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1 point

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7th Oct 2021

>. You’re referring to Spivak’s Calculus book?

Yes, this or the third edition.

> Anything after algebra?

At this point the best use of your time in preparation for higher level math is doing more and more proofs. I suppose you could do some discrete math? Say a number theory book or a book on graph theory/combinatorics?

1 point

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27th Mar 2021

Yes - I think that studying the Bible in-depth can be very fruitful, and while I don't think it's necessary for every Christian to do an in-depth study, I think that for those of us (somewhat) interested in can be fruitful. What I'm saying is that I don't think that studying Greek (or Hebrew) is the most efficient means of doing so - especially when there are Greek scholars who have really squeezed everything out of the texts.

>When a mathematician encourages me to learn more about the foundations of mathematics,

Well, now you're speaking my language. I'm a mathematician, and I wouldn't recommend someone to learn the foundations of mathematics. I don't really even like this stuff and I can't say I've benefited much from knowing any of it. Indeed, if someone wanted to learn more about mathematics, I'd start them with this calculus book and them move them on to an algebra / analysis text, etc. That's kind of the level that I think laypeople should be working with.

1 point

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31st Oct 2020

If you are curious about the theoretical aspect of calculus, I recommend you check out Spivak's book on the subject. Unlike most university calculus textbook, Spivak treats the subject rigorously and proves every theorem in calculus.

But I will slightly disagree on the notion that calculus is all algebra/trig. The notion of limit is something completely new, and trying to apply it, as well as derivatives and integrals, correctly is a skill that requires more than just algebra and trigonometry.

You will also see variety of applications of calculus as the time goes on (e.g. optimization, volumes, etc.), and half of the problem is on setting up the problem. To do so, you have to understand exactly what derivatives and integrals are, and how they can be applied to the problem you are working on. I think this is more than "just algebra/trig."

And calculus is also the first time you see some result that isn't necessarily intuitive or obvious. Take a look at the fundamental theorem of calculus, for instance. It says that the "area under the curve" of a function has something to do with the "antiderivative" of the function, even though they appear quite unrelated at the first sight.

Yes, I understand that the standard calc curriculum in university still emphasizes computations, and I remember getting bored at them at some point as well. But if you look things carefully, calculus is definitely more than just algebra and trigonometry.

1 point

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30th Sep 2019

None of the questions you asked is “silly” or “simple.” There’s a whole lot going on in calculus, most of which is typically explained in a real analysis course. Rigorous proofs of things like the mean value theorem or various forms of integration are challenging, but they will provide the clarity you’re looking for.

I recommend that you check out something like Spivak’s Calculus, which is going to give a more rigorous intro to the subject. Alternately, you can just find a good analysis or intro to proofs class somewhere. It’s a fascinating subject, so good luck!

1 point

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31st Mar 2017

Well, funnily enough I started university as a physics major. I'll be graduating as a physics/applied maths double major :D. It was really funny, applied maths got it's hooks in me and from there I started getting really interested in how all of the methods we were implementing worked (as I said in the other comment :D)

So, I started with a couple of text books. Namely Calculus by Michael Spivak and some linear algebra textbook I can't remember the name of (sorry, I'm sure you can get recommendations online.) I, uhh, may have gotten some pdfs through less than legal channels as well.

Even though I already understood a lot of the content covered in those textbooks quite well, I thought it was a good idea to just set a baseline, y'know? From there I went onto lecture notes. I think MIT publishes all of it's lecture notes online for free, and IIRC they're quite good. Spend as much time as you need. All of the "interesting" stuff in maths usually needs to have a very solid foundation in more "boring" stuff (I think learning about the fundamentals is fun though :D)

If there's something in particular you're interested in, then you can look up suggestions for reading material online and maybe what you should study before jumping into that particular topic. The reading material may be a textbook, or some lecture notes or something else. (For example, I'm slowly working my way through Topology by James Munkres)

Anyway, that's my advice. Start at the bottom, go slow and look up recommended texts online.

