Applications
Inner products
Computing with dot products
Computing with inner products
Orthogonal and orthonormal sets
Projection and distance
Gram-Schmidt process
Orthogonal matrices
Applications
Linear transformations
Properties
Evaluating linear transformations
Associated matrices
One-to-one and onto
Kernel and image
Inverses
Determinants
Computing determinants
Properties
Applications
Precalculus <<<
Functions <<<
Definition, concept <<< Algebra__Functions__Definition, concept
Function notation <<< Algebra__Functions__Function notation
Graphs <<< Algebra__Functions__Graphs
Domain and range <<< Algebra__Functions__Domain and range
Piecewise functions <<< Algebra__Functions__Piecewise functions
Compositions and combinations of functions <<< Algebra__Functions__Compositions and combinations of functions
Difference quotient <<< Algebra__Functions__Difference quotient
Interpretation and applications <<< Algebra__Functions__Interpretation and applications
Transformations of functions and graphs <<<
Shifts: vertical and horizontal <<< Algebra__Transformations of functions and graphs__Shifts: vertical and horizontal
Scale changes: vertical and horizontal <<< Algebra__Transformations of functions and graphs__Scale changes: vertical and horizontal
Shift and scale change <<< Algebra__Transformations of functions and graphs__Shift and scale change
Reflect <<< Algebra__Transformations of functions and graphs__Reflect
Reflect and shift <<< Algebra__Transformations of functions and graphs__Reflect and shift
Reflect and scale change <<< Algebra__Transformations of functions and graphs__Reflect and scale change
Symmetry: even, odd, neither <<< Algebra__Transformations of functions and graphs__Symmetry: even, odd, neither
Three or more transformations <<< Algebra__Transformations of functions and graphs__Three or more transformations
Vertical shifts <<< Algebra__Transformations of functions and graphs__Vertical shifts
Horizontal shifts <<< Algebra__Transformations of functions and graphs__Horizontal shifts
Vertical stretches and compressions <<< Algebra__Transformations of functions and graphs__Vertical stretches and compressions
Horizontal stretches and compressions <<< Algebra__Transformations of functions and graphs__Horizontal stretches and compressions
Reflections and symmetry <<< Algebra__Transformations of functions and graphs__Reflections and symmetry
Graphs <<< Algebra__Transformations of functions and graphs__Graphs
Linear equations and functions <<<
Finding the slope <<< Algebra__Linear equations and functions__Finding the slope
Parallel and perpendicular lines <<< Algebra__Linear equations and functions__Parallel and perpendicular lines
Equations of lines: slope-intercept form <<< Algebra__Linear equations and functions__Equations of lines: slope-intercept form
Equations of lines: point-slope form <<< Algebra__Linear equations and functions__Equations of lines: point-slope form
Equations of lines: standard form <<< Algebra__Linear equations and functions__Equations of lines: standard form
Equations of lines: general <<< Algebra__Linear equations and functions__Equations of lines: general
Linear functions <<< Algebra__Linear equations and functions__Linear functions
Linear equations <<< Algebra__Linear equations and functions__Linear equations
Linear inequalities <<< Algebra__Linear equations and functions__Linear inequalities
Graphs of lines <<< Algebra__Linear equations and functions__Graphs of lines
Applications and models <<< Algebra__Linear equations and functions__Applications and models
Quadratic equations and functions <<<
Solve by factoring <<< Algebra__Quadratic equations and functions__Solve by factoring
Completing the square <<< Algebra__Quadratic equations and functions__Completing the square
Quadratic formula <<< Algebra__Quadratic equations and functions__Quadratic formula
Complex roots <<< Algebra__Quadratic equations and functions__Complex roots
Solving equations <<< Algebra__Quadratic equations and functions__Solving equations
Forms: vertex, factored, general <<< Algebra__Quadratic equations and functions__Forms: vertex, factored, general
Inequalities <<< Algebra__Quadratic equations and functions__Inequalities
Graphs <<< Algebra__Quadratic equations and functions__Graphs
Applications and models <<< Algebra__Quadratic equations and functions__Applications and models
Operations on polynomial and rational expressions <<<
Polynomials: add, subtract <<< Algebra__Operations on polynomial and rational expressions__Polynomials: add, subtract
Polynomials: multiply <<< Algebra__Operations on polynomial and rational expressions__Polynomials: multiply
