Thomas' Calculus

Citation

Hass, J. R., Heil, C. E., Bogacki, P., & Weir, M. D. (2022). Thomas' Calculus (15th ed.). Pearson.

Chapter Summary

Pearson's contents list runs to seventeen printed chapters, two further chapters that are online only (Chapters 18 and 19), and appendices. Chapters 1–11 make up the single-variable course and Chapters 12–16 the multivariable course; Pearson also sells the two halves separately as Single Variable and Multivariable volumes. According to the publisher's description, the fifteenth edition adds exercises, updates many applications and revises figures and wording for clarity, and the two online chapters are new.

Chapters 1–2: Functions; Limits and Continuity

The book opens with a short chapter on functions: graphs, combining functions, shifting and scaling graphs, trigonometric functions and a section on graphing with software. Chapter 2 starts from rates of change and tangent lines, then covers the limit of a function and the limit laws, the precise definition of a limit, one-sided limits, continuity, and limits involving infinity and asymptotes.

Chapters 3–4: Derivatives; Applications of Derivatives

Chapter 3 moves from the derivative at a point to the derivative as a function, differentiation rules, rates of change, trigonometric derivatives, the chain rule, implicit differentiation, derivatives of inverse functions and logarithms, related rates, and linearization and differentials. Chapter 4 turns to extreme values, the Mean Value Theorem, the first derivative test, concavity and curve sketching, applied optimization, Newton's method and antiderivatives.

Chapters 5–6: Integrals; Applications of Definite Integrals

Integration is introduced through area and finite sums, sigma notation, the definite integral and the Fundamental Theorem of Calculus, followed by substitution and the area between curves. Chapter 6 applies the definite integral to volumes (cross-sections and cylindrical shells), arc length, areas of surfaces of revolution, work and fluid forces, and moments and centres of mass.

Chapters 7–9: Transcendental Functions; Techniques of Integration; First-Order Differential Equations

Chapter 7 treats inverse functions, natural logarithms, exponential functions, exponential change and separable differential equations, L'Hôpital's rule, inverse trigonometric and hyperbolic functions, and relative rates of growth. Chapter 8 collects the techniques of integration (parts, trigonometric integrals and substitutions, partial fractions, tables and computer algebra, numerical integration, improper integrals) and ends with a section on probability. Chapter 9 introduces first-order differential equations: slope fields and Euler's method, linear equations, applications, autonomous equations and phase planes for systems.

Chapters 10–11: Infinite Sequences and Series; Parametric Equations and Polar Coordinates

Chapter 10 builds from sequences and infinite series through the integral, comparison, ratio and root tests, alternating series and conditional convergence, to power series and Taylor and Maclaurin series, including their convergence and applications. Chapter 11 covers parametrized plane curves, calculus with parametric curves, polar coordinates and their graphs, areas and lengths in polar coordinates, and conic sections, including conics in polar form.

Chapters 12–13: Vectors and the Geometry of Space; Vector-Valued Functions and Motion in Space

The multivariable course begins with three-dimensional coordinates, vectors, the dot and cross products, lines and planes, and cylinders and quadric surfaces. Chapter 13 differentiates and integrates vector functions: curves in space and their tangents, projectile motion, arc length, curvature and normal vectors, tangential and normal components of acceleration, and velocity and acceleration in polar coordinates.

Chapters 14–15: Partial Derivatives; Multiple Integrals

Chapter 14 covers functions of several variables, limits and continuity in higher dimensions, partial derivatives, the chain rule, directional derivatives and gradients, tangent planes and differentials, extreme values and saddle points, Lagrange multipliers, Taylor's formula for two variables, and partial derivatives with constrained variables. Chapter 15 covers double and triple integrals, including polar, cylindrical and spherical coordinates, applications, and substitutions in multiple integrals.

Chapters 16–17: Integrals and Vector Fields; Second-Order Differential Equations

Chapter 16 develops line integrals of scalar functions, vector fields and line integrals (work, circulation and flux), path independence and conservative fields, Green's theorem, surfaces and area, surface integrals, Stokes' theorem, and the divergence theorem presented as a unified theory. Chapter 17 returns to differential equations with second-order linear equations, nonhomogeneous equations, applications, Euler equations and power-series solutions.

Chapters 18–19 (online) and Appendices

Chapter 18, Complex Functions, covers complex numbers, limits and continuity, complex derivatives, the Cauchy–Riemann equations, complex series and conformal maps. Chapter 19, Fourier Series and Wavelets, covers periodic functions, summing sines and cosines, vectors and approximation in three and more dimensions, approximation of functions, and the Haar system and wavelets. Appendix A collects real numbers, induction, lines, circles and parabolas, proofs of limit theorems, the theory of the real numbers and two technical results. Appendix B, online, covers determinants, extreme values for functions of more than two variables, and gradient descent.