As for tutoring, pure maths is sort of a hobby for me, so I probably wouldn't be the best teacher. If you have questions about PDEs though I'd be happy to look at them, since they're more what I'm "supposed" to know haha.

I hope this was helpful.

1 point

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28th Oct 2016

And all of this stuff would have to arise from a situation where "numbers" x and y is. The first sum is a finite number of steps. I've heard that Spivak's Calculus is a good idea to look at mathematicians and get a PhD position if you don't remember it.

1 point

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4th Jan 2016

I think all three of the above summer programs offer scholarships, so you might make some inquiries before ruling those out entirely. But if that turns out to be nonviable for you, then you'll still have a number of other good options by which you can learn lots of interesting math.

One additional general point: with the advent of the internet, it has become *orders of magnitude* easier to find quality resources and recommendations. The trick will be that there's so much *period*, so you may feel overwhelmed.

As for textbooks, Spivak's <em>Calculus</em> has a deservedly good reputation. From your vantage point, I think it has two primary strengths:

It has

*many many*exercises, from the straightforward to the challenging.It's been around for awhile, so it's easier to work on your own with this while getting outside help from others.

For example, on reddit alone, there have been *two* separate subreddits for those trying to learn from Spivak's *Calculus*: /r/spivak and /r/SpivakStudyGroup.

Oh, but "Calculus III", assuming that means multivariable calculus, isn't covered in Spivak, as I recall.

I think that the people best situated to offer you concrete advice would be your current teachers and possibly someone there who can have a face-to-face conversation with you. If there's a nearby college, university, or even community college, you might see whether one of the math professors would be available to offer more specific advice.

But for now, it sounds like you have a solid plan for going forward, so it'll be mostly a matter of following through. Good luck!

1 point

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30th Jun 2015

http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918

Spivaks calculus is an analytical approach to teaching calculus. Its pure math for the beginner. Not for the faint of heart! It's problem solving, not computation.

Might be a free pdf of it too if you google that. Pretty sure its not in print.

1 point

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7th Jun 2015

You've probably been hearing about this which is a good book for an introduction to single-variable calculus if you're particularly advanced. It splits the difference between an intro calc and an intro-real analysis text.

1 point

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24th Sep 2012

This isn't quite what you asked for, but consider taking a look at Spivak's <em>Calculus</em>. It's a rigorous, proof-based presentation of calculus, and working through the problems in that book will give you a far more solid understanding of how calculus works than just doing more computations. You can't write proofs by memorization, so that'll ensure that you really know the material on a deeper level.

I don't know where to find detailed solutions for the problems in Spivak, but you learn more from working out the proofs yourself.

1 point

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24th Oct 2012

Spivak's Calculus 4th edition link Designed specifically for the advanced student who hasn't formally seen calculus but wants to go at it from a deeper level than you would see in an AP calculus textbook or a 1st year college calculus course.

Read the first review on Amazon, pretty much explains why it's good.

1 point

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1st Aug 2012

You might want to take a look at Spivak's <em>Calculus</em>. It builds up the foundations of calculus rigorously, so you can see exactly what's going on "behind the scenes".

To briefly try to answer your questions, integration is the area under a curve. That's the way to think of it. The space under a curve has dimension one higher than the curve, so in a sense, you're increasing the dimension. Integrating several times is just repeating this process.

How do you calculate the area under a curve? The method usually taught first, Riemann integration, works by splitting the area into successively thinner rectangles under the curve; the integral is the limit as the width of the rectangles goes to zero. This is visually intuitive, and what's going on behind the scenes is pretty much just formalizing this. (There's another method, Lebesgue integration, which is more widely used in real analysis; it's less intuitive than Riemann integration, but it can be applied to less "nice" functions and is easier to use in proofs.)

As for higher dimensions, it's practically impossible to visualize them properly. Usually, you work with them formally and use analogies to lower dimensions for your visualizations. (Often, two or three dimensions is enough to get a reasonably intuition for what's going on.) I'm not sure how calculus specifically will help you with that, though you'll work with higher dimensions a fair amount if you take a class in real analysis, which is essentially the modern mathematical theory of calculus.

