Appendix I — OpenStax Chapter-to-Chapter Mapping
OpenStax Calculus (Strang & Herman; published by OpenStax / Rice University) is a free, openly licensed (CC BY-NC-SA) three-volume calculus sequence. Because it costs nothing and is widely adopted, it is the natural free comparison for this textbook, and many instructors assign it as a no-cost companion. This appendix lets you cross-reference this book against OpenStax in either direction.
A note on OpenStax's structure and numbering. OpenStax splits the calculus sequence across three volumes, and each volume restarts its chapter numbering at 1. This is the single biggest source of confusion when cross-referencing, so all citations below name the volume first, then the chapter/section.
- Volume 1 (Chapters 1–6) covers a standard Calculus I course: functions and graphs, limits, derivatives, applications of derivatives, and an introduction to integration.
- Volume 2 (Chapters 1–7) covers Calculus II: integration techniques, applications of integration, differential equations, sequences and series, and parametric/polar curves. Volume 2 re-presents the integration material from the end of Volume 1 in its opening chapters, so there is deliberate overlap at the Volume 1 / Volume 2 seam.
- Volume 3 (Chapters 1–6) covers Calculus III (multivariable and vector calculus): parametric/polar (revisited), vectors in space, vector-valued functions, multivariable differentiation, multiple integration, and vector calculus.
OpenStax sections are numbered chapter.section within each volume (e.g., V1 §3.3 is Volume 1, Chapter 3, Section 3). Where a section number below is marked (verify), treat the chapter assignment as reliable but confirm the exact section number against your edition before printing it on a syllabus — OpenStax has issued multiple editions and section numbering has occasionally shifted.
Main mapping: this book → OpenStax
One row per chapter of this book. "OpenStax Volume & Section(s)" gives the closest corresponding location; "Notes" flags coverage differences.
Part I — Foundations (Ch. 1–5)
| This book (Ch. # — Title) | OpenStax Volume & Section(s) | Notes |
|---|---|---|
| 1 — Why Calculus | V1, Preface & §2.1 (intro framing) | No direct OpenStax equivalent; OpenStax opens with function review, not a motivational "why." This book's Ch. 1 is original framing. |
| 2 — Functions & Models | V1, Ch. 1 (Functions and Graphs), §1.1–1.5 | Strong match. OpenStax 1.4–1.5 cover inverse, exponential, log, and trig functions. |
| 3 — The Limit | V1, Ch. 2 (Limits), §2.2–2.4 | OpenStax 2.2 (intuitive limit), 2.3 (limit laws), 2.4 (continuity overlaps). |
| 4 — Continuity | V1, §2.4 (Continuity); IVT in §2.4 | This book separates continuity into its own chapter; OpenStax folds it into Ch. 2. |
| 5 — Rates of Change | V1, §2.1 (A Preview of Calculus) & §3.1 (Defining the Derivative) | OpenStax 2.1 tangent/velocity preview; 3.1 the derivative as a rate. |
Part II — The Derivative (Ch. 6–12)
| This book (Ch. # — Title) | OpenStax Volume & Section(s) | Notes |
|---|---|---|
| 6 — The Derivative | V1, Ch. 3 (Derivatives), §3.1–3.2 | Definition, derivative as a function, differentiability. |
| 7 — Differentiation Rules | V1, §3.3–3.6, §3.9 | Power/product/quotient (3.3), trig (3.5), chain rule (3.6), derivatives of exp/log (3.9). |
| 8 — Implicit & Related Rates | V1, §3.8 (Implicit Differentiation) & §4.1 (Related Rates) | Note: OpenStax places implicit diff in Ch. 3 but related rates in Ch. 4. |
| 9 — Applications of Derivatives | V1, §4.3–4.6 (Maxima/Minima, Mean Value Thm, derivative tests, curve sketching) | Broad match across several Ch. 4 sections. |
| 10 — Optimization | V1, §4.7 (Applied Optimization Problems) | Direct match. |
| 11 — Linear Approximation & Newton's Method | V1, §4.2 (Linear Approximations & Differentials) & §4.9 (Newton's Method) | L'Hôpital's Rule (V1 §4.8) is nearby; this book may treat it within Ch. 9 or 11. |
| 12 — Antiderivatives | V1, §4.10 (Antiderivatives) | OpenStax closes Ch. 4 with antiderivatives, just before the integration chapter. |
Part III — The Integral (Ch. 13–19)
| This book (Ch. # — Title) | OpenStax Volume & Section(s) | Notes |
|---|---|---|
| 13 — The Definite Integral | V1, §5.1–5.2 (Approximating Areas; The Definite Integral); also V2, §1.1–1.2 | Re-presented at start of Volume 2. |
