Appendix H — Stewart Chapter-to-Chapter Mapping

If you are coming to this book from James Stewart's Calculus: Early Transcendentals (9th edition, Cengage) — whether as an instructor adopting a new text mid-sequence or as a student cross-referencing an assigned reading — this appendix is your translation table. It answers the practical question every switcher asks first: "I used to cover Stewart §4.7 here; where does that live in this book?"

The two books cover the same standard single- and multivariable calculus core, so the mapping is dense and the overlap is high. But the organizing logic differs. Stewart groups material into 16 large chapters built around mathematical machinery (e.g., "Differentiation Rules," "Techniques of Integration"). This book uses 40 shorter chapters built around concepts and the order in which understanding accrues — separating, for instance, the limit (Ch. 3) from continuity (Ch. 4), and the definite integral as a limit of sums (Ch. 13) from the Fundamental Theorem that makes it computable (Ch. 14). The result is a finer-grained sequence that maps to Stewart cleanly but rarely one-to-one.

How to read the tables. Section numbers below refer to Stewart, Early Transcendentals, 9th edition. Section numbering shifts between editions (the 7th, 8th, and 9th editions renumber several sections, and the non-ET "Early Transcendentals vs. standard" split differs again), so treat any specific decimal (e.g., "§3.10") as a 9e landmark rather than a universal address. Chapter-level numbers (1–16) are stable across recent ET editions. Where this book consolidates several Stewart sections, or splits one Stewart section across several chapters, the Notes column says so.

A reverse table (Stewart → this book) and a summary of where the two books genuinely diverge follow the main mapping.

Main Mapping: This Book → Stewart ET 9e

Part I — Foundations (Ch. 1–5)

This book (Ch. # — Title) Corresponding Stewart 9e section(s) Notes
1 — Why Calculus Preface / Ch. 1 front matter; "A Preview of Calculus" Motivational chapter with no direct Stewart analog. Stewart's brief "Preview of Calculus" is the closest match; this book expands it into a full chapter framing change, accumulation, and the four anchor problems.
2 — Functions & Models §1.1–1.5 Direct parallel to Stewart Ch. 1 (functions, transformations, exponentials, inverse/log, models). Stewart also folds parametric/curve-fitting previews here; this book defers parametrics to Ch. 25.
3 — The Limit §2.1–2.3, §2.6 (limits at infinity) Stewart bundles limits and derivatives into one chapter (Ch. 2); this book splits them. Includes the ε–δ definition (Stewart §2.4) at the "Formal" rigor level — see note on Ch. 3/4 split below.
4 — Continuity §2.5, plus §2.4 (precise definition) Stewart treats continuity as one section (§2.5); this book gives it a full chapter and pulls the ε–δ material (Stewart §2.4) alongside limits/continuity. IVT lives here, as in Stewart.
5 — Rates of Change §2.7–2.8 Derivative as rate of change and the derivative as a function. Stewart §2.7 (derivatives and rates) and §2.8 (the derivative as a function). This book foregrounds the rate-of-change interpretation before the rules, consistent with its "mathematics of change" theme.

Part II — Differentiation (Ch. 6–12)

This book (Ch. # — Title) Corresponding Stewart 9e section(s) Notes
6 — The Derivative §2.7–2.8 (definition), §3.1 Formal definition and basic interpretation; overlaps with Ch. 5. This book introduces gradient descent here as an anchor example (derivative = direction to step downhill); Stewart has no comparable thread.
7 — Differentiation Rules §3.1–3.6 Power, product, quotient, chain, trig, exponential/log derivatives. Strong one-to-one correspondence with the bulk of Stewart Ch. 3.
8 — Implicit & Related Rates §3.5 (implicit), §3.9 (related rates) This book pairs implicit differentiation with related rates in one chapter; Stewart separates them (§3.5 sits early in Ch. 3, §3.9 near the end). Logarithmic differentiation (Stewart §3.6) sits in Ch. 7 here.
9 — Applications of Derivatives §3.7–3.8 (rates in science), §4.1–4.5 Maxima/minima, MVT, curve sketching, l'Hôpital. Covers Stewart §4.1 (max/min), §4.2 (MVT), §4.3 (shape of a graph), §4.4 (l'Hôpital), §4.5 (curve sketching). Optimization word problems split out to Ch. 10.
10 — Optimization §4.7 Applied optimization word problems. Direct match to Stewart §4.7; this book adds more economics/data-science framing (marginal analysis, loss minimization) than Stewart's mostly geometric/physical set.
11 — Linear Approximation & Newton's Method §3.10 (linear approx & differentials), §4.8 (Newton's method) Combines Stewart's §3.10 and §4.8. Differentials introduced here. Euler's formula is mentioned here as a forward anchor (full derivation in Ch. 24).
12 — Antiderivatives §4.9 Antiderivatives as the reverse of differentiation. Direct match to Stewart §4.9. Placed at the end of Part II (differentiation) here, as in Stewart, bridging into integration.

