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Calculus, 4th edition

Michael Spivak · 17 HN comments
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This edition differs from the third mainly in the inclusion of additional problems, as well as a complete update of the Suggested Reading, together with some changes of exposition, mainly in Chapters 5 and 20.
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Jul 16, 2020 · dilap on Topology Illustrated (2015)
i learned from this back in college:

one of my favorite text books ever. (tho it was paired w/ one of my favorite teachers ever, which i'm sure helped a lot.)

I think this would be a good book in a year or two. For now, if I can introduce basic concepts like slopes, derivatives, limits, and similar, simply and intuitively, that would be ideal.

Thus far, my favorite is Cartoon Guide to Calculus.

"Spivak's Calculus" is a reference to the book "Calculus" by Michael Spivak.

Many consider the book's presentation of the topic utterly beautiful, bordering even on the spiritual.

As always, it depends ...

If you just want to learn the maths relevant to your specific interests then you can simply pick a page on wikipedia, build a tree of topics you need to cover, then start to knock them off one by one, building a web of knowledge as you go. Then ask questions on making sure you take the advice about how to ask questions the smart way[0][1].

If, on the other hand, you want to get into studying maths generally and build your maths study skills, then I would recommend starting with a really good maths text book and work through it, doing all the exercises, reviewing earlier material, and taking it seriously. Two options are Spivak[2][3] which claims to be about calculus, but is really about analysis, or "Sets and Groups" by Green[4]. The latter is great to create the underlying basic knowledge you need for cryptography, but more, it teaches you how to do maths properly.

You could also just pick something you think is interesting on Khan Academy[5] and go for it.

But having said all that, it's tough to get back into maths, and you need to make sure you really understand your motivation. Most people don't want to write a book, they want to have written a book. Most people don't want to study maths, they want to have studied maths. If you're not serious, you won't succeed, especially with no one to track your progress, answer questions, and generally encourage, coax, support, and inspire you, it will be tough.

How well motivated are you?







You can try "Data Science from Scratch" [0] to get some taste. It uses Python to teach essentials of data science, and ML altorithms. The code quality is very good, and there is an introduction to Statistics, Maths and Python to start.

Then you can continue with improving your maths (Linear Algebra [1], Calculus [2], [3]) and moving on with Statistical Learning [4] [5]. I am personally going now through this plan.







An Introduction to Statistical Learning [0] is also good. It's a little less technical than The Elements of Statistical Learning. We used it for our statistical learning course at my university. The full PDF is available for free as well [1].



I agree. As far as calculus goes, I am more enamored with books like Spivak's ( that take a proof-centric approach to teach calculus from first principles.

Incidentally, for those who want to learn linear algebra for CS in a mooc setting there are 3 classes running at this very moment: (from UT Austin) (from Davidson) (from Brown)

The first 2 use matlab (and come with a free subscription to it for 6 months or so), the last python. One interesting part of the UT Austin class is that it teaches you an induction-tinged method for dealing with matrices that let you auto-generate code for manipulating them: .

And of course there are Strang's lectures too, but those are sufficiently linked to elsewhere.

My calc I course in university was applied calculus without a text. I had to go back and redo single variable by reading Spivak (and Polya's How to Solve It) to figure out the proofs in Concrete Math by D. Knuth, et. al.
Apr 12, 2014 · gms on Ask HN: Math books like SICP?
It's difficult to find what you are looking for, since maths is a much older field. However, Spivak's 'Calculus' is pretty close:
I clicked on this topic in order to make this suggestion. To expand on it a little bit, I was a math major and read Spivak's 'Calculus' after I had already taken real analysis. I found it delightful - it really approaches the topics from first principles and unlike many calculus textbooks actually goes through the effort of presenting proofs of the theorems. Highly recommended.

As some recreational reading, less suiting the original request, I very much enjoyed David Foster Wallace's 'Everything and More: A Compact History of Infinity' ( DFW is not for everyone, but I enjoyed it a lot. Maybe just check it out of the library first to see if it's for you.

