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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 | |
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 | |
Power sets | |
Cardinality | |
Relations between sets | |
Subset | |
Properties of relations | |
Equivalence relations | |
Functions | |
Definition of function | |
Injective, surjective, bijective | |
Image and inverse image | |
Propositional logic | |
Translation | |
Operations on propositions | |
Truth tables | |
Rules of inference | |
Boolean circuits | |
First order logic | |
Predicates | |
Translation | |
Semantics of quantifiers | |
Proofs | |
Fuzzy logic | |
Puzzles | |
Pattern matching | |
Numeric | |
Non-numeric | |
Financial mathematics | |
Annuities | |
Income streams | |
Loans | |
Perpetuities | |
Sinking funds | |
Mixed methods | |
Bonds | |
Prices and coupon rates | |
Book value | |
Other bonds | |
Yield rates | |
Equations of value | |
Dollar weighted rate of return | |
Time weighted rate of return | |
Interest | |
Simple interest | |
Compound interest | |
Continuous interest | |
Force of interest | |
Effective and nominal rates of interest | |
Present and future value of money | |
Options | |
Introduction to options | |
Put-call parity | |
Binomial trees | |
Hedging strategies | |
Black-Scholes and the Greeks | |
Expected and contingent payments | |
Contingent payments | |
Expected payments | |
Equities | |
Introduction to stocks | |
Forwards and futures | |
Complex analysis | |
Arithmetic | |
Conversion to a + bi form | |
Addition/subtraction | |
Multiplication | |
Complex conjugates | |
Division | |
Multiple operations | |
Modulus/norm | |
Conversion to/from polar form | |
Powers and roots | |
Complex equations | |
Linear | |
Quadratic | |
Non-linear | |
Complex plane | |
Regions and domains | |
Complex functions | |
Complex functions as mappings | |
Limits | |
Analytic functions | |
Differentiability and analyticity | |
Harmonic functions | |
Entire functions | |
Applications | |
Elementary functions | |
Exponential function | |
Integration in the complex plane | |
Cauchy's theorem | |
Series and residues | |
Sequences | |
Taylor series | |
Laurent series | |
Zeroes and poles | |
Residues | |
Arithmetic | |
Integers | |
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Multiplication | |
Integer division | |
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Fractions/rational numbers | |
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Interpreting fractions | |
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Addition/subtraction | |
Multiplication | |
Division | |
Multiple operations | |
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Absolute value | |
Ratio/proportions | |
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Decimals | |
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Addition/subtraction | |
Multiplication | |
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Multiple operations | |
Converting between fractions and decimals | |
Inequalities | |
Absolute value | |
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Percents | |
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Irrational numbers | |
Interpreting irrational numbers | |
Simplify radical numbers | |
Rational exponents | |
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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 | |
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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 | |
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Group axioms | |
Subgroups | |
Cyclic groups | |
Permutation groups | |
Product of groups | |
Cosets, Lagrange's theorem, and normality | |
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Homomorphisms | |
Group actions | |
Rings | |
Ring axioms | |
Units and zero divisors | |
Ideals and homomorphisms | |
Quotient rings and polynomial rings | |
Cryptography | |
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