Part III — Integration (Ch. 13–19)

This book (Ch. # — Title) Corresponding Stewart 9e section(s) Notes
13 — The Definite Integral §5.1–5.2 Area, distance, Riemann sums, the definite integral as a limit of sums. Direct match to Stewart §5.1–5.2. Introduces the area-under-the-normal-curve anchor.
14 — FTC §5.3–5.4 The Fundamental Theorem (both parts), indefinite integrals, net change. This book treats FTC as its conceptual keystone (see continuity theme 3); Stewart §5.3 (FTC) and §5.4 (indefinite integrals / net change theorem).
15 — Integration Techniques I (u-sub, parts) §5.5 (substitution), §7.1 (parts) Substitution rule and integration by parts. Stewart places substitution in Ch. 5 (§5.5) and parts at the start of Ch. 7 (§7.1); this book unites them as "techniques I."
16 — Integration Techniques II (trig, trig sub, partial fractions, numerical) §7.2–7.5, §7.7 Trig integrals (§7.2), trig substitution (§7.3), partial fractions (§7.4), strategy (§7.5), approximate/numerical integration (§7.7). Tables/CAS (§7.6) folded into the Python thread.
17 — Improper Integrals §7.8 Direct match to Stewart §7.8. Given its own chapter here for emphasis on convergence (sets up series in Part IV).
18 — Applications of Integration §6.1–6.5, §8.1–8.5 Areas between curves, volumes (disks/shells), work, average value (Stewart Ch. 6), plus arc length, surface area, physics/engineering and probability applications (Stewart Ch. 8). Consolidates two Stewart chapters into one broad applications chapter.
19 — Differential Equations §9.1–9.5 (and §3.8 exponential growth) Modeling with ODEs, direction fields, separable equations, exponential/logistic growth, linear equations. Stewart Ch. 9. This book develops the SIR epidemic model in full here; Stewart covers logistic growth (§9.4) but not SIR. Series solutions (Stewart §17, in some editions) not covered.

Part IV — Sequences & Series (Ch. 20–24)

This book (Ch. # — Title) Corresponding Stewart 9e section(s) Notes
20 — Sequences §11.1 Sequences and their limits. Direct match to Stewart §11.1.
21 — Series §11.2 Series, partial sums, geometric and telescoping series, the test for divergence. Stewart §11.2.
22 — Convergence Tests §11.3–11.7 Integral test, comparison tests, alternating series, ratio/root tests, strategy. Consolidates Stewart §11.3 (integral), §11.4 (comparison), §11.5 (alternating), §11.6 (ratio/root), §11.7 (strategy).
23 — Power & Taylor Series §11.8–11.10 Power series, representing functions as series, Taylor and Maclaurin series. Stewart §11.8–11.10. Returns to the normal-curve anchor (Taylor approximation of $e^{-x^2}$).
24 — Applications of Series §11.11 Applications of Taylor polynomials (approximation, error bounds). Stewart §11.11. Climaxes the Euler's-formula anchor ($e^{i\pi}+1=0$ via Taylor series) — Stewart mentions the connection only briefly.