Polynomials: divide <<< Algebra__Operations on polynomial and rational expressions__Polynomials: divide
Rational expressions: multiply, divide <<< Algebra__Operations on polynomial and rational expressions__Rational expressions: multiply, divide
Rational expressions: add, subtract <<< Algebra__Operations on polynomial and rational expressions__Rational expressions: add, subtract
Simplify rational expressions <<< Algebra__Operations on polynomial and rational expressions__Simplify rational expressions
Partial fractions <<< Algebra__Operations on polynomial and rational expressions__Partial fractions
Polynomial equations and functions <<<
Polynomial equations <<< Algebra__Polynomial equations and functions__Polynomial equations
Polynomial functions <<< Algebra__Polynomial equations and functions__Polynomial functions
Inequalities involving polynomials <<< Algebra__Polynomial equations and functions__Inequalities involving polynomials
Remainder and factor theorems <<< Algebra__Polynomial equations and functions__Remainder and factor theorems
Zeros and multiplicities <<< Algebra__Polynomial equations and functions__Zeros and multiplicities
Counting zeros <<< Algebra__Polynomial equations and functions__Counting zeros
Graphs of polynomials <<< Algebra__Polynomial equations and functions__Graphs of polynomials
Applications and models <<< Algebra__Polynomial equations and functions__Applications and models
Complex roots <<< Algebra__Polynomial equations and functions__Complex roots
Variation and power functions <<<
Direct variation <<< Algebra__Variation and power functions__Direct variation
Inverse variation <<< Algebra__Variation and power functions__Inverse variation
Mixed variation <<< Algebra__Variation and power functions__Mixed variation
Power functions <<< Algebra__Variation and power functions__Power functions
Applications of power functions <<< Algebra__Variation and power functions__Applications of power functions
Systems of equations and inequalities <<<
Linear systems <<< Algebra__Systems of equations and inequalities__Linear systems
Nonlinear systems <<< Algebra__Systems of equations and inequalities__Nonlinear systems
Inequalities <<< Algebra__Systems of equations and inequalities__Inequalities
Functions with fractional exponents and radical functions <<<
Functions with fractional exponents <<< Algebra__Functions with fractional exponents and radical functions__Functions with fractional exponents
Radical functions <<< Algebra__Functions with fractional exponents and radical functions__Radical functions
Equations <<< Algebra__Functions with fractional exponents and radical functions__Equations
Applications <<< Algebra__Functions with fractional exponents and radical functions__Applications
Rational equations and functions <<<
Simplifying <<< Algebra__Rational equations and functions__Simplifying
Rational equations <<< Algebra__Rational equations and functions__Rational equations
Rational functions <<< Algebra__Rational equations and functions__Rational functions
Rational inequalities <<< Algebra__Rational equations and functions__Rational inequalities
Graphs of rational functions <<< Algebra__Rational equations and functions__Graphs of rational functions
Asymptotes <<< Algebra__Rational equations and functions__Asymptotes
Applications and models <<< Algebra__Rational equations and functions__Applications and models
Inverse functions <<<
1-1 functions <<< Algebra__Inverse functions__1-1 functions
Finding the inverse function <<< Algebra__Inverse functions__Finding the inverse function
Interpreting inverse functions <<< Algebra__Inverse functions__Interpreting inverse functions
Exponential and logarithmic expressions and functions <<<
Exponential functions <<< Algebra__Exponential and logarithmic expressions and functions__Exponential functions
Properties of logarithms <<< Algebra__Exponential and logarithmic expressions and functions__Properties of logarithms
Logarithmic functions <<< Algebra__Exponential and logarithmic expressions and functions__Logarithmic functions
Exponential and logarithmic equations <<< Algebra__Exponential and logarithmic expressions and functions__Exponential and logarithmic equations
Inequalities <<< Algebra__Exponential and logarithmic expressions and functions__Inequalities
Graphs <<< Algebra__Exponential and logarithmic expressions and functions__Graphs
Applications and models - population growth <<< Algebra__Exponential and logarithmic expressions and functions__Applications and models - population growth
Applications and models - radioactive decay <<< Algebra__Exponential and logarithmic expressions and functions__Applications and models - radioactive decay