Key Concepts

Chapters 1–6: Single-variable calculus

• Precise definition of a limit: the formal definition of the limit, given its own section in Chapter 2 after the limit laws.
• Mean Value Theorem: the result in Chapter 4 on which the first derivative test and monotonicity arguments rest.
• Fundamental Theorem of Calculus: the link between the definite integral and antiderivatives, stated in Chapter 5.

Chapters 7–11: Transcendental functions, techniques, series and curves

• Separable differential equations and slope fields: introduced with exponential change in Chapter 7 and extended in Chapter 9.
• L'Hôpital's rule: handling indeterminate forms, in Chapter 7.
• Taylor and Maclaurin series: Chapter 10 treats them together with the convergence tests and their applications.
• Polar coordinates and conics: Chapter 11 extends curve description beyond rectangular coordinates.

Chapters 12–17: Multivariable and vector calculus

• Dot and cross products: the algebraic tools of Chapter 12.
• Gradient and directional derivative: Chapter 14's account of how a function of several variables changes in a given direction.
• Lagrange multipliers: constrained extremum problems in Chapter 14.
• Green's, Stokes' and divergence theorems: Chapter 16 presents the last as a unified theory with the earlier two.

Critical Analysis

Strengths

• One book for the whole sequence. The contents run from functions and limits through vector calculus and include first-order and second-order differential equations (Chapters 9 and 17), so a single text can serve a three-semester or three-quarter sequence, and the printed split into Single Variable and Multivariable volumes suits shorter courses.
• Technology built into the structure. The table of contents includes a section on graphing with software (1.4), a section on integral tables and computer algebra systems (8.6), a section on numerical integration, and "Technology Application Projects" listed at the end of each of Chapters 11–17 in the contents.
• Review material at chapter ends. The contents list "Questions to Guide Your Review", practice exercises, and additional and advanced exercises at the end of each of Chapters 11–17, which gives students a way to test themselves before an exam. The back matter also lists answers to odd-numbered exercises and an applications index.
• Proofs kept accessible. Proofs of limit theorems, the theory of the real numbers and the mixed derivative theorem sit in Appendix A, so the main chapters can stay focused on the concepts and techniques while the details remain available to those who want them.

Limitations

• Light on theory. Because the more delicate proofs live in the appendices, the book is a calculus text rather than an introduction to analysis; students who want a rigorous development of limits and continuity will need a separate analysis text.
• Assumes a precalculus foundation. The opening chapter has four sections (functions, combining and shifting graphs, trigonometric functions, graphing with software), and Appendix A reviews real numbers, induction and lines, circles and parabolas; students with weak algebra will need to shore it up on their own.
• Some topics are online only. The complex functions and Fourier series and wavelets chapters, and Appendix B, are online, so a reader with only the printed volume does not have them.

Where It Fits in the Course

Thomas' Calculus is the main full-sequence textbook in this course list. Calculus Volume 1 by Gilbert Strang and Edwin Herman is a free, openly licensed text covering single-variable calculus; it is a natural companion or alternative for the first two semesters, though it stops before the multivariable material. Calculus Made Easy is a short informal primer on the core ideas of differentiation and integration, useful as a gentle first read before or beside Chapters 2–5 but not a substitute for the exercise practice that Thomas offers. Calculus: Concepts and Methods and Advanced Calculus are listed as more rigorous, more extensive treatments, and belong after the Thomas sequence rather than instead of it.

How to Use This Book

Thomas' Calculus is meant for students in science, engineering and mathematics who have completed precalculus (algebra, functions and trigonometry). It is best used as the spine of a two- or three-semester course, with a pencil in hand: the exercise sets and chapter-end review sections are where the learning happens.

A practical plan:

• Semester one (Chapters 1–6): work through functions, limits, derivatives and the first integral applications in order; the Mean Value Theorem (4.2) and the Fundamental Theorem of Calculus (5.4) are the two sections to master before moving on.
• Semester two (Chapters 7–11): transcendental functions, integration techniques, differential equations and series. Chapter 10 has ten sections and introduces a series of convergence tests; it pays to practise them until choosing among them is routine.
• Semester three (Chapters 12–17): vectors and three-dimensional geometry come first and are needed for everything after; Chapters 14–16 build in order from partial derivatives to multiple integrals to line and surface integrals.
• Optional: Section 8.9 (probability), Chapter 17 (second-order equations) and the online Chapters 18–19 depend on the course; the Technology Application Projects suit students with access to a computer algebra system.

Use Appendix A when a proof is wanted, and the answers to odd-numbered exercises to check practice work as you go.