2 points

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16th May 2021

Cheers, thanks for the extra info.

Just like another commenter has mentioned, she really ought to try Michael Spivak's Calculus. It will be hard -- perhaps even too much so, in which case maybe she wants to try a proof-writing book like Hammack's Book of Proof first -- but it is the correct introduction to "real" calculus/math if she is really interested in getting serious about math.

Best of luck to her and to you. Math is a trying but beautiful endeavour to pursue, whether it's as a hobby (for me) or as work (for many others whose math skills far exceed mine).

2 points

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22nd Sep 2015

Yes, typically. The distinction really only exists in introductory calculus courses and is more of a convention or a tendency than some clear-cut, formal rule. The distinction between "early" and "late" transcendentals is a curricular one, not a mathematical one, and it means roughly something like this:

**Early transcendentals**— introduce polynomials, rational functions, trigonometric functions (sine, cosine, etc.) exponentials, and logarithms up front and use them as examples to explore derivatives and integrals**Late transcendentals**— develop the concepts of derivatives and the integrals using only polynomials + rational functions and introduce exponentials, logarithms, and trigonometric functions later on

So, I would expect those two books to explore the same "conceptual terrain" but with different order and, potentially, a different relative emphasis on those concepts.

As /u/voluminous_lexicon said, though, they'll have different exercises and those exercises will be phrased in terms of the precise material covered in the book. There's really no escaping that.

I'd check out AbeBooks, one of the best places to buy textbooks, IMO. You can check out the listings for Thomas' Calculus: Early Transcendentals vs. Thomas' Calculus yourself. Even the brand new copies have almost identical prices on AbeBooks.

If it matters, I personally prefer the "late transcendentals" approach. Exponential, logarithmic, and trigonometric functions all have very clear, simple, and *precise* definitions in terms of integrals. Without those precise definitions, the rules surrounding those functions seem kind of arbitrary and numbers like e (aka "Euler's number" aka the base of the natural logarithm) take on a mystical quality.

For example, the natural logarithm of a positive number *x* can be *defined* as the integral of f(t) = t^(-1) from 1 to *x*, i.e.,

> ln(x) := ∫* 1 x* t^(-1) dt

where `:=`

means "defined to be equal." From this definition you can prove all the usual properties of the logarithm, e.g., that ln(xy) = ln(x) + ln(y) for all x,y > 0. You can define the function exp(x) = e^(x) as the inverse function of the natural logarithm. Etc. etc.

I never encountered this approach until my first-year university calculus source, which used Michael Spivak's *Calculus* for the textbook. Before that, I always found the rules of exponents, logarithms, and trig functions intelligible but seemingly arbitrary, requiring me to memorize lots of little algebraic rules to manipulate them.

1 point

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28th Sep 2022

> I’m in greece

Oh, right: you'd already told me this before! Apologies for my faulty memory.

Michael Spivak's <em>Calculus</em> has a well-earned reputation for being a strong introduction to calculus, especially for those seeking a deep exploration of theory rather than just computations. If your class is following Spivak, then having a copy on hand will likely be a valuable resource. And the exercise you shared above seems more in the spirit of a challenging problem from that text rather than something from, say, the popular (but less advanced) text by James Stewart.

There's a likely tradeoff, though. Spivak's *Calculus* is hundreds of pages long, and some of it will likely be dense. You've already expressed some worry of feeling overloaded right now, and I wouldn't want to add to that. I recommend skimming it, assuming you can find a copy, then deciding for yourself whether having yet another calculus textbook will ultimately be more illuminating or simply add to your paradox of choice-related issues.

Whatever you decide about Spivak's text, I think a lot of my other recommendations probably still make sense: take advantage of whatever resources your class and school provide (from office hours to free peer tutoring to study groups). If you're at or near the top of your class, then you're less likely to get value from that *in the form* of stronger students helping share their strategies and such. But forcing yourself to *explain* your solutions to others, *within whatever policies your class and school have regarding academic integrity* (in the sense of not helping others cheat), can have a very positive effect in terms of clarifying your own thinking and improving your ability to communicate your mathematical ideas.

