What is the Best Way to Prepare for Spivak's 'Calculus'?

[WannabeFeynman]

Hello. I wanted to complete my preparation in algebra 2 and precalculus so that I can tackle Spivak's "Calculus." I have a few options in front of me. I can either go:
1. through the Art of Problem Solving series rather quickly to be prepared. The issue with that is that some members here scoffed at those books, which led me to assume they wouldn't be the more rigorous preparation I can get.

2. Another option is Algebra by Gelfand, in conjunction with "Geometry" by Lang, Jacobs and Kiselev. After completing those, I can go through "Basic Mathematics" by Lang and the Art of Problem Solving Precalculus book.

Is the second option more rigorous and would better prepare me for higher mathematics (I am planning to pursue a Mathematics degree)? Also, I would appreciate if someone would recommend some other textbooks to use as well.

 

[IGU]

If AoPS works for you, what do you care what other people think?

I missed seeing comments of the AoPS scoffers. Pointers?

There are many, many good ways to prepare. Pick one that works for you and just do it. My kid loved Apostol. No prep beyond AoPS Into to Algebra and Intro to Geometry.

 

[WannabeFeynman]

Some people think it focuses too much on competition, lacks on rigour in some places etc. I find AoPS a decent set of books but I think the second option (some other books with AoPS) is a better option - albeit more time consuming.

Your child went to Apostol's Calculus with just knowledge of AoPS' Introduction to Algebra and Geometry?

 

[IGU]

Some people think it focuses too much on competition, lacks on rigour in some places etc. I find AoPS a decent set of books but I think the second option (some other books with AoPS) is a better option - albeit more time consuming.

Well, I haven't seen such comments on this forum. In any case, the books don't focus on competition. Lots of kids who use AoPS are into math competitions, and their forums reflect that, and they have various competition related classes. But I'm not sure that's relevant to using their books for self study.

Your child went to Apostol's Calculus with just knowledge of AoPS' Introduction to Algebra and Geometry?

I tell you this only because adequate preparation for Apostol should also be adequate for Spivak.

He self-studied calculus from Apostol I & II, as mentioned here. That was from early 2010 to early 2011, before AoPS had published their Calculus book. At that point he'd also self studied AoPS's Intro to Number Theory and Intro to Counting and Probability books, but those don't contain much relevant to learning calculus. But do be warned that what my kid (now 17) does in mathematics isn't all that generalizable to humans.

A possibly interesting aside is that he used his mother's calculus books. She got them when she was at Caltech and took Math 1 from Tom Apostol himself. If I'd had Spivak lying around instead then he'd probably have learned from that. Both are excellent books.

 

[dumplump]

I don't understand some people's obsession with AOPS. I bought those books and I was not terribly impressed by them. They seemed to focus too much on weird topics. I would take these weird topics to my TA, he was a math major, and he wouldn't even understand what the text was talking about. Spivak as first introduction, as in beginner level, is usually a bad idea, but that depends on the person.

 

[deluks917]

I do not understand why you need any preparation. The book has no pre-requisites besides basic High School Algebra. What you probably need is a lot of time to think about the material and ponder the problems. In my opinion you are much better off skipping any prep work and instead giving yourself a long time to work through Spivak.

I actually self studied the book and it took me about 16 weeks working hard 6-10 hours a day to learn the material in the first 24 chapters (this is everything before complex number/Construction of the reals in my edition). However no other book has repaid the effort I put into it as much as Spivak Calculus. But if your goal is to learn Spivak, I would not attempt to do it even sort of fast.

If I was teaching from Spivak the only "prep" I would "wish" my students had was a very basic understanding of double integrals. Since this allows a much easier to understand proof of the Taylor Theorem. But this is an extremely minor point.

 

[dumplump]

I do not understand why you need any preparation. The book has no pre-requisites besides basic High School Algebra. What you probably need is a lot of time to think about the material and ponder the problems. In my opinion you are much better off skipping any prep work and instead giving yourself a long time to work through Spivak.