I appreciated DFW's book for even attempting to do a popular treatment of what we would now call the history of real analysis. But there were some serious technical problems with it:
What would be a good book series (preferably a classic one that's stood the test of time) on math (Algebra, various Calculus topics, Statistics, etc)?

I'd like to edit this some more during the edit window for this comment. To start, the books by Israel M. Gelfand, originally written for correspondence study.

An acclaimed calculus book is Calculus by Michael Spivak.

Also very good is the two-volume set by Tom Apostol.

Those are all lovely, interesting books. A good bridge to mathematics beyond those is Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard and Hubbard.

A very good book series on more advanced mathematics is the Princeton University Press series by Elias Stein.

Is this the kind of thing you are looking for? Maybe I can think of some more titles, and especially series, while I am still able to edit this comment.

You may want to try working through Spivak's _Calculus_ [1] textbook. It is a bit more involved than most calculus textbooks used in universities today, so it will likely not feel like you are simply repeating something that you have already done. I would recommend this book to anybody who wants to brush up on calculus after studying it before.


I have Spivak's book actually, I got the hardcover from a used book store several years ago. It did seem a bit daunting to me from what I recall, I had used Tom Apostle's book when studying calculus back then.

I'll give more time to working through Spivak, hopefully it feels more approachable this time around.

Cal Newport, the author of the submitted blog post, draws comments both here on HN and on his own blog pointing out that deep understanding of a subject doesn't necessarily equate to VISUAL thinking about a subject. There is a big literature on "learning styles" and some attempts by some schoolteachers to categorize children by what their preferred learning styles are. When I have taken learning style questionnaires, and when I have asked my wife (a piano performance major and private music teacher) about this, the answer on learning styles is "all of the above." I personally think, based on my observations of successful learners of a variety of subjects, that learning styles are themselves learnable, and a learner with a deep knowledge of a particular subject will know multiple representations of that subject. My wife has had many piano performance courses, and also music theory and ear training courses, and has learned visual representations of music both in the form of standard musical notation and in the form of "music mapping,"

which she has found very helpful.

As for mathematics, the subject I teach now, I have always cherished visual representations of mathematical concepts, for example those found in W. W. Sawyer's book Vision in Elementary Mathematics

But other mathematicians who taught higher mathematics, for example Serge Lang, recommended memorizing some patterns of multiplying polynomials by oral recitation, just like reciting a poem.

The acclaimed books on Calculus by Michael Spivak

and Tom Apostol

are acclaimed in large part because they use both well-chosen diagrams and meticulously rewritten words to deepen a student's acquaintance with calculus, related elementary calculus concepts to the more advanced concepts of real analysis.

Chinese-language textbooks about elementary mathematics for advanced learners, of which I have many at home, take care to introduce multiple representations of all mathematical concepts. The brilliant book Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma

demonstrates with cogent examples just what a "profound understanding of fundamental mathematics" means, and how few American teachers have that understanding.

Elementary school teachers having a poor grasp of mathematics and thus not helping their pupils prepare for more advanced study of mathematics continues to be an ongoing problem in the United States.

In light of recent HN threads about Khan Academy,

I wonder what Khan Academy users who also have read the submitted blog post by Cal Newport think about how well students using Khan Academy as a learning tool can follow Newport's advice to gain insight into a subject. Is Khan Academy enough, or does it need to be supplemented with something else?

Having just gone through the entire statistics playlist of Khan Academy (about 10 hours of video) in about a week, I think I can offer some data here. I ran a running instance of clojure next to KA, and I wouldn't go on to the next video until I'd replicated what Sal had done in Clojure. This seemed to be extremely helpful, especially because there are no practice problems for the stats videos.

I think a quick-and-easy way to get feedback is essential. For some lessons there are practice problems, but for others, a student who wanted to maximize learning/minute spent watching video would be wise to at least open up Excel or something.