| 14 — FTC | V1, §5.3 (Fundamental Theorem of Calculus); also V2, §1.3 | Substitution appears in V1 §5.5 / V2 §1.5. |
| 15 — Integration Techniques I | V2, §3.1 (Integration by Parts), §3.2 (Trig Integrals) | Volume 2's Ch. 3 is "Techniques of Integration." |
| 16 — Integration Techniques II | V2, §3.3 (Trig Substitution), §3.4 (Partial Fractions), §3.5 (Tables/CAS) | Numerical integration sits in V2 §3.6. |
| 17 — Improper Integrals | V2, §3.7 (Improper Integrals) | Direct match. |
| 18 — Applications of Integration | V1, Ch. 6 (Applications of Integration) & V2, Ch. 2 | Area between curves, volumes, arc length, work, physics/probability apps spread across V1 Ch. 6 and V2 Ch. 2. |
| 19 — Differential Equations | V2, Ch. 4 (Introduction to Differential Equations), §4.1–4.5 | Direction fields, separable, linear first-order; the SIR model here is an extension OpenStax does not develop in depth. |
Part IV — Sequences & Series (Ch. 20–24)
| This book (Ch. # — Title) | OpenStax Volume & Section(s) | Notes |
|---|---|---|
| 20 — Sequences | V2, §5.1 (Sequences) | Volume 2 Ch. 5 is "Sequences and Series." |
| 21 — Series | V2, §5.2 (Infinite Series) | Geometric and telescoping series. |
| 22 — Convergence Tests | V2, §5.3–5.6 (Divergence/Integral, Comparison, Ratio/Root, Alternating Series) | Multiple sections; absolute vs. conditional convergence in 5.5. |
| 23 — Power & Taylor Series | V2, Ch. 6 (Power Series), §6.1–6.4 | Power series, Taylor/Maclaurin series, operations. |
| 24 — Applications of Series | V2, §6.4 (Working with Taylor Series) | This book broadens applications (e.g., Euler's formula, numerical methods) beyond OpenStax's treatment. |
Part V — Curves & Coordinates (Ch. 25–27)
| This book (Ch. # — Title) | OpenStax Volume & Section(s) | Notes |
|---|---|---|
| 25 — Parametric Curves | V2, §7.1–7.2 (Parametric Equations; Calculus of Parametric Curves); also V3, §1.1–1.2 | OpenStax presents parametrics in both Vol 2 Ch. 7 and Vol 3 Ch. 1. |
| 26 — Polar Coordinates | V2, §7.3–7.5 (Polar Coordinates; Area & Arc Length; Conics in Polar); also V3, §1.3–1.5 | Same dual placement. |
| 27 — Conic Sections | V2, §7.5 (Conic Sections) (verify); also V3, §1.5 (verify) | Conics appear within the parametric/polar chapter rather than as a standalone chapter. |
Part VI — Multivariable Calculus (Ch. 28–33)
| This book (Ch. # — Title) | OpenStax Volume & Section(s) | Notes |
|---|---|---|
| 28 — Vector-Valued Functions | V3, Ch. 2 (Vectors in Space) & Ch. 3 (Vector-Valued Functions) | This book likely compresses OpenStax's vectors-in-space (V3 Ch. 2: dot/cross products, lines/planes) and vector-valued functions (V3 Ch. 3) into one chapter. |
| 29 — Functions of Several Variables | V3, §4.1–4.3 (Functions of Several Variables; Limits & Continuity; Partial Derivatives) | Volume 3 Ch. 4 is "Differentiation of Functions of Several Variables." |
| 30 — Multivariable Chain Rule & Gradient | V3, §4.4–4.6 (Tangent Planes/Linear Approx; Chain Rule; Directional Derivatives & Gradient) | Direct match. |
| 31 — Optimization in Several Variables | V3, §4.7–4.8 (Maxima/Minima; Lagrange Multipliers) | This book threads gradient descent / ML here, which OpenStax does not cover. |
| 32 — Multiple Integrals | V3, §5.1–5.5 (Double/Triple Integrals; polar, cylindrical, spherical) | Volume 3 Ch. 5 is "Multiple Integration." |
| 33 — Change of Variables & Jacobians | V3, §5.7 (Change of Variables in Multiple Integrals) | Direct match; moments/centers of mass are V3 §5.6. |
Part VII — Vector Calculus (Ch. 34–38)
| This book (Ch. # — Title) | OpenStax Volume & Section(s) | Notes |
|---|---|---|
| 34 — Vector Fields | V3, §6.1 (Vector Fields) | Volume 3 Ch. 6 is "Vector Calculus." |
| 35 — Line Integrals (Green's) | V3, §6.2–6.4 (Line Integrals; Conservative Fields/FTC for Line Integrals; Green's Theorem) | Combines several Ch. 6 sections. |
| 36 — Surface Integrals | V3, §6.5–6.6 (Divergence & Curl; Surface Integrals) | Direct match. |
| 37 — Stokes' & Divergence Theorems | V3, §6.7–6.8 (Stokes' Theorem; The Divergence Theorem) | Direct match; these close OpenStax Volume 3. |
| 38 — Generalizing FTC | No OpenStax equivalent | This book's differential-forms synthesis (unifying FTC, Green's, Stokes', Divergence as one generalized Stokes' theorem) goes beyond OpenStax entirely. |