I actually self studied the book and it took me about 16 weeks working hard 6-10 hours a day to learn the material in the first 24 chapters (this is everything before complex number/Construction of the reals in my edition). However no other book has repaid the effort I put into it as much as Spivak Calculus. But if your goal is to learn Spivak, I would not attempt to do it even sort of fast.

You need decent preparation to handle spivak. If a person picks up spivak without understanding how to write proofs, or understand the logic of compiling proofs, then that will just discourage most from trying to learn the subject. Most college analysis courses offer spivak as the text for entry level analysis. This probably has to do with the general goal of the text. Most want to learn calculus not to prove the theorems, most want to learn calculus to apply its concepts. I would dare say that most high school students have not had to prove any theorem that they were taught in class. Recommending to someone, with zero understanding of how to structure and write proofs, a book that is mainly about proving theorems for the topic they want to learn is counterproductive. That is why there are other calculus texts that are mainstream, such as Stewart, Anton, etc.
I agree that spivak will help someone understand calculus, but to suggest it as the first exposure to calculus would probably do more harm then good.

 

[deluks917]

You need decent preparation to handle spivak. If a person picks up spivak without understanding how to write proofs, or understand the logic of compiling proofs, then that will just discourage most from trying to learn the subject.

In fact I personally did this. I had some exposure to single variable calculus (though at a very low level) before reading Spivak. But I had absolutely no understanding of proofs. I explicitly remember thinking to myself "I don't really get what a proof is." When I read the chapters on induction and limits I was deeply confused. I mentally "kind of" got the idea but I felt like their was some magic involved in the proofs. It felt like induction was "cheating" somehow. But in both cases after I had worked through enough problems it clicked and understood what an induction argument was and how limits worked. Both of these realizations have honestly carried me through years and years of math. I still say if you really understand the ideas in Spivak, much of measure theory/integration is fairly clear (the integration part, Spivak does not really help with understanding the construction of the Lebesgue measure).

My preparation for Spivak was very bad imo. To give you an idea of how lost I was I will give a funny example. The first time I looked at a proof of the binomial theorem it took me to days to even understand the proof and I immediately forgot it. (The next time it took a little under a day's work. The third time I tried to understand the proof it was basically obvious). Before reading Spivak I had been convinced math was uninteresting so I knew little math.

I strongly disagree you need much of any preparation (besides algebra) to understand SPivak. Even if the material is deeply confusing at first you can learn it by working the problems and thinking through the material. If one can trust oneself not to quit too rapidly the answer book for Spivak is an extremely good resource (my rule was not to consider looking until I had tried for 4+ hours, usually longer). And most importantly the answer book let's you check your work on the previous problems. The hardest thing for me when I was the chapters on limits/continuous functions. They took me almost a month to grasp. But after all my work I never again struggled with the concept of a limit (of course some specific limits are pretty tricky!).

Based on my personal experience, the stated purpose of Spivak in writing the book and Mathwonk's posts on what Spivak was used for in the past I do not think Spivak requires much preparation. The book is self contained. Prep will mostly just waste time you need to use to think through the material in "Calculus."

Here is another person giving a different explanation of what I mean. Read the first answer in this thread: http://math.stackexchange.com/quest...wledge-is-necessary-to-begin-spivaks-calculus

 

[dumplump]

Y

In fact I personally did this. I had some exposure to single variable calculus (though at a very low level) before reading Spivak. But I had absolutely no understanding of proofs. I explicitly remember thinking to myself "I don't really get what a proof is." When I read the chapters on induction and limits I was deeply confused. I mentally "kind of" got the idea but I felt like their was some magic involved in the proofs. It felt like induction was "cheating" somehow. But in both cases after I had worked through enough problems it clicked and understood what an induction argument was and how limits worked. Both of these realizations have honestly carried me through years and years of math. I still say if you really understand the ideas in Spivak, much of measure theory/integration is fairly clear (the integration part, Spivak does not really help with understanding the construction of the Lebesgue measure).