Applications and models – general <<< Algebra__Exponential and logarithmic expressions and functions__Applications and models - general
Finite sequences and series <<<
Notation <<< Algebra__Finite sequences and series__Notation
Arithmetic <<< Algebra__Finite sequences and series__Arithmetic
Binomial theorem <<< Algebra__Finite sequences and series__Binomial theorem
Summation formulas <<< Algebra__Finite sequences and series__Summation formulas
Conic sections <<<
Parabolas <<< Algebra__Conic sections__Parabolas
Circles <<< Algebra__Conic sections__Circles
Ellipses <<< Algebra__Conic sections__Ellipses
Hyperbolas <<< Algebra__Conic sections__Hyperbolas
Intersections of conics <<< Algebra__Conic sections__Intersections of conics
Polar or parametric form <<< Algebra__Conic sections__Polar or parametric form
Geometric and algebraic foundations for trigonometry <<<
Similar figures <<< Trigonometry__Geometric and algebraic foundations for trigonometry__Similar figures
The Pythagorean theorem & its converse <<< Trigonometry__Geometric and algebraic foundations for trigonometry__The Pythagorean theorem & its converse
Radians, converting radians & degrees <<< Trigonometry__Geometric and algebraic foundations for trigonometry__Radians, converting radians & degrees
Arc length, sector area, angular and linear velocity <<< Trigonometry__Geometric and algebraic foundations for trigonometry__Arc length, sector area, angular and linear velocity
Reference angles (using coterminal angles) <<< Trigonometry__Geometric and algebraic foundations for trigonometry__Reference angles (using coterminal angles)
Trigonometric functions <<<
Unit circle <<< Trigonometry__Trigonometric functions__Unit circle
Sine & cosine functions - definitions, graphs, & properties <<< Trigonometry__Trigonometric functions__Sine & cosine functions - definitions, graphs, & properties
Tangent & cotangent functions - definitions, graphs, & properties <<< Trigonometry__Trigonometric functions__Tangent & cotangent functions - definitions, graphs, & properties
Secant & cosecant functions - definitions, graphs, & properties <<< Trigonometry__Trigonometric functions__Secant & cosecant functions - definitions, graphs, & properties
Inverse trigonometric functions - definitions, graphs, & properties <<< Trigonometry__Trigonometric functions__Inverse trigonometric functions - definitions, graphs, & properties
Combinations and compositions of functions <<< Trigonometry__Trigonometric functions__Combinations and compositions of functions
Trigonometric functions of special angles <<< Trigonometry__Trigonometric functions__Trigonometric functions of special angles
Trigonometric functions of non-special angles <<< Trigonometry__Trigonometric functions__Trigonometric functions of non-special angles
Modeling with trigonometric functions <<< Trigonometry__Trigonometric functions__Modeling with trigonometric functions
Triangle trigonometry <<<
Sine, cosine, and tangent of an angle in a right triangle <<< Trigonometry__Triangle trigonometry__Sine, cosine, and tangent of an angle in a right triangle
Applications of special triangles & right triangles <<< Trigonometry__Triangle trigonometry__Applications of special triangles & right triangles
Analytic trigonometry <<<
Double-angle & half-angle formulas <<< Trigonometry__Analytic trigonometry__Double-angle & half-angle formulas
Addition & subtraction formulas <<< Trigonometry__Analytic trigonometry__Addition & subtraction formulas
Using and proving basic identities <<< Trigonometry__Analytic trigonometry__Using and proving basic identities
Using and proving general identities <<< Trigonometry__Analytic trigonometry__Using and proving general identities
Solving trigonometric equations exactly <<< Trigonometry__Analytic trigonometry__Solving trigonometric equations exactly
Solving trigonometric equations numerically <<< Trigonometry__Analytic trigonometry__Solving trigonometric equations numerically
Solving trigonometric inequalities exactly <<< Trigonometry__Analytic trigonometry__Solving trigonometric inequalities exactly
Solving trigonometric inequalities numerically <<< Trigonometry__Analytic trigonometry__Solving trigonometric inequalities numerically
Product-to-sum & sum-to-product formulas <<< Trigonometry__Analytic trigonometry__Product-to-sum & sum-to-product formulas
Polar coordinates & vectors <<<
Polar and rectangular coordinates <<< Trigonometry__Polar coordinates & vectors__Polar and rectangular coordinates
Curves <<< Trigonometry__Polar coordinates & vectors__Curves
Inequalities <<< Trigonometry__Polar coordinates & vectors__Inequalities
