How to Self-Study Calculus: Topics, Order & Book Guide
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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.
Table of Contents
Calculus
After high-school mathematics comes calculus. This includes the following major areas.
- 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:
- Continuity
- Limits
- Derivatives
- Rules for differentiation
- Mean value theorem and consequences
- Geometrical meaning of derivatives
- Curve sketching
- Rate of change
- Integration — Integration is the inverse process of differentiation. Integration is used to find areas, lengths, and much more.Prerequisites: Differentiation
Important topics:
- Indefinite integrals
- Rules for indefinite integration
- Definite integration
- Rules for definite integration
- Fundamental theorem of calculus
- Applications of integration to find areas, volumes, and lengths
- Applications to physics
- 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:
- Convergence of sequences
- Convergence of series
- Special sequences and series
- Convergence tests for series
- Taylor series
- Integration and differentiation with series
- Multivariable calculus — Everything from single-variable calculus extends to multiple dimensions.Prerequisites: Single-variable calculus
Important topics:
- Basic spatial geometry, e.g. parametrization of lines and curves
- Limits and continuity in multivariable functions
- Differentiation of multivariable functions
- Integration of multivariable functions
- Multivariable Taylor series
- Gradients and tangent planes
- Maximization problems including Lagrange multipliers
- Different coordinate systems
- 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.
Advanced education and experience with mathematics
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Apostol's calculus books are fantastic for a first course on analysis, i.e. for a SECOND phase on calculus in the standard pedagogical sequence for most people who want to study mathematics formally. They are too dense to be useful for a first course on single or multi-variable calculus.
Dear micromass,
Have you read Apostol's calculus books? I want to know how they compare to the two Nitecki books you suggested in your guide? And also how Freidberg's Linear Algebra compares to Shilov's book on the same topic.
I wanted to go through calculus and then Linear Algebra following either of two paths:
a) Keisler's Infinitesmal approach>>>Nitecki Deconstructing Calculus>>>Nitecki Calculus in 3D>>>Freidberg's Linear Algebra
OR
b) Simmon's Calculus with analytic geometry>>>Apostol Vol 1>>>>Apostol Vol 2>>>>Shilov's Linear Algebra
It's a very nice book. But don't use it as first calculus book, since it's too difficult for that. It is very suitable as a second course though, if you enjoy the book.Can I skip first two chapters of third book if I followed the first book ?
“Hi,
What is your opinion on Courant’s introduction books on calculus ?
Thanks”
It’s a very nice book. But don’t use it as first calculus book, since it’s too difficult for that. It is very suitable as a second course though, if you enjoy the book.
Send me a PM, I might be able to help :)
“What is your goal? What kind of book do you want?”
I really want to understand the math behind quantum mechanics… :frown:
“Well, I’ll just have to buy another book then… :frown:”
What is your goal? What kind of book do you want?
Well, I’ll just have to buy another book then… :frown:
“I just ordered the book of Mary Boas. How does that compare with these books?”
It doesn’t compare at all with these books. They are very different. First of all, Boas does not cover single variable calculus. It starts with series and multivariable calculus. So it assumes you know integrals and derivatives already.
Second and most important, Boas is for physicists who don’t really care much about the underlying math. So if you want to know the math in detail, then Boas is not good. If you simply wish to use it as a tool, then Boas is truly an excellent resource.
Boas is a math methods for physics and engineering. It has less emphasis on theory, and goes over different subjects such as LA, DE’s, vector calc, basically everything an undergrad physics major will need. If you’re a physics major It will benefit you tremendously to work through it.
“Hmm interesting, of all those topics (it took me 2 semesters to get over them) the courses i took on the matter never talked about multi variable Taylor series, Laplace transform, or system of ODEs :c, maybe i should try to learn those on my own.
Also, about “vector calculus” section, does that mean Green’s, Gauss’ and Stokes’ theorem?
Very good, organized, and easy to read.
