How to Self-Study Calculus: Topics, Order & Book Guide

We often get questions here from people self-studying mathematics. One common question is: “What mathematics should I study and in what order?” To answer that, I have made a list of topics a mathematician should ideally know and their prerequisites.

Calculus

After high-school mathematics comes calculus. This includes the following major areas.

1. Differentiation — Differentiation is finding the tangent line to a specific function. You can deduce surprisingly many facts from this procedure.Prerequisites: Basic high-school mathematics

   Important topics:

   1. Continuity
   2. Limits
   3. Derivatives
   4. Rules for differentiation
   5. Mean value theorem and consequences
   6. Geometrical meaning of derivatives
   7. Curve sketching
   8. Rate of change
2. Integration — Integration is the inverse process of differentiation. Integration is used to find areas, lengths, and much more.Prerequisites: Differentiation

   Important topics:

   1. Indefinite integrals
   2. Rules for indefinite integration
   3. Definite integration
   4. Rules for definite integration
   5. 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 are important for approximating functions (for example, sine and logarithm can be approximated very well using series).Prerequisites: Differentiation and integration

   Important topics:

   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 — Everything from single-variable calculus extends to multiple dimensions.Prerequisites: Single-variable calculus

   Important topics:

   1. Basic spatial geometry, e.g. 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

Calculus — Book Recommendations

The best calculus book is undoubtedly:

Elementary Calculus — An Infinitesimal Approach (Keisler)

Freely available here: https://www.math.wisc.edu/~keisler/calc.html

This book takes you from elementary calculus to the standard topics in multivariable calculus and presents both the standard and nonstandard (infinitesimal) approaches.

The nonstandard approach came first historically and involves infinitesimal numbers. Numbers are so small that they’re not real numbers anymore. The tools of infinitesimals were used by many great mathematicians such as Euler and Gauss. More recently, mathematicians have preferred the standard real-number approach; however, Robinson showed infinitesimals can be made rigorous, and they remain useful in physics and engineering (and provide intuition in pure mathematics).

Keisler treats both approaches, so you will be able to read a standard calculus or analysis book afterward.

This book covers:

• Limits
• Differentiation
• Integration
• Series
• Vectors
• Partial differentiation
• Multiple integrals
• Vector calculus
• Some differential equations

If you’re familiar with basic high-school math, you will have no problems with this book. Many concepts like logarithms and trigonometric functions are reviewed along the way. A familiarity with proofs is recommended.

If you want a somewhat more rigorous treatment after Keisler, consider the following.

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

http://www.amazon.com/Calculus-Deconstructed-First-Year-Mathematical-Association/dp/0883857561

This constructs calculus rigorously. The theory is built carefully and the exercises are interesting, with notable historical discussions.

This book covers:

• Sequences and their limits
• Continuity
• Differentiation
• Integration
• Power series

You can read this book after a thorough exposure to a book like Keisler.

After seeing single-variable calculus more rigorously, you might want to study multivariable calculus more deeply. One readable resource is:

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

Freely available here: http://www.tufts.edu/~znitecki/Hardcore2.pdf

If you enjoyed Nitecki’s previous recommendation, you will likely enjoy this one as well. It starts from the beginning of multivariable calculus and proceeds far; the end discusses forms, a useful modern tool. Everything is rigorously proved (some proofs are placed in an appendix).

This book covers:

• Coordinates and vectors (introductory linear algebra)
• Curves in space
• Differentiation of real-valued functions
• Integration of real-valued functions
• Vector fields and forms

Read this book if you already know some rigorous single-variable calculus.

Finally, you may want to learn some differential equations:

Differential Equations (Ross)

http://www.amazon.com/Differential-Equations-Shepley-L-Ross/dp/0471032948

This is a well-written book that covers main solution techniques and some theory. It is accessible and enjoyable for a subject that is often presented less attractively.

The book covers:

• Analytic solutions of first-order and higher-order ODEs
• Series solutions
• Systems of linear ODEs
• Approximate methods for ODEs
• Laplace transform
• Existence and uniqueness
• Sturm–Liouville theory and Fourier series
• Nonlinear differential equations
• Partial differential equations

You can read this book after a first encounter with single-variable calculus, although some topics require additional background.

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