This is an excellent idea and an opportunity to kill two birds with one stone (improving my mathematics and learning clojure have been on the cards for a while now). I'm a professional programmer, but thanks to the over-specialisation of the British education system, only did two years of mathematics in high school. Increasingly in my work I find myself struggling with relatively basic statistics concepts and think it's about time to try to educate myself.
I'm sorry but it read a bit much like 'hey if you can visualize it you can learn it' but that is a cruel joke to someone who can't visualize anything. Sort of like laughing at someone for not being able to see the number in a color blindness test.

Some people just don't visualize. Not even a little bit. And I'm not sure its "just because they never learned to." Myself, I've always seen the 'picture' in my head and even dream in full technicolor (like this means anything) but my wife of 20+ years just can't. She is definitely smart, graduated with a CS degree from USC and is a much better planner than I will ever be, but those questions where you see a flat piece of paper with a bunch of dotted lines on it and you need to guess the shape it will be if they were all folded, just can't see it.

When I was growing up I used to think they only put those kinds of questions on tests so that everyone could get a few answers right, they were just that easy for me.

So Newport's thesis that if you can visualize it you can gain 'insight' is no doubt true for some people, but it certainly isn't a panacea for teaching complex subjects.

If you've ever seen the online math courses that Stanford did [1] under the EPGY program, it has some excellent tools that seem to work well for a variety of learning styles. Worth a look, and just down right priceless if you're home schooling your kids.


I think you make an important point, but I'm not sure the post takes any particular focus on visual representation other than a graph is generally an easier way to intuit what a derivative is. His repeated use of the word concept suggests insight for him requires a more general abstraction.

As an aside, why do or did people claim there is visual learning aside from spatial learning? I don't experience visual and spatial imagination as different things. (With reasoning about time always assumed.)

why do or did people claim there is visual learning aside from spatial learning?

I'm pretty sure that those are distinct neurological processes, as revealed by the differing individual deficits that patients can have after suffering strokes. But I don't have the medical references at hand, and you have certainly seen many sources that combine writing about both, as I have.

[...] other than a graph is generally an easier way to intuit what a derivative is.

Even that is a matter of personal preference. I honestly believe it's easier to get the concept of a derivative by linking it to instantenous velocity.

There are many different ways of thinking about mathematical concepts like derivatives. The more you know, the more deeply you know them, the better.

Here's a random example: Marsden and Weinstein define derivatives in their out-of-print textbook Calculus Unlimited without limits. The tangent to a graph at the point x is the boundary between two line pencils, one of lines entering the epigraph at x, the other of lines leaving. There's no limit-taking of chords. It's a simple and neat definition that connects with classical notions of tangency.

In his essay On Proof and Progress in Mathematics, Thurston lists a dozen other definitions or conceptions of derivatives in his personal arsenal, some very sophisticated. But even those among his definitions that are elementary and have roughly the same scope there is a difference in their psychological affordances, and that can make all the difference.

Although, as with anything I believe, new information could change my mind, I don't currently think "learning styles" exist in any meaningful sense. Some information or concepts are better learned through, e.g., visual aids, but that's the nature of the information not the learner.

This video by Harvard-educated cognitive psychologist and professor Daniel Willingham is relevant:

Most books have problems and most have solutions guides for about half of the problems. I'm surprised nobody's mentioned Spivack yet:

   Anybody have insight into how to actualize these 
   nuggets into some semblance of a self-learning course?
Buy Calculus by Micheal Spivak. Solve at least one problem every day. Make it ritual and a daily requirement. Watch MIT lectures for corresponding chapter you are on.

To learn this, don't trouble over the path and reason at present. Buy the book and start. Right now.

Buy it. To learn this- buy it and start. Right now.

Which MIT lectures would you recommend that use Spivak?
Strang's Calculus book is available for free. Is the Spivak book a much better resource? I'm not familiar with either one, but I've seen both recommended before.

