# Mathematical and historical notes

1. Pythagorean theorem: the marked corner is between the two perpendicular legs; square-root vincula are rendered with KaTeX. Earlier Babylonian knowledge is acknowledged.
2. Complex numbers: depressed cubic x³+px=q and Bombelli's x³=15x+4 are both shown. Paired cube roots 2±i sum to 4. Cardano and Bombelli have separate roles.
3. Logarithms: positive arguments, common base; multiplication becomes addition.
4. Calculus: x(t)=t²; finite secant converges to derivative 2t. Newton and Leibniz are credited.
5. Gravitation: inverse-square force; circular-orbit illustration assumes a central mass and is schematic.
6. Normal distribution: binomial N=100, p=.5 has mean 50 and standard deviation 5; the normal curve is an approximation.
7. Wave equation: ideal linear, nondispersive one-dimensional waves; fixed-end standing wave is introduced at its narration cue.
8. Euler: convex polyhedra, V−E+F=2. A face diagonal changes E and F by one each.
9. Fourier: harmonics 1,3,5 have amplitudes 1,.45,.2; the finite demonstration illustrates decomposition, not every possible signal.
10. Navier–Stokes: incompressible Newtonian fluid, constant density and viscosity. The divergence-free streamline field is an illustrative analytic field, not a numerical solution of the displayed PDE.
11. Maxwell: modern SI differential notation; the wave uses in-phase transverse E and B. Vacuum c=1/√(μ₀ε₀).
12. Thermodynamics: two isolated equal-heat-capacity bodies approach 315 K from 350/280 K. Energy is conserved and total entropy rises.
13. Relativity: equal opposite light pulses have zero net momentum. Positive mass lost is m_initial−m_final=2ε/c². The Sun ledger is schematic.
14. Quantum mechanics: the free Gaussian packet is separate from the hydrogen model. Hydrogen E_n≈−13.6 eV/n²; n=3→2 emits about 1.89 eV, 656 nm. Visible wavelengths are approximate.
15. Shannon: binary entropy uses base-two logarithms; H(.95)≈.286 bits. This is information uncertainty, not thermodynamic entropy.
16. Chaos: Lorenz's continuous system is distinct from the logistic map. Logistic stable, period-two and chaotic examples use R=2.8,3.2,4. Lorenz is not credited with inventing the logistic map.
17. Black–Scholes: European call, constant volatility/rate, ideal continuous trading and model assumptions. Example S=K=100, r=.05, σ=.2, τ=1: C≈10.4506, Δ≈.63683. Paths and hedge are illustrations, not forecasts or investment advice.

## References

- Maxwell and waves: https://openstax.org/books/university-physics-volume-2/pages/16-1-maxwells-equations-and-electromagnetic-waves
- Field phase: https://openstax.org/books/university-physics-volume-2/pages/16-2-plane-electromagnetic-waves
- Entropy: https://openstax.org/books/university-physics-volume-2/pages/4-6-entropy
- Hydrogen data: https://pml.nist.gov/PhysRefData/Handbook/Tables/hydrogentable1.htm
- Spectra: https://science.nasa.gov/mission/webb/science-overview/science-explainers/spectroscopy-101-how-absorption-and-emission-spectra-work/
- Black–Scholes: https://ocw.mit.edu/courses/15-450-analytics-of-finance-fall-2010/0d1260b891a96241316d883d4f5bfaec_MIT15_450F10_lec02.pdf
- Cubic history: https://www.nationalacademies.org/read/11540/chapter/8
- Biographies and individual image rights: IMAGE-ATTRIBUTION.md.