Math Foundations

Refreshers for everything the other books lean on.

A shared companion. Short, focused chapters on the mathematical machinery the other books invoke — calculus, vector calculus, complex exponentials, ODEs, PDEs, Fourier, linear algebra, dimensional analysis. Each chapter is a single page: link in from anywhere, scan, return.

Chapters

Chapter 0Pre-calculus: trigonometry and logarithmsRadians, the unit circle, identities; logs, exponentials, and decibel thinking
Chapter 1Single-variable calculusDerivatives, integrals, Taylor expansion — the working subset
Chapter 2Partial derivatives and vector calculusGradient, divergence, curl, the Laplacian
Chapter 3Complex exponentials and phasorsEuler's formula and why we use it for oscillations
Chapter 4Linear algebraVectors, matrices, eigenvalues, inner products, the spectral theorem
Chapter 5Linear ordinary differential equationsFirst- and second-order, homogeneous and forced
Chapter 6Linear partial differential equationsSeparation of variables and characteristics
Chapter 7Fourier series and the Fourier transformSinusoids as a basis, frequency as a dual coordinate
Chapter 8Dimensional analysisUnits, scaling, order-of-magnitude estimation
Chapter 9Numerical methodsHow the simulations on this site actually work
Chapter 10Statistics and probabilityRandom variables, the Gaussian, Brownian motion, Poisson, Bayes
Chapter 11Chaos and nonlinear dynamicsSensitive dependence, bifurcations, strange attractors, Lyapunov exponents
Chapter 12Calculus of variationsExtremising functionals — the Euler–Lagrange equation and the principle of least action
Chapter 13Operators and observablesLinear operators on function spaces: adjoints, commutators, ladder operators, and canonical quantization
Chapter FCheat sheetEvery key formula, in order, at a glance
GlossaryGlossaryTerms used in this book
Chapter HHistoryA chronological narrative
Chapter SStudySpaced-repetition review
BibliographyBibliographySources and further reading