If I’ve learned real analysis, should I go back and (re)learn calculus properly

calculusreal-analysisself-learningsoft-question

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)?

Update

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.

Best Answer

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?

Related Question