The good news is that this sounds like a class where you're going to learn an *enormous* amount of math, especially by the standards of typical introductory calculus classes. As you've already found, though, that means this class will be challenging in a way unlike other math classes you've likely taken up to this point. Don't let that, by itself, scare you! But I'd definitely take seriously that this class will be challenging, and strategies that have worked in earlier math classes may need to be modified for this one. In particular, I'd recommend against last-minute 12-hour study sessions before an exam, since studying on a regular basis will almost certainly be more effective *and* more sustainable.

But as I mentioned above, I'd try to get informed recommendations from as wide as possible a range of people who've been where you are, or who've helped people in your situation. Again, good luck!

1 point

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23rd Jan 2022

If you'd like a particular reference, Michael Spivak wrote a classic "calculus-on-Euclidian-space" called Calculus, but has what I understand to be a pretty solid introduction to what you're looking for called Calculus on Manifolds. (Links to the Amazon pages)

1 point

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14th Jan 2022

Oh alright, in that case I would strongly suggest that you take a look at Thomas' Calculus (the editions does not matter) I teached an advanced placement class with this, and I left my students with strong computational and proof skills, in the case that you want to emphasis more analysis Spivak's Calculus

Also these books are really easy to get :)

1 point

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2nd Dec 2021

They probably meant this book. It's expensive but without much work you can find a high quality PDF online

1 point

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6th Jul 2021

You will be asked to confront a lot of these ideas in a major way in the course of earning a bachelor degree in math, if that is your current plan. But if you wanted to whet your appetite, you could try to work through an advanced calculus book (this is essentially a term for calculus with honest proofs). Along the way to setting up the tools to attack calculus rigorously, you will be asked to confront many of the ideas you've taken for granted. Two the standard choices of books are those written by Spivak or Apostol. Make sure to actively do the exercises! Also, expect this to take a long time to work through (either of these books could easily keep you busy for the next 6-12 months, if not more). For your trouble, if you are planning to do a math major, you would find that working through either book would cover many of the things you will need to learn.

1 point

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24th Nov 2020

1 point

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10th Oct 2020

Alfred North and Bertrand Russell have *Principia Mathematica* in which they attempt to derive mathematics from first, logical principles. Bertrand Russell also has *Introduction to Mathematical Philosophy*, in which he attempts to make the earlier work more accessible.

Though it is not necessarily philosophical, I cannot help but mention Dr. Spivak's *Calculus, 4th Edition*, in which he *authoritatively* builds calculus from first principles. If you've ever wondered what mathematicians meant by "rigour", or "mathematical beauty", look no further! This book has both in spades.

1 point

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24th May 2018

Question about Spivak's Calculus and Ross' Elementary Classical Analysis:

Are they books treating mathematics on the same level? Do they treat the rigorous theoretical foundation and computational techniques equally well? Can each one be an alternative to the other? Could someone please give brief comparative reviews/comments on them?

This question is also on r/learnmath: HERE.

1 point

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2nd Jan 2018

Hi, first of all congratulations on your drive and ambition! It's always nice to see young people pursuing their passion for learning.

That said, you haven't actually studied *mathematics* yet, so it might be a bit premature to say you want to become a maths professor, all the courses you've done up to this point including your AP Calculus class have been (increasingly difficult) forms of arithmetic, and you won't know if you really enjoy math or are good at it until you've studied some at least some algebra and analysis. Being good at and enjoying the low level maths you've done is a (somewhat) necessary but not sufficient condition for being good at and enjoying higher level maths.

