If you've learned real analysis, what, if any, is the value in going back and learning calculus "properly", via, let's say, Spivak or Courant?

I've learned real analysis using the texts of Ross and of Abbott (not yet Rudin). My initial study of calculus was Stewart-style: Here's the procedure, do it, get an A. That leaves many gaps, both in depth of understanding and even in fluency (it's hard to memorize a procedure that you don't really understand). Should I invest time in relearning single variable calculus from a classic text such as Spivak or Courant? If so, how should I approach it, and what should my goals be?

Similarly for multivariable calculus: I learned it using Stewart and did very well in the class. Which means I learned next to nothing. And I certainly didn't encounter things like the Implicit Function Theorem. Should I go back and relearn multivariable calculus, using, e.g. Hubbard or Shifrin?

Do I move forward (Rudin) or back (single or multivariable calculus)?


littleO asks "Once you’ve learned real analysis... doesn’t that mean you’ve already learned single variable calculus properly... [since it] proves the main results of single variable calculus". In one sense that's true. But, in another sense, the focus of a real analysis is on proving theorems, whereas a book like Spivak has many (challenging) computational problems. Would you believe that someone can prove the Cauchy sequences converge implies the Nested Interval Theorem, but perhaps get confused when doing integration by substitution? Or that they find a fresh presentation of the development of the integral of $x^n$ using first principles (Courant develops it using limits of series, even before introducing the Fundamental Theorem of Calculus), to be very enlightening? So even if you've proven the main results, there still a lot to be gained from learning, the "right" way, how to develop the basic procedures from fundamentals, and how to use them to solve problems.

Put another way, we have three things:

  1. Procedure focused. E.g. Stewart.
  2. Theorem & proof focused. E.g. real analysis.
  3. Using procedures, but a) developing them from first principles and b) using them masterfully. E.g. Spivak, Apostol, Courant.

I've found that #1 alone is insufficient: it leaves big gaps, and even the procedures don't stick. #2 is excellent - and there's no limit on how far and deep you can go. But something tells me that before I do so, I need to go back and master the procedures, including their development and usage. I'm not sure if that's correct and if so how I should do it without being stuck in repetition and without failing to advance.

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    $\begingroup$ Once you’ve learned real analysis from Abbot, doesn’t that mean you’ve already learned single variable calculus properly? Abbot rigorously proves the main results of single variable calculus. Then Baby Rudin will go on to develop multivariable calculus rigorously, but Hubbard & Hubbard or Shifrin are likely more readable. $\endgroup$
    – littleO
    Commented Jun 9, 2023 at 0:33
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    $\begingroup$ This is a great question. Ever since I took real analysis, I've been complaining loudly that it should come before calculus. I can't comprehend why that isn't a more popular view among mathematicians. $\endgroup$
    – Archr
    Commented Jun 9, 2023 at 0:34
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    $\begingroup$ Go back and forth, in my opinion. I regularly refer to my freshman calculus texts because I want to ensure I have those foundations (plus, I am a TA for those courses!). It especially helps when relearning sequences and series. $\endgroup$ Commented Jun 9, 2023 at 0:35
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    $\begingroup$ My experience with baby Rudin is that it shows its age in the modern age. There are better books pedagogically nowadays. I personally liked Terence Tao's Analysis I and II. $\endgroup$
    – balddraz
    Commented Jun 9, 2023 at 0:36
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    $\begingroup$ Thanks. I’d say you don’t have to commit to either approach. There are many paths you can traverse over the knowledge landscape, as long as you’re willing to backtrack and fill in gaps as necessary. It’s certainly worth spending some time on Spivak. Don’t forget that you can learn in a “big picture first” style, scrolling over the knowledge landscape and zooming in and out as necessary. I say yes to Spivak’s book Calculus. $\endgroup$
    – littleO
    Commented Jun 9, 2023 at 1:17

2 Answers 2


I’d say you don’t have to commit to either approach. There are many paths you can traverse... as long as you’re willing to backtrack and fill in gaps as necessary... Don’t forget that you can learn in a “big picture first” style, scrolling over the knowledge landscape and zooming in and out as necessary. I say yes to Spivak’s book Calculus.

--littleO (in comments)

Go back and forth... I regularly refer to my freshman calculus texts because I want to ensure I have those foundations...

--Sean Roberson (in comments)

To expand on the above, it's worth quoting Stanford's Ravi Vakil (emphasis added):

Mathematics is so rich... that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils... Caution: this backfilling is necessary.

