Basics

• Functions and their properties

  Functions built from the ground up: the genesis of the concept (Oresme, Euler, Dirichlet), from relation to function, the vertical line test, domain and codomain, a full catalogue of elementary functions with derivations, properties (monotonicity, extrema, parity, periodicity, convexity, asymptotes), graph transformations, composition, the inverse function, injection/surjection/bijection, functions of several variables in 3D, and the idea of the fourth dimension with the generalisation to $\mathbb{R}^n$. Every concept and every example with its own figure.

• Limits and continuity

  Limits built from the ground up: genesis (Zeno's paradoxes, the infinitesimals of Newton and Leibniz, the rigour of Cauchy and Weierstrass), the limit of a sequence with the ε–N definition, the limit of a function with the ε–δ definition, Heine's definition, the laws of limits with justification, one-sided limits, indeterminate forms, the squeeze theorem with proof, the limit of sin x / x, limits at infinity and the number e, continuity and its types, the intermediate value theorem, limits of functions of several variables in 3D, and the generalisation of continuity to ℝⁿ. Every concept and every example with its own figure.

• Derivatives — the rate of change

  The derivative built from the ground up: genesis (Fermat's tangent problem, Newton's fluxions, Leibniz's dx), the difference quotient and the limit, the derivative from the definition step by step, differentiability versus continuity (proof), the rules of differentiation with a derivation of the product rule, the chain rule, derivatives of elementary functions, higher-order derivatives, monotonicity and convexity, Fermat's theorem and Lagrange's mean value theorem, Taylor's formula, economic applications (profit maximisation, elasticity), partial derivatives and the gradient in 3D, and the generalisation to ℝⁿ. Every concept and every example with its own figure.

• Integrals — summing infinitely many terms

  The integral built from the ground up: genesis (Archimedes' method of exhaustion, the problem of area, Newton and Leibniz), the Riemann sum with lower and upper sums, the definite integral as area, the antiderivative and the indefinite integral, the fundamental theorem of calculus with justification, integration by substitution and by parts (with geometric interpretation), improper integrals, economic applications (total cost, consumer surplus, present value), double integrals as volume in 3D, and the generalisation to ℝⁿ. Every concept and every example with its own figure.

• Linear algebra — vectors and matrices

  Linear algebra built from the ground up: genesis (systems of equations, Gaussian elimination, Cayley's matrices, determinants), vectors and their operations, the dot product, norm and angle, linear combination and basis, matrices and multiplication with a full computation, the matrix as a linear transformation, the determinant as area, the inverse matrix, eigenvalues and eigenvectors, the geometry of systems of equations, regression in matrix notation and orthogonal projection in 2D and 3D, and the generalisation to ℝⁿ. Every concept and every example with its own figure.

• Descriptive statistics

  Descriptive statistics built from the ground up: genesis (Graunt, Quetelet, Galton, Pearson), types of variables, measures of location (the mean as a balance point, median, mode) with an analysis of outliers, measures of variability (variance, standard deviation, coefficient of variation), quantiles and the box plot, skewness and kurtosis, the correlation coefficient, standardisation, and multivariate data in ℝⁿ. Every concept and every example with its own figure.

• Variance and standard deviation

  Variance and standard deviation built from the ground up: the genesis of the concept of spread, why the mean is not enough, the derivation of the formula step by step (why square), variance as the average area of squares of deviations, the standard deviation and its interpretation, population versus sample and Bessel's correction with justification, Chebyshev's inequality, covariance and the covariance ellipse, and the generalisation to ℝⁿ. Every concept and every example with its own figure.

• Correlation and the Pearson coefficient

  Correlation built from the ground up: genesis (Galton, regression to the mean, Pearson), the definition of the coefficient, the sign of the product of deviations, the picture of r on scatter plots, computation step by step, the geometric interpretation r = cos θ in ℝⁿ, the restriction to linear relationships, Anscombe's quartet, correlation versus causation (the confounder), correlation versus the regression slope, and Spearman's rank correlation. Every concept and every example with its own figure.

• Probability distributions

  Probability distributions built from the ground up: the genesis of probability theory, discrete and continuous random variables, the density function and the cumulative distribution function, the normal distribution, Student's t-distribution and its heavy tails, the chi-square distribution as a sum of squared normal variables, the F-distribution, the relationships between distributions, applications in hypothesis testing, and the generalisation to ℝⁿ. Every concept and every example with its own figure.

• The normal distribution

  The normal (Gaussian) distribution built from the ground up: genesis (de Moivre, Laplace, Gauss, Quetelet), the origin of the bell curve as a limit of the binomial distribution, the density formula and the role of the parameters μ and σ, the 68-95-99.7 rule, standardisation with proof, the z-score, tails and critical values, the bivariate distribution in 3D, elliptical contours, and the multivariate distribution in ℝⁿ. Every concept and every example with its own figure.

• The central limit theorem

  The central limit theorem built from the ground up: genesis (the Galton board, the de Moivre–Laplace theorem, Lindeberg and Lévy), the law of large numbers, the statement of the theorem, an illustration of the convergence of the distribution of the mean to the normal regardless of the population, the standard error and the square-root-of-n rule, the multivariate CLT in 3D, the significance for econometric inference, and verification by simulation. Every concept and every example with its own figure.

• The Method of Least Squares: History, Theory and Proofs

  A comprehensive account of the method of least squares: from the eighteenth-century problem of the figure of the Earth and its precursors (Cotes, Mayer, Bošković), through Legendre's first publication (1805), Gauss's original derivation (1809) and Laplace's probabilistic foundation, to the Gauss-Markov theorem (1823) and the birth of regression — with dates, names, original-language texts, full proofs and portraits.