Geometry
Euclidean plane
Shapes
Perimeter
Area
Surface area
Volume
Similar figures <<< Trigonometry__Geometric and algebraic foundations for trigonometry__Similar figures
Properties of shapes
Angles
Parallel lines with transversals
Bisectors
Complementary/supplementary
Polygons
Transformations
Rotation and reflection
Circle geometry
Arcs and chords
Inscribed and circumscribed
Circumference and area
Vector geometry
Vectors and vector arithmetic
Dot product, length, and unit vectors
Cross product
Lines
Planes
Lines with planes
Coordinate systems
Probability
Sample Space
Outcomes & events
Probability: direct computation, inclusion/exclusion
Conditional probability -- direct
Independence
Bayes theorem -- inverse probability
Odds
Random variables
Expectation
Variance, standard deviation
Median, mode
Discrete: probability mass function
Continuous: density function, cumulative distribution function
Generating function
Discrete distributions
Bernoulli
Binomial
Normal approximation to binomial
Geometric
Negative binomial
Poisson
Hypergeometric
Multinomial
Continuous distributions
Uniform
Exponential
Gaussian normal
Application of a normal distribution
Weibull
Gamma
Other distribution
Laws, theory
Chebychev's inequality
Weak law of large numbers
Central limit theorem
Several variables
Joint distribution
Marginal distributions
Covariance & correlation
Stochastic process
Markov chain
M/M/1 queue
Statistics
Experimental design
Concepts
Sample survey methods
Concepts
Ratio estimators
Regression estimators
Stratified sampling
Cluster sampling
Sampling bias
Capture-recapture method
Response bias
Sampling for a proportion
Exploratory data analysis/descriptive statistics
Classifying data
Summary statistics
Graphical representations
Description of distributions
Summarizing data in tables
Sampling distributions
Sample mean
Sample proportion
General
Simulation
Confidence intervals
Concepts
One sample mean - z
One sample mean - t
One sample proportion
Two sample proportion
Paired samples
Independent samples - z
Independent samples - t
Variance
Hypothesis tests
Concepts
One sample mean - z
One sample mean - t
One sample proportion
Two sample proportion
Paired samples
Independent samples - z
Independent samples - t
Type I/type II errors and power
One sample variance
Chi-squared test for independence
Chi-squared test for goodness of fit
Fisher's exact test
Two sample variances
One-way ANOVA
Multi-way ANOVA
Simple linear regression
Correlation
Regression
Prediction
Residuals
Hypothesis tests
Confidence intervals
Diagnostics
Nonparametric methods
Sign test
Rank sum test
Signed rank test
Runs test
Kruskal-Wallis test
Kolmogorov-Smirnov test
Permutation/randomization methods
Confidence intervals
Relative efficiency
Bayesian inference
Choice of prior
Improper priors
Posterior distribution
Credibility intervals
Hypothesis tests
Time series
Descriptive methods
Autocorrelation
Runs test
Seasonal variation
ARIMA models
Forecasting
Frequency domain
Point estimation
Biasedness
Consistency
Sufficiency
Maximum likelihood estimation
Multiple regression
Nonlinear regression
Indicator variables
Parameter estimates
Confidence intervals
Hypothesis tests
Multiple selection
Model selection
Generalized linear methods
Logistic regression
Loglinear for contingency tables
Ordinal regression
Combinatorics
Counting
Factorial arithmetic
Principles (addition, subtraction, multiplication)
Permutations
Combinations
Pigeonhole principle
Inclusion/exclusion
Stars and bars
Recursive
Multiple techniques
Recurrence relations
Concepts
Solving homogeneous
Solving nonhomogeneous
Generating functions
Graph theory
Terminology
Matrices
Adjacency
Incidence
Walks
Eulerian
Hamiltonian
Coloring
Trees
Spanning trees
Algorithms
Kruskal's
Dijkstra's
Statics
Particles
Free-body diagrams
Forces
Resultants
Resolution into components
2D equilibrium
3D equilibrium
Rigid bodies
Free-body diagrams
Moments and couples
Resultants
2D reactions
2D equilibrium
3D reactions
3D equilibrium
Constraints and statical determinacy
Structures
Simple trusses
Method of joints
Method of sections
Zero-force members
Space trusses
Compound trusses
Frames and machines
Types of supports
Internal forces in members
Cables
Cables: concentrated loads
Cables: distributed loads
Parabolic cables
Catenary
Distributed forces: centroid and center of gravity
Center of mass
Centroids
Composite bodies
Theorems of Pappus and Guldinus
Resultant of general distributed loading
Fluid pressure
Distributed forces: moment of inertia
Area moment of inertia
Parallel axis theorem: area moment of inertia
Radius of gyration: area moment of inertia