Applying it to the OP, this would suggest:

  • You can't really understand elementary calculus until you've learned real analysis
  • At the same time, you can't postpone elementary calculus until you've mastered real analysis (attempts to do so usually fail)
  • That's fine - learn elementary calculus, learn its procedures, and go on to real analysis
  • But, once you do, "this backfilling is necessary" - you need to go back and opportunistcally backfill your elementary calculus. E.g. how integration by substitution really works, or, e.g. how log can be defined from first principles, or, e.g. how integrals can and were computed prior to the Fundamental Theorem of Calculus
  • This backfilling generally should not be systematic. Rather, when an elementary topic comes up, and you sense it a bit hazy, don't just quickly look it up - but rather take it as an opportunity to stop, detour a bit, and invest in a proper backfill. Then move on.

I'd appreciate comments on the above. Does this sound like right approach?

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    $\begingroup$ I always tell students to read a textbook backwards and backfill as needed. $\endgroup$
    – Deane
    Commented Jun 11, 2023 at 20:31
  • $\begingroup$ @Deane Read a single book backwards? How? $\endgroup$ Commented Jun 11, 2023 at 20:32
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    $\begingroup$ @SRobertJames roughly speaking, find a topic you like or jumps out at you (which usually happens in later chapters), and gloss through the introduction of that chapter, and see what theorems/concepts from previous chapters are invoked. Also, gloss through the problems of each chapter to get a feel for things. Of course, to do this successfully also requires a bit of back and forth: if you really know nothing, then the middle/end of the book is completely unintelligible, so you start from the front. But once you get a few things, you can start this process. Learning is very much nonlinear :) $\endgroup$
    – peek-a-boo
    Commented Jun 11, 2023 at 21:03
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    $\begingroup$ that’s how I do things anyway… $\endgroup$
    – peek-a-boo
    Commented Jun 11, 2023 at 21:04
  • $\begingroup$ @SRobertJames, peekaboo explained it well. Except you usually need to look at other sources too. You look for something that speaks to you. $\endgroup$
    – Deane
    Commented Jun 12, 2023 at 0:55

This answer is based on my own learning style. We should understand that everyone learns differently, and there is no magic-wand solution.

Freshman calculus in the US is inherently procedural - not to say it's good or bad, but we're laying the foundation for something that students in math, physics, engineering, and related disciplines may be using through the rest of their degree. We solve simple problems and then use those techniques to try to gain insight on harder problems. When I got to real analysis as a junior, I actually didn't rely solely on a lot of freshman calculus at the time, especially with the theorems on convergence of sequences and uniform continuity. The way of working in real analysis is much different than freshman calculus, but it did help me understand some of the results I've encountered before (e.g., Mean Value Theorem, Intermediate Value Theorem).

In my comment, I say that I regularly refer back to my freshman calculus materials because 1) I serve as a teaching assistant for those courses and 2) I also privately tutor these courses. If I told a struggling student in calculus that we need to justify the exchange of sum and integral in power series arguments every single time, we'd get nowhere. Thus, I sprinkle some of the overarching results from analysis as needed if it enriches their learning. An example of this is in the chapter on sequences and series as seen in e.g. OpenStax Calculus Volume 2, Stewart, Thomas, and similar texts.

In my view, learning real analysis at the junior/senior level has some merit, but I don't think it will enrich your knowledge of calculus. This is why I say go back and forth. For example, the OpenStax Calculus Volume 2 text has a few results on series that I wasn't previously exposed to, so I went to my analysis texts to see if they were referenced there and if I could prove it. In particular, it's this version of the Limit Comparison Test:

Let $a_n, b_n \geq 0$ for all $n \geq 1.$

  • If $\lim_{n \to \infty} \frac{a_n}{b_n} = L \neq 0,$ then both $\sum_{n = 1}^\infty a_n$ and $\sum_{n = 1}^\infty b_n$ both converge or both diverge.
  • If $\lim_{n \to \infty} \frac{a_n}{b_n} = 0$ and $\sum_{n = 1}^\infty b_n$ converges, then $\sum_{n = 1}^\infty a_n$ converges.
  • If $\lim_{n \to \infty} \frac{a_n}{b_n} = \infty$ and $\sum_{n = 1}^\infty b_n$ diverges, then $\sum_{n = 1}^\infty a_n$ diverges.

The last two conclusions were fuzzy and I proved them for myself (if I remember right, you need to use $\lim \sup$ or $\lim \inf$). I then found a way to explain the result to my students intuitively. For procedural problems such as taking derivatives or integrals, I don't rely on my knowledge from analysis, but instead in following the procedures from freshman calculus. At that stage, it's important - follow the rules of the game and don't read too deeply into those basic examples.

(My first analysis course was out of Abbott, reading Chapters 1-5 and some of 6. Then in my first graduate program, I took a senior level course to fulfill a prerequisite which used Carothers' Real Analysis, and we focused on Chapters 13-14, 16-19.)


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