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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 | |