Part V — Parametric & Polar (Ch. 25–27)

This book (Ch. # — Title) Corresponding Stewart 9e section(s) Notes
25 — Parametric Curves §10.1–10.2 Curves defined parametrically; calculus with parametric curves (tangents, area, arc length). Stewart §10.1–10.2.
26 — Polar Coordinates §10.3–10.4 Polar coordinates and areas/lengths in polar form. Stewart §10.3–10.4.
27 — Conic Sections §10.5–10.6 Conics and conics in polar coordinates. Stewart §10.5–10.6. This book gives conics a standalone chapter; Stewart keeps them as the tail of Ch. 10.

Part VI — Multivariable Differential & Integral (Ch. 28–33)

This book (Ch. # — Title) Corresponding Stewart 9e section(s) Notes
28 — Vector-Valued Functions §13.1–13.4 (and §12.1–12.6 as needed) Vector functions, derivatives/integrals, arc length, curvature, motion in space. Stewart Ch. 13. The vector/geometry-of-space prerequisites (Stewart Ch. 12: dot/cross products, lines, planes, quadric surfaces) are folded in here as needed rather than given a separate chapter.
29 — Functions of Several Variables §14.1–14.4 Functions of several variables, limits/continuity, partial derivatives, tangent planes and linear approximation. Stewart §14.1–14.4.
30 — Multivariable Chain Rule & Gradient §14.5–14.6 Chain rule, directional derivatives, gradient vector. Stewart §14.5–14.6. Climaxes the gradient-descent anchor (gradient + ML); Stewart presents the gradient geometrically but not as an optimization-engine for machine learning.
31 — Optimization in Several Variables §14.7–14.8 Maxima/minima of multivariable functions; Lagrange multipliers. Stewart §14.7 (max/min) and §14.8 (Lagrange).
32 — Multiple Integrals §15.1–15.7 Double and triple integrals over rectangles/general regions; integrals in polar, cylindrical, and spherical coordinates; applications. Stewart §15.1–15.7.
33 — Change of Variables & Jacobians §15.9 (and §15.6–15.8 context) General change of variables in multiple integrals; the Jacobian. Stewart §15.9. (Cylindrical/spherical coordinate integration, Stewart §15.7–15.8, treated in Ch. 32 here as special cases.)

Part VII — Vector Calculus (Ch. 34–38)

This book (Ch. # — Title) Corresponding Stewart 9e section(s) Notes
34 — Vector Fields §16.1 Vector fields; divergence and curl introduced (Stewart §16.5 introduces div/curl, but this book previews them here). Stewart §16.1.
35 — Line Integrals (Green's) §16.2–16.4 Line integrals, the Fundamental Theorem for line integrals, Green's Theorem. Stewart §16.2 (line integrals), §16.3 (FTLI / conservative fields), §16.4 (Green).
36 — Surface Integrals §16.6–16.7 Parametric surfaces and areas; surface integrals. Stewart §16.6–16.7.
37 — Stokes' & Divergence Theorems §16.5, §16.8–16.9 Curl and divergence (§16.5), Stokes' Theorem (§16.8), the Divergence Theorem (§16.9).
38 — Generalizing FTC §16.10 (summary) + beyond Stewart §16.10 ("Summary") gathers the vector-calculus theorems as variants of the FTC. This book goes further, synthesizing them as instances of the generalized Stokes' theorem via differential forms — material outside Stewart's scope.

Part VIII — Synthesis (Ch. 39–40)

This book (Ch. # — Title) Corresponding Stewart 9e section(s) Notes
39 — Modeling Portfolio (no direct analog) Capstone integrating the progressive modeling project (SIR, gradient descent, etc.). Stewart has applied "Discovery/Lab" projects scattered through chapters but no cumulative modeling capstone.
40 — The Big Picture (no direct analog) Retrospective synthesis chapter tying together the six recurring themes and all four anchor examples. No Stewart counterpart.

