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 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 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 Inhomogeneous 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 Basic concepts Circuit elements System of units Capacitance and inductance Capacitor and inductor combinations Capacitors Inductors Magnetically coupled networks Mutual inductance The ideal transformer Nodal and loop analysis techniques Loop analysis Nodal analysis Operational amplifiers Fundamental op-amp circuits Polyphase circuits Source/load connections Resistive circuits Circuits with dependent sources Circuits with series-parallel resistor combinations Kirchhoff's law Ohm's law Series and parallel resistor combinations Single-loop circuits Single-node-pair circuits 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 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Multiplication Division Multiple operations Inequalities Absolute value Ratio/proportions Estimation Applications Decimals Interpreting decimals Addition/subtraction Multiplication Division Multiple operations Converting between fractions and decimals Inequalities Absolute value Estimation Rounding Applications Percents 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 Divisibility Definitions Division algorithm Prime factorization GCDs and LCMs Congruences Modular arithmetic Linear congruences Fast powering Chinese remainder theorem Multiplicative orders Fermat's little theorem Diophantine equations Pythagorean triples Fermat's last theorem Abstract algebra Fields and polynomials Fields Polynomials Groups Group axioms Subgroups Cyclic groups Permutation groups Product of groups Cosets, Lagrange's theorem, and normality Quotient groups Homomorphisms Group actions Rings Ring axioms Units and zero divisors Ideals and homomorphisms Quotient rings and polynomial rings Cryptography Classic ciphers Shift cipher Affine cipher Rail fence cipher Real analysis Limits and accumulation 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