How to Self-Study Calculus: Topics and Book Guide

To self-study calculus, work through four core areas in order: differentiation, integration, sequences and series, then multivariable calculus. Each builds directly on the last, so skipping ahead creates gaps. A strong starting text is Keisler’s “Elementary Calculus: An Infinitesimal Approach,” which is freely available online and covers both standard and infinitesimal methods.

Key Takeaways

• Calculus study progresses through four sequential stages: differentiation, integration, sequences and series, and multivariable calculus.
• H. Jerome Keisler’s textbook “Elementary Calculus: An Infinitesimal Approach” is distributed free of charge at math.wisc.edu and covers material from basic calculus through vector calculus.
• Ziv Nitecki’s “Calculus Deconstructed: A Second Course in First-Year Calculus” rebuilds single-variable calculus with rigorous proofs and is intended for readers who already know Keisler-level material.
• Nitecki’s follow-up text, “Calculus in 3D: Geometry, Vectors and Multivariate Calculus,” is also freely available as a PDF through Tufts University and extends into differential forms.
• Shepley L. Ross’s “Differential Equations” covers first-order and higher-order ODEs, Laplace transforms, and Sturm-Liouville theory, and can be started right after a first pass through single-variable calculus.

What Are the Four Main Areas of Calculus to Study?

1. Differentiation

Differentiation is the process of finding the tangent line to a specific function at a given point. A surprising number of facts about a function’s behavior can be deduced from this single procedure. The only prerequisite is basic high-school mathematics.

Important topics within differentiation include:

1. Continuity
2. Limits
3. Derivatives
4. Rules for differentiation
5. Mean value theorem and its consequences
6. Geometrical meaning of derivatives
7. Curve sketching
8. Rate of change

2. Integration

Integration is the inverse process of differentiation and is used to find areas, lengths, and many other quantities. Differentiation is a prerequisite for this stage.

Important topics within integration include:

1. Indefinite integrals
2. Rules for indefinite integration
3. Definite integration
4. Rules for definite integration
5. The fundamental theorem of calculus
6. Applications of integration to find areas, volumes, and lengths
7. Applications to physics

3. Sequences and Series

Sequences and series matter because they let you approximate functions closely. Sine and logarithm functions, for example, can both be approximated very well using series expansions. Differentiation and integration are prerequisites for this stage.

Important topics within sequences and series include:

1. Convergence of sequences
2. Convergence of series
3. Special sequences and series
4. Convergence tests for series
5. Taylor series
6. Integration and differentiation with series

4. Multivariable Calculus

Every technique from single-variable calculus extends into multiple dimensions in this stage. Single-variable calculus is a prerequisite.

Important topics within multivariable calculus include:

1. Basic spatial geometry, including parametrization of lines and curves
2. Limits and continuity in multivariable functions
3. Differentiation of multivariable functions
4. Integration of multivariable functions
5. Multivariable Taylor series
6. Gradients and tangent planes
7. Maximization problems, including Lagrange multipliers
8. Different coordinate systems
9. Vector calculus

Which Calculus Textbooks Are Worth Using?

Elementary Calculus: An Infinitesimal Approach (Keisler)

This textbook is freely available at math.wisc.edu and takes a reader from elementary calculus through the standard topics in multivariable calculus, presenting both the conventional real-number approach and the nonstandard infinitesimal approach.

The infinitesimal approach came first historically and relies on infinitesimal numbers, quantities smaller than any positive real number. Mathematicians including Euler and Gauss used infinitesimal reasoning in their work. Later mathematicians shifted toward the standard real-number approach, but Abraham Robinson later showed that infinitesimals can be made logically rigorous, and they remain useful in physics, engineering, and as a source of intuition in pure mathematics.

Keisler covers both approaches side by side, so a reader who finishes this book will be prepared to move on to a standard calculus or analysis text. The book covers limits, differentiation, integration, series, vectors, partial differentiation, multiple integrals, vector calculus, and some differential equations. A reader familiar with basic high-school math should have no trouble with it, since logarithms and trigonometric functions are reviewed along the way, though some familiarity with mathematical proofs is recommended.

Calculus Deconstructed: A Second Course in First-Year Calculus (Nitecki)

This book, published by the Mathematical Association of America and available through Amazon, constructs calculus rigorously. The theory is built up carefully, and the exercises include notable historical discussions.

It covers sequences and their limits, continuity, differentiation, integration, and power series. It is meant to be read after thorough exposure to a text like Keisler’s.

Calculus in 3D: Geometry, Vectors and Multivariate Calculus (Nitecki)

This book is freely available as a PDF from Tufts University. Readers who enjoyed Nitecki’s first-year calculus text are likely to enjoy this one too. It starts from the beginning of multivariable calculus and goes far, ending with a discussion of differential forms, a useful modern mathematical tool. Everything is proved rigorously, with some proofs placed in an appendix.

It covers coordinates and vectors (introductory linear algebra), curves in space, differentiation of real-valued functions, integration of real-valued functions, and vector fields and forms. It is intended for readers who already know some rigorous single-variable calculus.

Differential Equations (Ross)

Shepley L. Ross’s “Differential Equations” is available through Amazon and covers the main solution techniques along with some supporting theory. It is written accessibly for a subject often presented less attractively.

The book covers analytic solutions of first-order and higher-order ordinary differential equations (ODEs), series solutions, systems of linear ODEs, approximate methods for ODEs, the Laplace transform, existence and uniqueness theorems, Sturm-Liouville theory and Fourier series, nonlinear differential equations, and partial differential equations. It can be started after a first encounter with single-variable calculus, though some topics require additional background.

Frequently Asked Questions

What order should I study calculus topics in?

Study differentiation first, then integration, then sequences and series, then multivariable calculus. Each stage depends on solid understanding of the one before it, since integration is defined as the inverse of differentiation and series convergence relies on limit concepts from differentiation.

Do I need to know proofs before starting calculus?

Basic high-school mathematics is enough to begin a book like Keisler’s “Elementary Calculus.” A familiarity with mathematical proofs is recommended but not strictly required for that text, though more rigorous follow-up books like Nitecki’s “Calculus Deconstructed” assume more comfort with proof-based reasoning.

What is an infinitesimal in calculus?

An infinitesimal is a quantity smaller than any positive real number, used historically by mathematicians such as Euler and Gauss before the standard real-number approach to calculus became dominant. Abraham Robinson later showed that infinitesimals can be treated with full logical rigor.

Can I learn multivariable calculus and differential equations for free?

Yes. Nitecki’s “Calculus in 3D: Geometry, Vectors and Multivariate Calculus” is freely available as a PDF from Tufts University, and Keisler’s “Elementary Calculus” is freely available from the University of Wisconsin. Ross’s “Differential Equations” is not free and is sold through retailers such as Amazon.

Which book should I read after finishing an infinitesimal calculus course?

After finishing a book like Keisler’s, a rigorous second course such as Nitecki’s “Calculus Deconstructed: A Second Course in First-Year Calculus” is a reasonable next step, followed by “Calculus in 3D” for multivariable material.