Reverse Mapping: Stewart ET 9e → This Book

Stewart 9e chapter This book chapter(s)
1 — Functions and Models Ch. 2
2 — Limits and Derivatives Ch. 3 (limits), Ch. 4 (continuity), Ch. 5 (rates), Ch. 6 (derivative definition)
3 — Differentiation Rules Ch. 7 (rules), Ch. 8 (implicit & related rates), Ch. 11 (§3.10 linear approx)
4 — Applications of Differentiation Ch. 9 (§4.1–4.5), Ch. 10 (§4.7 optimization), Ch. 11 (§4.8 Newton), Ch. 12 (§4.9 antiderivatives)
5 — Integrals Ch. 13 (§5.1–5.2), Ch. 14 (§5.3–5.4 FTC), Ch. 15 (§5.5 substitution)
6 — Applications of Integration Ch. 18
7 — Techniques of Integration Ch. 15 (§7.1 parts), Ch. 16 (§7.2–7.5, 7.7), Ch. 17 (§7.8 improper)
8 — Further Applications of Integration Ch. 18
9 — Differential Equations Ch. 19
10 — Parametric Equations and Polar Coordinates Ch. 25 (parametric), Ch. 26 (polar), Ch. 27 (conics)
11 — Infinite Sequences and Series Ch. 20 (sequences), Ch. 21 (series), Ch. 22 (tests), Ch. 23 (power/Taylor), Ch. 24 (applications)
12 — Vectors and the Geometry of Space Folded into Ch. 28 (as prerequisite material)
13 — Vector Functions Ch. 28
14 — Partial Derivatives Ch. 29, Ch. 30 (chain rule/gradient), Ch. 31 (optimization/Lagrange)
15 — Multiple Integrals Ch. 32, Ch. 33 (change of variables/Jacobian)
16 — Vector Calculus Ch. 34, Ch. 35, Ch. 36, Ch. 37, Ch. 38

Topics This Book Covers That Stewart Does Not

  • Gradient descent and machine-learning optimization. Introduced as an anchor in Ch. 6 and developed through Ch. 30, the derivative/gradient is explicitly framed as the engine behind ML training. Stewart presents the gradient geometrically but never connects it to learning algorithms.
  • The SIR epidemic model. Developed fully in Ch. 19 and revisited in the Ch. 39 capstone. Stewart covers logistic growth (§9.4) but not compartmental disease models.
  • Broader application range. Biology, economics, and data science applications appear throughout with the same weight Stewart reserves mostly for physics and geometry (e.g., marginal analysis, pharmacokinetics, loss minimization, probability distributions).
  • Differential forms and the generalized Stokes' theorem. Ch. 38 synthesizes Green's, Stokes', and the Divergence Theorem as a single statement via differential forms. Stewart's §16.10 only informally notes the family resemblance.
  • Computational Python throughout. Every chapter verifies hand computation with sympy/numpy/scipy and visualizes with matplotlib. Stewart treats CAS and computing as optional supplements (the occasional "CAS required" exercise), not as a continuous thread.
  • Explicit three-rigor-level structure. Each concept is presented intuitively, computationally, and formally (with ε–δ proofs in Math Major Sidebars). Stewart interleaves rigor without flagging the levels.
  • Two synthesis chapters (39–40). A cumulative modeling capstone and a thematic retrospective have no Stewart counterpart.

Stewart Topics This Book Treats Differently or Lightly

  • Vectors and the geometry of space (Stewart Ch. 12). Dot/cross products, lines, planes, and quadric surfaces are not given a standalone chapter; they are introduced as prerequisite material inside Ch. 28. Instructors who spend a full week on Stewart Ch. 12 should plan to supplement.
  • Cylindrical and spherical coordinate integration (Stewart §15.7–15.8). Treated as special cases within Ch. 32 rather than as separate sections, with the general Jacobian change-of-variables (§15.9) elevated to its own chapter (Ch. 33).
  • Tables of integrals / CAS for integration (Stewart §7.6). Folded into the running Python thread rather than presented as a standalone technique.
  • Exercise volume. This book aims to match Stewart's conceptual breadth but carries fewer drill exercises per section; instructors who relied on Stewart's large drill banks may want to pair with the OpenStax problem sets (see Appendix I).
  • Conic sections (Stewart §10.5–10.6). Given a full standalone chapter (Ch. 27) here, slightly more prominence than Stewart's end-of-chapter placement.

Cross-reference note: a parallel mapping against OpenStax Calculus appears in Appendix I. Section-level Stewart numbers above are 9e landmarks; verify against your specific edition's table of contents before building a week-by-week crosswalk.