Chapter 25 — Further Reading

An annotated guide for going deeper into parametric curves. Each entry says what it adds and where to look, with explicit section mapping for the two reference texts this book is measured against (see _continuity.md §8 and the appendix mappings).

Mapped to the standard texts

Stewart, J. (2021). Calculus: Early Transcendentals (9th ed.). Cengage. The closest companion to this chapter. - §10.1 — Curves Defined by Parametric Equations. Sketching, orientation, and eliminating the parameter (our §25.2), with the cycloid derived as a worked example. - §10.2 — Calculus with Parametric Curves. The slope $dy/dx = \dot y/\dot x$, the second derivative (with Stewart's own warning against $\ddot y/\ddot x$), arc length, and surface area of revolution — our §25.3–25.5, in the same order. Stewart's exercise sets in these two sections are the natural place to drill the mechanics beyond our Chapter 25 problem set. (Full chapter-by-chapter mapping in appendices/appendix-h-stewart-chapter-mapping.md.)

Strang, G., & Herman, E. Calculus, Volume 2 (OpenStax, free). Excellent, freely available, and application-rich. - §7.1 — Parametric Equations. Parametrization and orientation; the cycloid and its history; eliminating the parameter — our §25.1–25.2. - §7.2 — Calculus of Parametric Curves. Derivatives, tangent lines, arc length, and surface area — our §25.3–25.5, with extra worked projectile and area examples that complement our §25.6. (Full mapping in appendices/appendix-i-openstax-chapter-mapping.md.)

Apostol, T. M. (1967). Calculus, Volume I (2nd ed.). Wiley. A more rigorous voice: Apostol treats parametric curves alongside vector functions and is precise about when a parametrization is "regular" (no cusps), sharpening the §25.3 cusp discussion for math-major readers.

Spivak, M. (2008). Calculus (4th ed.). Publish or Perish. Develops curves in tandem with the careful definition of arc length as a supremum of inscribed polygons — the rigorous foundation under our intuitive "chop, approximate, sum" derivation of §25.4.

The cycloid and the calculus of variations

Stillwell, J. (2010). Mathematics and Its History (3rd ed.). Springer. The single best narrative source for §25.7: Galileo weighing paper cutouts, Roberval's area proof, Wren's $8r$ arc length, and Johann Bernoulli's 1696 brachistochrone challenge that drew Newton, Leibniz, and L'Hôpital into a public contest. Readable with only this chapter's background.

Simmons, G. F. (2007). Calculus Gems: Brief Lives and Memorable Mathematics (MAA). Its essays on the cycloid and the brachistochrone link the geometry to the physics of least time and gently motivate the Euler–Lagrange equation gestured at in our §25.7 Math Major Sidebar.

Lawrence, J. D. (1972). A Catalog of Special Plane Curves (Dover). A compact field reference: the cycloid, trochoid, astroid, cardioid, and Lissajous figures each given in parametric form with arc length and enclosed area — the practical companion to §25.8 when you need a classical curve's formula quickly and cheaply.

Projectile motion and trajectory physics

Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics (10th ed.). Wiley. The standard introductory treatment of two-dimensional motion. Its projectile chapter develops exactly the $x(t)$, $y(t)$ decomposition and the range and apex formulas of our §25.6 and Case Study 1, with abundant worked numbers.

Marion, J. B., & Thornton, S. T. (2003). Classical Dynamics of Particles and Systems (5th ed.). Cengage. Goes a level deeper, adding linear and quadratic air resistance and showing rigorously how the optimal launch angle slides below $45^\circ$ — the honest sequel to the vacuum model's closing caveat.

Hahn, A. J. (1998). Basic Calculus: From Archimedes to Newton to Its Role in Science (Springer). Traces the historical road from Galileo's discovery that the trajectory is a parabola to Newton's calculus, situating the §25.6 model in the story of how ballistics and calculus matured together.

Toward what comes next

For Chapter 26 (Polar): any of the Stewart §10.3–10.4 or OpenStax §7.3–7.4 readings preview how letting the parameter be an angle yields a specialized, beautiful calculus of area and length.

For Chapter 28 (Vector-valued functions): Apostol and Spivak both bridge directly from the plane curves of this chapter to $\mathbf{r}(t)$, velocity, speed, and curvature — the natural sequel once you are comfortable with $\dot y/\dot x$.

A practice recommendation

The single most useful habit: practice translating among the three descriptions of §25.10 — explicit, implicit, and parametric — for the same curve, and notice which makes a given question (slope? length? area?) easiest. Then implement two or three classical curves (cycloid, astroid, a Lissajous figure) in matplotlib using the §25.9 pattern and watch them draw as $t$ increases; seeing the orientation appear in real time fixes the path-not-graph idea better than any static figure.