Haven't read his calculus book, but Strang's linear algebra book is the best math book I've ever read. It's actually readable! Certain chapters are available on line. Based on it I would definitely try his Calculus book if I needed one.
Spivak's book is more of an analysis text. If you are interested in mathematics, go with Spivak. If you're more interested in engineering or physics, you may be happier with Strang.
Strang's Calculus book is fantastic. I bought it long ago merely to have it (having already learned Calculus).
Jul 27, 2010 · thefool on Ask HN: How Do I Get Math?
Get this book: and work through it if you want a rigorous treatment of MV calc and intro analysis.

Or this book: for regular calculus. Read them, work through the problems, get an answer key, email your solutions to professors asking them to look at them.

You will get a good introduction to analysis (and exposure to many other parts of math) from those two books.

Spivak's Calculus, is a good bet. But it's very theory-involved.
Michael Spivak's Calculus text:

Jun 14, 2009 · tokenadult on Textbooks must die
"A textbook author in Toronto made enough money from his calculus textbook to afford a $20 million house."

I know that calculus textbook.

It's workmanlike, but certainly not inspiring like Spivak's textbook.

I think it would be enlightening if you could provide the textbooks you buy.

Interpreting that as a request to name the textbooks I find useful, I'll do that here.

Elementary mathematics:

Primary Mathematics


Miquon Math

Secondary mathematics:

The Gelfand Correspondence Program series


Basic Mathematics by Serge Lang


The Art of Problem Solving expanded series

When a student has those materials well in hand, it is time to work on AMC and Olympiad style problem solving,

and also the best calculus textbooks, such as those by Spivak or Apostol.

Elementary reading:

By far the best initial reading text is

Let's Read: A Linguistic Approach

but there are many other good reading series, including

Primary Phonics


Teach Your Child to Read in Ten Minutes a Day

(I devote more time than that to reading instruction, typically, because I use multiple materials)

and quite a few others. There is more junk than good stuff among elementary reading materials, alas.

A snapshot of my bookshelf's "math" section, which really hasn't changed much since I was in high school and hadn't taken calculus:

W.W. Sawyer, What is Calculus About? and Mathematician's Delight

Courant and Robbins, What is Mathematics?

Hogben, Mathematics for the Million

Steinhaus, Mathematical Snapshots

Ivars Peterson, The Mathematical Tourist

Davis and Hersh, The Mathematical Experience

Polya, How to Solve It

Huff, How to Lie With Statistics

McGervey, Probabilities in Everyday Life

Raymond Smullyan: The Lady or the Tiger, Alice in Puzzle-Land, others

Anything by Martin Gardner. I happen to have picked up Mathematical Magic Show and Mathematical Circus, but I'm sure there are many other collections.

I also recommend cryptography stuff. David Kahn's The Codebreakers is not really a math book, but it is awesome and it stars mathematicians, as does Simon Singh's The Code Book. You could read Schneier's Applied Cryptography.

This is HN, so I would be remiss if I didn't point out that you can learn a lot of fun and useful math by reading SICP, Knuth, or any good algorithms book.

If anybody out there knows a good, spirited statistics book addressed to someone who knows calculus, tell me. I keep planning to go through Fundamentals of Applied Probability Theory but I never get around to it; see "Related Resources" here:

Having said all of that: I have a Ph.D. in physics/EE, so I've got to tell you, if you haven't tried calculus you haven't lived. ;) I'm not sure how to go about learning calculus in a fun way for a mathematician -- I took fairly standard first- and second-year college courses in calculus and physics and learned it that way. The folks on Amazon seem kind of enthusiastic about Spivak:

/Great/ list. Thanks!

I just bought the Polya book a few days ago, and the majority of the rest are going on my wishlist.

I took calculus my junior year, but again: I don't trust my high school experience to tell me whether or not I like something.

Thanks a ton for the names. I'll check out the library later this afternoon!

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