Polar moment of inertia
Composite areas
Product of inertia
Area moment of inertia about an inclined axis
Mass moment of inertia
Radius of gyration: mass moment of inertia
Parallel axis theorem: mass moment of inertia
Composite masses
Friction
Dry friction
Wedges
Screws
Journal bearings
Thrust bearings
Flat belts
Rolling resistance
Virtual work
Work
Principle of virtual work
Conservative forces, potential energy, and equilibrium
Stability of equilibrium
Dynamics
Kinematics of particles
Rectilinear motion
Graphical solutions of rectilinear motion problems
Curvilinear motion: 2D rectangular coordinates
Curvilinear motion: normal and tangential coordinates
Curvilinear motion: polar coordinates
Curvilinear motion: 3D rectangular coordinates
Curvilinear motion: cylindrical coordinates
Curvilinear motion: spherical coordinates
Motion of connected particles
Relative motion: translating axes
Kinetics of particles
Free-body diagrams
2D equations of motion
3D equations of motion
Dynamic equilibrium (d'Alembert's principle)
Central-force motion and space mechanics
Work of a force
Principle of work and energy
Kinetic energy
Potential energy
Conservative forces
Conservation of energy
Power and efficiency
Linear impulse and momentum
Angular impulse and momentum
Principle of impulse and momentum
Conservation of momentum
Impact
Kinetics of systems of particles
Equations of motion
Principle of work and energy
Principle of impulse and momentum
Conservation of energy and momentum
Steady mass flow
Variable mass
Planar kinematics of rigid bodies
Rectilinear and curvilinear translation
Rotation about a fixed axis
General plane motion
Absolute motion
Relative motion: velocity, translating axes
Relative motion: velocity, instantaneous centers
Relative motion: acceleration, translating axes
Relative motion: velocity, rotating axes
Relative motion: acceleration, rotating axes
Planar kinetics of rigid bodies
Free-body diagrams
Mass moment of inertia <<< Statics__Distributed forces: moment of inertia__Mass moment of inertia
Radius of gyration: mass moment of inertia <<< Statics__Distributed forces: moment of inertia__Radius of gyration: mass moment of inertia
Parallel axis theorem: mass moment of inertia <<< Statics__Distributed forces: moment of inertia__Parallel axis theorem: mass moment of inertia
Composite masses <<< Statics__Distributed forces: moment of inertia__Composite masses
Equations of motion
Sliding, rolling, tipping
Work of a force
Principle of work and energy
Systems of rigid bodies
Kinetic energy
Potential energy
Conservative forces
Conservation of energy
Power and efficiency
Linear impulse and momentum
Angular impulse and momentum
Principle of impulse and momentum
Conservation of momentum
Systems of rigid bodies
Impact: central
Impact: eccentric
3D kinematics of rigid bodies
Rotation about a fixed point
General motion
Eulerian angles
Helical axes
Absolute motion
Relative motion: translating axes
Relative motion: rotating axes
3D kinetics of rigid bodies
Free-body diagrams
Principal axes and moments of inertia
Equations of motion
Euler's equations of motion
Gyroscopes, axisymmetric bodies, arbitrary bodies
Principle of work and energy
Kinetic energy
Potential energy
Conservation of energy
Linear impulse and momentum
Angular impulse and momentum
Principle of impulse and momentum
Conservation of momentum
Impact
Lagrangian mechanics
Degrees of freedom, generalized coordinates, constraints
Virtual work, virtual displacements
Generalized forces
Lagrange equations
Constrained systems
Hamilton's principle
Generalized momentum
Quasi-coordinates
Mechanics of materials
Stress
Axial stress
Shear stress
Bearing stress
Simple structures
Load and resistance factors
Factor of safety
Stress on an arbitrary plane
Stress tensors
Strain
Axial strain
Axial stress-strain diagram
Hooke's law: isotropic materials in axial loading
Hooke's law: isotropic materials in 3D loading
Hooke's law: anisotropic materials
Hooke's law: orthotropic materials
Axial deformation
Statically indeterminate axial members
Thermal expansion
Poisson's ratio
Bulk modulus
Shear strain
Stress concentration
Plastic deformation
Residual stress
Strain energy
Shear stress-strain diagram
Non-linear material response
Torsion
Torsional shear strain
Torsional stress
Transmission shafts
Angle of twist
Statically indeterminate shafts
Non-circular shafts
Stress concentration
Plastic deformation
Hollow shafts
Elastoplastic materials
Residual stress
Bending
Bending strain
Bending stress
Non-symmetric bending
Bending deformation
Composite beams
Stress concentration
Plastic deformation
Curved beams
Elastoplastic materials
Eccentric axial loading