My preparation for Spivak was very bad imo. To give you an idea of how lost I was I will give a funny example. The first time I looked at a proof of the binomial theorem it took me to days to even understand the proof and I immediately forgot it. (The next time it took a little under a day's work. The third time I tried to understand the proof it was basically obvious). Before reading Spivak I had been convinced math was uninteresting so I knew little math.

I strongly disagree you need much of any preparation (besides algebra) to understand SPivak. Even if the material is deeply confusing at first you can learn it by working the problems and thinking through the material. If one can trust oneself not to quit too rapidly the answer book for Spivak is an extremely good resource (my rule was not to consider looking until I had tried for 4+ hours, usually longer). And most importantly the answer book let's you check your work on the previous problems. The hardest thing for me when I was the chapters on limits/continuous functions. They took me almost a month to grasp. But after all my work I never again struggled with the concept of a limit (of course some specific limits are pretty tricky!).

Based on my personal experience, the stated purpose of Spivak in writing the book and Mathwonk's posts on what Spivak was used for in the past I do not think Spivak requires much preparation. The book is self contained. Prep will mostly just waste time you need to use to think through the material in "Calculus."

Here is another person giving a different explanation of what I mean. Read the first answer in this thread: http://math.stackexchange.com/quest...wledge-is-necessary-to-begin-spivaks-calculus

you only acknowledging only part of the points I made. The point is that to suggest Spivak as the first introduction to calculus will most likely be off putting for people to learn it. Nowhere did I state that no one should use spivak. If a person wants to use spivak to learn calculus, then that is fine. To suggest as the first book, without any understanding of proof structure and logic, is probably one of the worst ways to introduce someone to calculus. That is why there are the mainstream texts which focus more on applications of calculus as less on the proofs of calculus.
Most colleges use spivak for pure math majors, and not for engineering or other science students. For math majors it is a great text because their study of mathematics would require them to write up proofs and theorems should they continue to do research in the future.

 

[deluks917]

I personally had absolutely no understanding of proof structure or logic (in the sense you mean) and I was fine. I cannot personally attest that someone with no knowledge of calculating derivatives would also be fine but probably they would be. My knowledge of calculus was very weak and not especially helpful for solving Spivak problems. Even his "computational" problems are super hard compared to most calc books. Actually some of the integrals are so tricky I would just skip them, in this day and age doing tricky integrals is not the most important skill.

For the record I recommend googling Paul's online calc notes for some easier problems in calculating integrals/derivatives. Spivak is a little light on "introductory" computational problems. But while I rec going to another source for more problems Spivak alone is a fine source. If you can do the computations in Spivak you can definitely do the computations in a "normal" calc book.

If someone intends to study engineering or physics (or mathematical biology) I think the concepts in calculus are so central that learning from Spivak is worthwhile. Calculus and Linear Algebra are the core foundation on which almost all these subjects depend. Spending time learning the theory of calculus is well spent imo. Though I would skip the construction fo the real numbers, unless reading those chapters seems fun. There is a point where learning too much theoretical math is no longer suffinetly useful to be worth the time, but spivak is before that point (I would recommend a rigorous treatment of single/multi-variale calculus and linear algebra to physics/engineering majors but not much "pure math" beyond that).

 

[member 620756]

If anyone is still even reading this thread, here is some advice. The 2nd option is the far more rigorous and challenging option. Algebra by Gelfand is the most respected rigorous pure math approach to high school algebra. Kiselev, Geometry by Jacobs, and Geometry by Lang are some of the most respected high school geometry texts out there. AoPs is good if you have no intent of becoming a pure mathematician or maybe a physicist, or something equally challenging. All I have to say is I have never seen a algebra book more engaging, rigorous, and awesome as Gelfand. High school geometry however is a toss up.