Chapter 39 — Further Reading

This is the capstone, so the reading is organized to support building a portfolio: first general modeling craft, then one cluster per track. Each entry says why it earns a place on your shelf, not merely that it exists. Full citations for textbook anchors (Stewart, OpenStax, Spivak) are in the book's master bibliography; here we point outward to the modeling literature.

Mathematical modeling — the craft itself

  • Meerschaert, Mathematical Modeling (4th ed.). The best single bridge from calculus to modeling. It is organized exactly around the §39.2 cycle — assumptions, building, solving, validating, refining — with worked one-variable, multivariable, and dynamic models. Read this first if you have never built a model end-to-end.
  • Bender, An Introduction to Mathematical Modeling (Dover, inexpensive). Short, opinionated, and strong on the judgment side: what to include, what to ignore, when a model is "good enough." Pairs naturally with §39.8 (validation and honesty).
  • Gershenfeld, The Nature of Mathematical Modeling. Broader and more advanced; valuable for seeing ODE models, optimization, and data fitting treated under one cover, which is exactly the §39.7 "one toolkit, four worlds" thesis.
  • Strogatz, Nonlinear Dynamics and Chaos (2nd ed.). The friendliest serious treatment of differential-equation models, phase portraits, and bifurcations. Chapters on population and epidemic models extend the SIR analysis of §39.3 into the qualitative theory.

Biology track — SIR, epidemics, population dynamics

  • Kermack & McKendrick (1927), "A Contribution to the Mathematical Theory of Epidemics," Proc. R. Soc. A. The original SIR paper and the source of the threshold theorem and the final-size relation derived in §39.3.5. Surprisingly readable; worth seeing the model in its birthplace.
  • Hethcote (2000), "The Mathematics of Infectious Diseases," SIAM Review 42(4). The standard modern survey: $R_0$, herd immunity, endemic equilibria, and dozens of compartmental variants. Your single best reference for SIR extensions (§39.3.6).
  • Keeling & Rohani, Modeling Infectious Diseases in Humans and Animals. The graduate reference. Its chapters on age structure and contact matrices $C_{ij}$ map directly onto the age-structured extension in §39.3.6, including the real COVID-era models.
  • Murray, Mathematical Biology I & II. Encyclopedic. Beyond epidemics, it covers logistic growth, predator–prey, and pattern formation — the other Biology-track portfolio options from the project description.

Economics track — production, optimization, welfare

  • Chiang & Wainwright, Fundamental Methods of Mathematical Economics (4th ed.). The classic. Lagrange multipliers, constrained optimization, and comparative statics — the exact toolkit of §39.4 — taught for economists, with the Cobb–Douglas case worked in full.
  • Varian, Microeconomic Analysis (3rd ed.). Graduate microeconomics; the authoritative treatment of production functions, cost minimization, and consumer/producer surplus (§39.4's welfare integral, Chapter 18).
  • Simon & Blume, Mathematics for Economists. Bridges calculus and economics rigorously; strong on why the Lagrange condition $\nabla Q = \lambda\nabla g$ is the tangency of an isoquant and a budget line (§39.4.2).

Physics track — orbits, motion, energy

  • Curtis, Orbital Mechanics for Engineering Students (4th ed.). The practical reference behind §39.5: vis-viva, Hohmann transfers, $\Delta v$ budgets, and the $J_2$ perturbation corrections mentioned as the model's limitation. Includes runnable code.
  • Bate, Mueller & White, Fundamentals of Astrodynamics (Dover, inexpensive). The classic, no-frills derivation of the two-body problem and transfer orbits; excellent for re-deriving vis-viva from energy conservation (the integral/work connection of §39.5.1).
  • Taylor, Classical Mechanics. For the broader physics track (oscillations, projectiles, central forces); ties the vector ODE $\ddot{\mathbf r} = -\mu\,\mathbf r/\|\mathbf r\|^3$ (Chapters 19, 28) back to Newton's laws and energy methods.

Data Science track — gradient descent, fitting, optimization

  • Boyd & Vandenberghe, Convex Optimization (free PDF, Cambridge). The definitive account of why convexity guarantees gradient descent reaches the global minimum — the theory behind the linear-fit validation in Case Study 2.
  • Goodfellow, Bengio & Courville, Deep Learning (free online). Chapters 4 and 6 cover gradient-based optimization and backpropagation, showing precisely how the three-line update of §39.6 scales to neural networks (the multivariable chain rule of Chapter 30, layer by layer).
  • James, Witten, Hastie & Tibshirani, An Introduction to Statistical Learning (free PDF). The most accessible treatment of least squares, the bias–variance tradeoff, and cross-validation — the overfitting-and-honesty discipline of §39.8. Includes Python and R labs.
  • Nocedal & Wright, Numerical Optimization (2nd ed.). For going deeper than plain gradient descent: line search, momentum, and the quasi-Newton methods behind scipy.optimize.minimize used in the economics verification (§39.4.3).

Software and reproducibility

  • The SciPy and NumPy documentation — especially scipy.integrate.solve_ivp (the SIR and orbital ODE solvers), scipy.optimize.minimize/brentq (the economics optimum and the SIR final-size root), and numpy.linalg.lstsq (the exact least-squares cross-check). Your portfolio's code should cite the exact functions it calls.
  • Project Jupyter documentation. A reproducible notebook is the modern modeling deliverable (§39.9, item 5). Make it run for someone else on the first try.