Inclined loads
Transverse shear
Shear and moment diagrams
Prismatic beams
Singularity functions
Non-prismatic beams
Shear stress
Shear flow
Shear center
Plastic deformation
Pressure vessels
Pressure vessels
Combined loading
Stress transformation
Transformation of plane stress
Principal stress and maximum shear stress
Mohr's circle for plane stress
Mohr's circle for 3D stress
Stress invariants
Strain transformation
Transformation of plane strain
Principal strain and maximum shear strain
Mohr's circle for plane strain
Mohr's circle for 3D strain
Strain rosettes
Failure theories
Maximum shear stress theory
Maximum distortion energy theory
Maximum normal stress theory
Mohr's criterion
Alternative yield criteria
Beam and shaft design
Prismatic beams
Non-prismatic beams
Transmission shafts
Combined loading
Beam deflection
Elastic curve
Statically indeterminate beams
Singularity functions
Superposition
Superposition for statically indeterminate beams
Moment-area method
Moment-area method for statically indeterminate beams
Method of integration for statically indeterminate beams
Buckling
Critical load
Euler buckling
Other end conditions
Eccentric loading
Column design under centric load
Column design under eccentric load
Local buckling
Inelastic buckling
Energy methods
Strain energy for axial stress
Strain energy for shear stress
Strain energy for combined stress
Impact loading
Work-energy method
Virtual work
Castigliano's theorem
Statically indeterminate structures
Internal energy density
Advanced topics
Beams on elastic foundations
Thick-walled cylinders
Flat plates
Fracture mechanics
Fatigue
Creep
Contact stresses
Finite element method
Vibrations
Elements
Spring elements
Damping elements
Mass and inertial elements
One DOF vibration: free
Undamped
Stability
Work-energy methods
Viscous damping
Coulomb damping
Hysteretic damping
One DOF vibration: harmonic excitation
Undamped
Viscous damping
Rotating unbalance
Base excitation
Coulomb damping
Hysteretic damping
Self-excitation and stability
One DOF vibration: general forced
Two DOF vibration
Multi-DOF vibration
Vibration suppression and control
Distributed systems
Vibration measurement and testing
Dynamic finite element analysis
Numerical methods in vibration
Non-linear vibration
Random vibration
Fluid mechanics
Definitions
Unit systems, conversion of units, dimensional homogeneity <<< Arithmetic__Units__Conversions
Thermodynamic properties of a fluid
Viscosity and other secondary properties
Streamlines, streaklines and pathlines
Pressure distributions
Hydrostatic pressure
Manometers
Hydrostatic forces: plane surfaces
Hydrostatic forces: curved surfaces
Hydrostatic forces: layered fluids
Buoyancy and stability
Pressure distribution in rigid-body motion
Integral relations for a control volume
Reynolds transport theorem
Continuity
Linear momentum
Bernoulli's equation
Angular momentum
Energy
Differential relations for fluid flow
Acceleration field
Continuity
Linear momentum
Energy
Stream function
Vorticity
Velocity potential
Dimensional analysis and similarity
Buckingham Pi theorem
Nondimensionalization of the fluid flow equations
Scaling and similarity
Viscous flow in ducts
Entry length
Laminar fully developed pipe flow
Turbulent pipe flow
Pressure head loss: major loss and friction factor
Pressure head loss: minor losses
Moody chart
Energy equation
Pumps and fans
Noncircular ducts
Multiple-pipe systems
Diffuser performance
Flow past immersed bodies
Momentum integral estimates
Boundary layers
Flat plate boundary layer
Boundary layers with pressure gradients
Forces on objects immersed in a flow
Potential flow
Elementary solutions
Superposition
Flow past closed-body shapes
Conformal mapping
Airfoil theory
Axisymmetric flow
Compressible flow
Speed of sound
Adiabatic and isentropic steady flow
Isentropic flow with area change
Converging and diverging nozzles
Shock waves: normal
Shock waves: oblique
Prandtl-Meyer expansion waves
Compressible duct flow with friction
Frictionless duct flow with heat transfer
Open-channel flow
Specific energy and critical depth
Uniform flow and the Chezy equations
Efficient uniform-flow channels
Gradually-varied flow
Hydraulic jump
Flow measurement and control by weirs
Turbomachinery
Pumps
Pumps: performance curves
Pumps: scaling laws and specific speed
Turbines
Electric circuits
AC steady-state analysis
Analysis techniques
Basic analysis using Kirchhoff's laws
Impedance and admittance
Sinusoids
Additional analysis techniques
Introduction
Maximum power transfer
Superposition
Thevenin's and Norton's theorems
Basic components and electric circuits
Units and scales