Cheers :D”
Yes, vector calculus is stuff like Stokes’ theorem.
Of course it is very likely that your courses did not cover everything of this. I don’t think it is really absolutely necessary to go back and learn them on your own (unless you enjoy learning this stuff of course, in which case: go ahead). If you ever meet one of those topics later, you can still go back and learn them.
Hmm interesting, of all those topics (it took me 2 semesters to get over them) the courses i took on the matter never talked about multi variable Taylor series, Laplace transform, or system of ODEs :c, maybe i should try to learn those on my own.
Also, about “vector calculus” section, does that mean Green’s, Gauss’ and Stokes’ theorem?
Very good, organized, and easy to read.
Cheers :D
The texts are just his recommendations. And it makes much more sense to introduce differentiation before integration.
“You seem to be presenting this as a “one true way”. ”
I don’t think I have said or implied anything remotely like that.
I will be learning calculus for the first time shortly and it’s nice to have a guide like this. Your posts are tremendously helpful for beginners like me, it is much appreciated!
Yes, very good! I will edit this in.
“Thanks a lot PWiz, I appreciate it. If you think I’ve missed something, please do tell!”
IMHO, “parametric equations” and “calculus in different coordinate systems” (or something along those lines) should be included in the post somewhere under the Multivariable section, but other than that, I think your post pretty much covers all the bases.
Thanks a lot PWiz, I appreciate it. If you think I’ve missed something, please do tell!
Nice one micromass! I’ve always thought about what this kind of list should constitute, and you’ve covered it really well. Before this, I was forced to say “you need Calc I, II, III and DEs to understand physics well” to my friends, but had a really hard time explaining the contents of each in detail. Well, I have a great reference now:woot:
Nice to see Keisler's "An infinitesimal approach to calculus" in the list. Great post.
Hi,What is your opinion on Courant's introduction books on calculus ? Thanks
Lest I be misunderstood in offering criticism, let me say thank you for doing this. It’s a meritorious effort and will be helpful to many, I’m sure. My impression regarding it being presented as the “one true way” came from these statements: The best calculus book is undoubtedly…. (highly controversial) So it is very beneficial to learn the nonstandard approach. (controversial at best) But I agree those are not representative of the whole piece. However, the impression I get is that you think these textbook suggestions are right for everybody. I’ve found that people have different styles and need different things. Some love examples, some hate them. Some need rigor and others prefer intuition. Some like exercises aplenty, and others prefer a few well-chosen problems. Some want an answer key and others find it too tempting and prefer it doesn’t exist. Some want their mathematics pure and others find it dry as dust if there isn’t real world motivation. It would help, I think, if you indicate who your recommendations are for. If you really think they’ll work for everybody, I’m suspicious.
You seem to be presenting this as a "one true way". For instance, you say differentiation is a prerequisite for learning integration. I'll note that Apostol does it the other way around, likely because historically that's the way it happened. Do you think that self-teaching from Apostol is a bad idea?I'm just thinking you might want to make the tone a bit more "here's one way to do it" than it is now. In any case, kudos on recommending free texts! That's certainly one thing Apostol's Calculus does not have going for it, regardless of how good it might be.
I'm afraid you posted a wrong link for the "Calculus in 3D: Geometry, vectors and multivariate calculus by Nitecki".This works better: http://www.tufts.edu/~znitecki/Hardcore2.pdf
I would like to share a recommendation: G.M. Fichtenholz "Differential and Integral Calculus". Fairly unpopular outside of the 'post-Soviet' countries, but it is among my personal favourites. A bit on the lengthy side, but it keeps a very approachable and 'eager to explain' tone just as easily when talking about basic differentiation and application of multi-variable functional series and transforms. Book genuinely 'feels' like a transcript from a very patient tutor. Plus it makes it a point to show worked-out examples to almost every single concept.It is also among the most complete resources when it comes for computational techniques, so if not for any other point it is still worth at least as a reference on solving problems.