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| Numbers, variables, and units | |
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| Concepts | |
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| Real numbers | |
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| Factorization, factors, and factorials | |
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| Decimal representation of numbers | |
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| Variables | |
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| The algebra of real numbers | |
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| Complex numbers | |
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| Units | |
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| Exercises | |
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| Algebraic functions | |
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| Concepts | |
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| Graphical representation of functions | |
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| Factorization and simplification of expressions | |
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| Inverse functions | |
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| Polynomials | |
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| Rational functions | |
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| Partial fractions | |
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| Solution of simultaneous equations | |
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| Exercises | |
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| Transcendental functions | |
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| Concepts | |
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| Trigonometric functions | |
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| Inverse trigonometric functions | |
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| Trigonometric relations | |
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| Polar coordinates | |
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| The exponential function | |
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| The logarithmic function | |
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| Values of exponential and logarithmic functions | |
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| Hyperbolic functions | |
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| Exercises | |
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| Differentiation | |
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| Concepts | |
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| The process of differentiation | |
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| Continuity | |
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| Limits | |
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| Differentiation from first principles | |
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| Differentiation by rule | |
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| Implicit functions | |
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| Logarithmic differentiation | |
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| Successive differentiation | |
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| Stationary points | |
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| Linear and angular motion | |
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| The differential | |
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| Exercises | |
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| Integration | |
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| Concepts | |
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| The indefinite integral | |
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| The definite integral | |
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| The integral calculus | |
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| Uses of the integral calculus | |
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| Static properties of matter | |
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| Dynamics | |
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| Pressure-volume work | |
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| Exercises | |
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| Methods of integration | |
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| Concepts | |
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| The use of trigonometric relations | |
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| The method of substitution | |
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| Integration by parts | |
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| Reduction formulas | |
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| Rational integrands. The method of partial fractions | |
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| Parametric differentiation of integrals | |
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| Exercises | |
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| Sequences and series | |
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| Concepts | |
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| Sequences | |
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| Finite series | |
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| Infinite series | |
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| Tests of convergence | |
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| MacLaurin and Taylor series | |
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| Approximate values and limits | |
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| Operations with power series | |
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| Exercises | |
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| Complex numbers | |
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| Concepts | |
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| Algebra of complex numbers | |
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| Graphical representation | |
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| Complex functions | |
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| Euler's formula | |
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| Periodicity | |
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| Evaluation of integrals | |
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| Exercises | |
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| Functions of several variables | |
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| Concepts | |
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| Graphical representation | |
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| Partial differentiation | |
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| Stationary points | |
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| The total differential | |
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| Some differential properties | |
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| Exact differentials | |
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| Line integrals | |
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| Multiple integrals | |
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| The double integral | |
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| Change of variables | |
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| Exercises | |
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| Functions in 3 dimensions | |
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| Concepts | |
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| Spherical polar coordinates | |
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| Functions of position | |
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| Volume integrals | |
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| The Laplacian operator | |
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| Other coordinate systems | |
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| Exercises | |
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| First-order differential equations | |
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| Concepts | |
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| Solution of a differential equation | |
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| Separable equations | |
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| Separable equations in chemical kinetics | |
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| First-order linear equations | |
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| An example of linear equations in chemical kinetics | |
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| Electric circuits | |
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| Exercises | |
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| Second-order differential equations. Constant coefficients | |
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| Concepts | |
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| Homogeneous linear equations | |
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| The general solution | |
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| Particular solutions | |
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| The harmonic oscillator | |
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| The particle in a one-dimensional box | |
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| The particle in a ring | |
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| Inhomogeneous linear equations | |
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| Forced oscillations | |
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| Exercises | |
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| Second-order differential equations. Some special functions | |
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| Concepts | |
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| The power-series method | |
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| The Frobenius method | |
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| The Legendre equation | |
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| The Hermite equation | |
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| The Laguerre equation | |
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| Bessel functions | |
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| Exercises | |
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| Partial differential equations | |
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| Concepts | |
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| General solutions | |
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| Separation of variables | |
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| The particle in a rectangular box | |
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| The particle in a circular box | |
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| The hydrogen atom | |
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| The vibrating string | |
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| Exercises | |
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| Orthogonal expansions. Fourier analysis | |
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| Concepts | |
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| Orthogonal expansions | |
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| Two expansions in Legendre polynomials | |
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| Fourier series | |
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| The vibrating string | |
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| Fourier transforms | |
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| Exercises | |
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| Vectors | |
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| Concepts | |
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| Vector algebra | |
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| Components of vectors | |
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| Scalar differentiation of a vector | |
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| The scalar (dot) product | |
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| The vector (cross) product | |
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| Scalar and vector fields | |
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| The gradient of a scalar field | |
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| Divergence and curl of a vector field | |
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| Vector spaces | |
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| Exercises | |
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| Determinants | |
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| Concepts | |
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| Determinants of order 3 | |
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| The general case | |
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| The solution of linear equations | |
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| Properties of determinants | |
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| Reduction to triangular form | |
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| Alternating functions | |
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| Exercises | |
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| Matrices and linear transformations | |
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| Concepts | |
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| Some special matrices | |
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| Matrix algebra | |
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| The inverse matrix | |
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| Linear transformations | |
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| Orthogonal matrices and orthogonal transformations | |
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| Symmetry operations | |
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| Exercises | |
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| The matrix eigenvalue problem | |
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| Concepts | |
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| The eigenvalue problem | |
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| Properties of the eigenvectors | |
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| Matrix diagonalization | |
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| Quadratic forms | |
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| Complex matrices | |
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| Exercises | |
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| Numerical methods | |
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| Concepts | |
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| Errors | |
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| Solution of ordinary equations | |
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| Interpolation | |
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| Numerical integration | |
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| Methods in linear algebra | |
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| Gauss elimination for the solution of linear equations | |
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| Gauss-Jordan elimination for the inverse of a matrix | |
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| First-order differential equations | |
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| Systems of differential equations | |
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| Exercises | |
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| Probability and statistics | |
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| Concepts | |
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| Descriptive statistics | |
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| Frequency and probability | |
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| Combinations of probabilities | |
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| The binomial distribution | |
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| Permutations and combinations | |
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| Continuous distributions | |
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| The Gaussian distribution | |
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| More than one variable | |
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| Least squares | |
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| Sample statistics | |
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| Exercises | |
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| Standard integrals | |
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| Solutions to exercises | |
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| Index | |