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Capacitors
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Magnetically coupled networks
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Source/load connections
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Kirchhoff's law
Ohm's law
Series and parallel resistor combinations
Single-loop circuits
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Wye-delta transformations
Steady-state power analysis
Average power
Complex power
Effective or rms values
Maximum average power transfer
The power factor
Voltage and current laws
The single-loop circuit
Thermodynamics
Definitions
Unit systems, conversion of units, dimensional homogeneity <<< Arithmetic__Units__Conversions
Systems
Hydrostatic pressure <<< Fluid mechanics__Pressure distributions__Hydrostatic pressure
Manometers <<< Fluid mechanics__Pressure distributions__Manometers
Chemical composition
Specific volume and density
Equilibrium
Energy
Internal energy
Enthalpy
Work: mechanical
Work: boundary
Work: shaft
Work: spring
Work: electrical
Properties
Gibbs phase rule
Phase diagrams
Pure substances: solids
Pure substances: compressed liquid
Pure substances: saturated liquid
Pure substances: two-phase mixture
Pure substances: saturated vapour
Pure substances: superheated vapour
Pure substances: phase change processes
Specific heat: cv
Specific heat: cp
Ideal gas: ideal gas law
Ideal gas: change in internal energy
Ideal gas: change in enthalpy
Compressibility factor
Polytropic process
Incompressible substance: change in internal energy
Incompressible substance: change in enthalpy
First law: closed systems
First law: closed systems
Continuity: open systems
Mass and volume flow rate
Steady-state
Transient
First law: open systems
Flow work
Steady-state
Transient
Steady flow devices: nozzles and diffusers
Steady flow devices: turbines
Steady flow devices: compressors and pumps
Steady flow devices: heat exchangers
Steady flow devices: throttling devices
Steady flow devices: mixing chambers
Steady flow devices: multiple devices
Second law
Efficiency and coefficient of performance
Kelvin-Planck statement
Clausius statement
Reversible and irreversible process
Carnot corollaries
Carnot cycle: heat engine
Carnot cycle: refrigeration and heat pump
Thermodynamic temperature scale
Entropy
Property diagrams involving entropy
Pure substance: solids
Pure substance: compressed liquid
Pure substance: saturated liquid
Pure substance: two-phase mixture
Pure substance: saturated vapour
Pure substance: superheated vapour
Pure substance: phase change processes
T dS equations
Ideal gas: change in entropy
Incompressible substance: change in entropy
Internally reversible processes
Directionality of processes
Entropy balance: closed systems
Entropy balance: open systems
Isentropic processes
Isentropic efficiencies
Entropy of mixing and solution
Exergy
Exergy of a system
Exergy transfer by heat, work and mass
Exergy balance: closed system
Exergy balance: open system
Exergetic efficiency
Thermoeconomics
Vapour power cycles
Rankine cycle
Rankine cycle: superheat, reheat and supercritical
Rankine cycle: regeneration
Gas power cycles
Otto cycle: air standard
Diesel cycle: air standard
Dual cycle: air standard
Stirling and Ericsson cycles
Brayton cycle: air standard
Brayton cycle: regeneration
Brayton cycle: regeneration, reheat and intercooling
Combined gas-vapour cycles
Aircraft turbine engines
Refrigeration and heat pump cycles
Vapour refrigeration and heat pump cycles
Gas refrigeration cycles
Ideal gas mixtures and psychrometrics
Describing mixture composition
Pressure, specific volume, temperature
Internal energy, enthalpy, entropy
Wet-bulb and dry-bulb temperatures
Specific and relative humidity of air
Dew point temperature
Psychrometric chart
Air conditioning process: mass and energy balance
Air conditioning process: conditioning moist air at constant composition
Air conditioning process: dehumidification
Air conditioning process: humidification
Air conditioning process: evaporative cooling
Air conditioning process: adiabatic mixing of two moist air streams
Air conditioning process: wet cooling towers
Reaction and combustion
Stoichiometry
Equilibrium constant
Heats of reaction
Heats of formation
Heats of combustion and heating value
Incomplete combustion
Adiabatic flame temperature
Explosion limits
Reactive energy balances: steady-state
Reactive energy balances: transient
Simultaneous reaction and phase equilibria
Entropy balance
Chemical exergy
Helmholtz free energy
Gibbs free energy
Departure functions
Internal energy departure function
Entropy departure function
Other departure functions
