Math Foundations

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.

For physical refreshers — mechanics, thermodynamics, fluid mechanics, elasticity, surface tension, kinetic theory — see the companion volume Physics Foundations.

Use it like a phrasebook. If a passage elsewhere in the bookshelf invokes a mathematical move you haven’t seen recently, follow the inline link, skim the relevant chapter, and resume the main flow. None of the entries assume you have never encountered the material; they assume you might want to remember it quickly.

Chapters

  1. Chapter 1Single-variable calculusDerivatives, integrals, Taylor expansion — the working subset
  2. Chapter 2Partial derivatives and vector calculusGradient, divergence, curl, the Laplacian
  3. Chapter 3Complex exponentials and phasorsEuler's formula and why we use it for oscillations
  4. Chapter 4Linear algebraVectors, matrices, eigenvalues, inner products, the spectral theorem
  5. Chapter 5Linear ordinary differential equationsFirst- and second-order, homogeneous and forced
  6. Chapter 6Linear partial differential equationsSeparation of variables and characteristics
  7. Chapter 7Fourier series and the Fourier transformSinusoids as a basis, frequency as a dual coordinate
  8. Chapter 8Dimensional analysisUnits, scaling, order-of-magnitude estimation
  9. Chapter 9Numerical methodsHow the simulations on this site actually work
  10. Chapter 10Statistics and probabilityRandom variables, the Gaussian, Brownian motion, Poisson, Bayes
  11. GlossaryGlossaryTerms used in this book
  12. BibliographyBibliographySources and further reading

In prose elsewhere on the bookshelf, an inline refresher link looks like:

The complex-exponential trick ([refresher](/foundations/complex-exponentials))
makes the driven oscillator's solution a one-line calculation.

The link can appear on any term: an operator, a technique, a theorem. The reader who already knows the material ignores the link. The reader who needs it follows, reads, and returns.