If you're serious though about progressing to the next level. Here are my recommended steps:

After finishing your AP Calc class, learn calculus for real. Buy a used copy of Spivak's Calculus https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ (or find a PDF of it somewhere) along with the answer book. Going through Spivak immediately after finishing your AP Calc class will be tough, not because you won't be able to handle the material, but because it will feel tedious in parts, you'll want to skim through it nodding your head and saying you understand it, and you will understand it, passively. You need to actually engage with the material though, make notes while you read, work through examples to make sure you

*really*understand it, then actually do exercises. Not all of them, but a lot of them, but focus on the proofs. Spivak's Calculus is pretty near an analysis textbook, so working through it will get you well prepared for what is to come.You didn't say if you had taken a Linear Algebra class or not. Doing Gilbert Strang's MIT OCW course in Linear Algebra is a good place to start. https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/. Linear algebra will be useful for, well, everything really, but it will really come in handy when doing multivariate calculus and differential equations, and it's a good place to continue practicing writing proofs. This is a lower division college linear algebra class, the kind you'd take in your first two years at university, which would be offered at a local community college, or just a bit beyond what I would expect an AP Linear Algebra class would cover. You'll need at least one further course in linear algebra beyond this before you start doing real maths.

At this point, if you've made it this far, I think working through a discrete math textbook would be a fun reward for your efforts. The one used in the class I took was Susan Epp's Discrete Mathematics with Applications, https://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0534359450/. It's probably not the

*best*text out there for the material, but it's pretty good, I enjoyed it, and it's available used very cheaply. Discrete math isn't so much a field of mathematics as much as it's a sampler plate of what is to come. There's lots of proofs of course (which is good), and you'll be exposed to some logic, set theory, sequences and induction and recursion, some number theory and methods of proof, graph theory, and counting and probability. It's a really, really, fun course, just make sure you eat your vegetables before having your dessert.After all that, you will be well prepared for trying to tackle your first Real Analysis or Algebra text. You may or may not want to plow through multivariate calculus and differential equations first, but those aren't

*strictly*necessary to understand the material. Ultimately you'll want and need to study them, but I think they're (mostly) used as pre-requisites to help develop the mathematical maturity and rigor needed to succeed in upper level maths.

1 point

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9th Aug 2017

You can start with Calculus by Spivak. If you're going to buy it then wait until after the Fall semester begins; the price is inflated right now because students need it for school.

This is a PDF of the third edition of the above book.

This is an excellent introduction to logic and proofs. You will want a strong understanding of how mathematicians communicate via proof and that book will really help.

Hope that helps!

1 point

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30th Apr 2017

No, he wrote a book on single-variable calculus, too: https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918

1 point

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8th Feb 2016

1 point

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21st Feb 2016

Unfortunately I can't. Most calc books are written in order to pump out as many copies and versions as possible, so it's not in their interest to write them well. I've heard that Spivak's Calculus is a good calculus book that actually teaches calculus, I've never picked it up but it's probably your best bet.

If you're comfortable with proofs, then Baby Rudin is solid.

1 point

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25th Jul 2015

What is your opinion of Calculus, 4th Edition by Michael Spivak? I'm going back to school to complete my degree in math, and I want to refresh and master calculus in order to better handle advanced classes. I've taken calculus classes on derivatives, integrals, and multivariable calc, but I've taken a long break. I have Essential Calculus: Early Trancendentals, 1st edition by James Stewart, but I've heard Spivak's book is one of the best.

1 point

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13th May 2015

These two books will take you from Calc 1 all the way to Diff Eq, and Linear Algebra. They are a little hardcore, so you might want to purchase How to Prove It to assist with the proofs.

1 point

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19th Feb 2015

Don't bother with a course - just learn it on your own. If you're going to need to learn and do proofs use http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918. If not, you can use pretty much any other textbook.

1 point

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20th Apr 2011

1 point

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12th Dec 2010

I would suggest Spivak for a rigorous, proof-based treatment of calculus and Sally's Tools of the Trade for an introduction to real analysis, linear algebra, and proofs all at the same time. My school uses both of these books for its introductory classes for math majors and they're very good. Note that Sally can be very terse, so it would be beneficial to find some online lectures and resources to supplement your reading.

0 points

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8th Nov 2012

I'm not very experienced, but I'd suggest Michael Spivak's <em>Calculus</em> since it was my school's introduction to real analysis.

There's also this resource from GA Tech. Not very rigorous, but cool nonetheless.