Phase equilibrium of pure substances
PVT behavior of pure substances
Clausius-Clapeyron equation
Vapour pressure estimation
Virial equation of state
Cubic equations of state
Generalized correlations for gases
Generalized correlations for liquids
Maxwell relations
Gibbs-Duhem equation
Fugacity and the fugacity coefficient
Calculating gas fugacity
Calculating liquid fugacity
Calculating solid fugacity
Multicomponent phase equilibrium
Energy changes due to mixing and solution
Binary vapour-liquid equilibrium (VLE)
Dew point and bubble point
Multicomponent vapour-liquid equilibrium (VLE)
Vapour-liquid-liquid equilibrium (VLLE)
Miscibility and liquid-liquid equilibrium (LLE)
Solubility and solid-liquid equilibrium (SLE)
Adsorption and solid-gas or solid-vapour equilibrium (SVE)
Mixing rules for equations of state
Fugacity and chemical potential from an equation of state
Activity models
Excess properties
Redlich-Kister and Margules models
Partial molar properties
van Laar model
Scatchard-Hildebrand theory
Flory-Huggins model
MOSCED and SSCED theories
UNIFAC
Wilson's equation
NRTL
UNIQUAC
Electrolyte solutions
Colligative properties
Speciation and dissociation constants
Redox reactions
Material and energy balances
Definitions
Unit systems, conversion of units, dimensional homogeneity <<< Arithmetic__Units__Conversions
Specific volume and density <<< Thermodynamics__Definitions__Specific volume and density
Flow rate <<< Thermodynamics__Continuity: open systems__Mass and volume flow rate
Chemical composition <<< Thermodynamics__Definitions__Chemical composition
Material balances on non-reactive processess
Degree of freedom analysis
Single unit
Multiunit
Recycle, purge and bypass
Material balances on reactive processess
Chemical reaction stoichiometry
Degree of freedom analysis
Reactive balances
Recycle, purge and bypass
Energy balances
Degree of freedom analysis
Non-reactive closed processes <<< Thermodynamics__First law: closed systems__First law: closed systems
Non-reactive open processes <<< Thermodynamics__First law: open systems__Steady-state
Reactive processes <<< Thermodynamics__Reaction and combustion__Reactive energy balances: steady-state
Coupled material and energy balances
Transient processes
Transient non-reactive material balances
Transient reactive material balances
Transient non-reactive open energy balances <<< Thermodynamics__First law: open systems__Transient
Transient reactive energy balances <<< Thermodynamics__Reaction and combustion__Reactive energy balances: transient
Transient coupled material and energy balances
Operations research
Linear programming
Constrained optimization - planar
Simplex method
Algorithms
Bin packing
Scheduling
Traveling salesman
Set theory and logic
Operations on sets
Boolean operations on sets
Membership tables
Venn diagrams
Products
Power sets
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Relations between sets
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Properties of relations
Equivalence relations
Functions
Definition of function
Injective, surjective, bijective
Image and inverse image
Propositional logic
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Truth tables
Rules of inference
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Proofs
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Financial mathematics
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Equations of value
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Calculations
Conversion between decimals and percents
Applications
Irrational numbers
Interpreting irrational numbers
Simplify radical numbers
Rational exponents
Inequalities
Complex numbers
Conversion to a + bi form <<< Complex analysis__Arithmetic__Conversion to a + bi form
Addition/subtraction <<< Complex analysis__Arithmetic__Addition/subtraction
Multiplication <<< Complex analysis__Arithmetic__Multiplication
Complex conjugates <<< Complex analysis__Arithmetic__Complex conjugates
Division <<< Complex analysis__Arithmetic__Division
Multiple operations <<< Complex analysis__Arithmetic__Multiple operations
Modulus/norm <<< Complex analysis__Arithmetic__Modulus/norm
Conversion to/from polar form <<< Complex analysis__Arithmetic__Conversion to/from polar form
Powers and roots <<< Complex analysis__Arithmetic__Powers and roots
Other bases
Converting
Addition/subtraction
Multiplication
Division
Units
Conversions
Interpretation
Number theory
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Definitions
Division algorithm
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Units and zero divisors
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