Part VIII — Synthesis (Ch. 39–40)
| This book (Ch. # — Title) | OpenStax Volume & Section(s) | Notes |
|---|---|---|
| 39 — Modeling Portfolio | No OpenStax equivalent | Capstone integrating the four-track modeling project (biology/economics/physics/data science). Original to this book. |
| 40 — The Big Picture | No OpenStax equivalent | Historical and conceptual synthesis. Original to this book. |
Reverse mapping: OpenStax → this book (quick reference)
OpenStax Volume 1 (Calculus I)
| OpenStax Vol 1 chapter | This book |
|---|---|
| 1. Functions and Graphs | Ch. 2 |
| 2. Limits | Ch. 3–4 |
| 3. Derivatives | Ch. 5–8 |
| 4. Applications of Derivatives | Ch. 8–12 |
| 5. Integration | Ch. 13–14 |
| 6. Applications of Integration | Ch. 18 |
OpenStax Volume 2 (Calculus II)
| OpenStax Vol 2 chapter | This book |
|---|---|
| 1. Integration (review) | Ch. 13–14 |
| 2. Applications of Integration | Ch. 18 |
| 3. Techniques of Integration | Ch. 15–17 |
| 4. Introduction to Differential Equations | Ch. 19 |
| 5. Sequences and Series | Ch. 20–22 |
| 6. Power Series | Ch. 23–24 |
| 7. Parametric Equations and Polar Coordinates | Ch. 25–27 |
OpenStax Volume 3 (Calculus III)
| OpenStax Vol 3 chapter | This book |
|---|---|
| 1. Parametric Equations and Polar Coordinates (revisited) | Ch. 25–27 |
| 2. Vectors in Space | Ch. 28 |
| 3. Vector-Valued Functions | Ch. 28 |
| 4. Differentiation of Functions of Several Variables | Ch. 29–31 |
| 5. Multiple Integration | Ch. 32–33 |
| 6. Vector Calculus | Ch. 34–37 |
(Note: this book's Ch. 38, 39, and 40 have no OpenStax counterpart — see below.)
What this book adds beyond OpenStax
OpenStax Calculus is mathematically complete and an excellent free baseline. This book is designed to exceed it on motivation, application breadth, and computation rather than to add new "core" calculus topics. Specifically:
- Application breadth across disciplines. OpenStax leans heavily on physics and geometry for its applications. This book threads four parallel modeling tracks — biology, economics, physics, and data science — so every tool is motivated from multiple directions. Examples like consumer/producer surplus, population dynamics, and curve fitting recur throughout.
- Gradient descent and machine learning (Ch. 31). The gradient is presented not only as the direction of steepest ascent but as the engine of modern optimization and ML training. OpenStax stops at the classical gradient and Lagrange multipliers.
- The SIR epidemic model (Ch. 19). This book develops a full coupled-ODE epidemiological model as an anchor application. OpenStax's differential-equations chapter stays with classical single-equation examples (growth/decay, logistic, simple physics).
- The differential-forms synthesis chapter (Ch. 38, "Generalizing FTC"). This book explicitly unifies the FTC, Green's, Stokes', and Divergence theorems under the generalized Stokes' theorem / differential forms. OpenStax presents these four theorems but does not synthesize them into a single framework.
- Computational Python integrated throughout. Rather than relegating technology to optional CAS sidebars (as OpenStax largely does), this book uses Python (NumPy/SymPy/Matplotlib-style workflows) as a first-class tool in worked examples and exercises across the whole sequence.
- Understanding-first motivation. Chapters 1 ("Why Calculus") and 40 ("The Big Picture") bookend the text with conceptual and historical framing that has no analogue in OpenStax, and a four-track Modeling Portfolio (Ch. 39) gives students a cumulative capstone deliverable.
Where OpenStax is a fine free substitute. For the core single-variable and multivariable calculus content — limits, derivatives, integrals, sequences/series, parametric/polar, and the standard vector-calculus theorems (this book's Chapters 2–37, excluding the discipline-specific applications above) — OpenStax covers essentially the same material to a comparable standard at zero cost. A student who only needs the mathematics, or who wants a free second explanation of any topic, can rely on OpenStax confidently; the volume/section pointers in this appendix make it straightforward to find the parallel treatment. What OpenStax does not replace are this book's cross-disciplinary applications, its computational integration, and its synthesis chapters (38–40).