How to learn differential and integral caculation?

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Without going to school? Overall I propably don't get the discipline and the energy to doing it, but maybe I could find the will to try. Any reference, something...

Try getting hold of some good text books. "Calculus, one and several variables" by Salas, Hille and Etgen is quite a comprehensive one accessible to beginners, that goes into the calculus of functions of a single variable as well as multivariate functions. I used it for both first and second year. It's still introductory though because there a more advanced topics in vector calculus and the calculus of complex functions.

MIT has video courses online in several branches of mathematics. Some of them are for beginning students.

See also:
http://freevideolectures.com/mathematics.html

ruveyn

That text presumes a certain degree of mathematical maturity and prior course work. I think video lectures are more effective for those who have not had proper instruction in math or who have been away from it for a long time.

ruveyn

I usually like the older texts better than the newer ones for some reason. Probably because the newer textbooks (especially the American ones) are often the size of telephone books and don't teach you, on careful inspection, all that much actually.

One book I can strongly recommend would be "A Course of Pure Mathematics" by G. H. Hardy (click the pages to flip them; if you want a hardcopy, it's been reprinted recently by Cambridge University Press). It does take some getting used to, but it teaches the subject (and much besides) really well, and that in a modest-sized volume. The exercises are hard most of the times, but almost always worthwhile. You may consider reading it first without doing the exercises though, then attempt them on a second reading, which is what I did. On amazon it gets good reviews.

To get the necessary background in algebra, I recommend reading the first volume of George Chrystal's "Algebra", if you can lay your hands on it. (Here is an online copy.) Don't bother to do all the exercises though; it would take way too long, and isn't really necessary. Three or four per chapter is fine in order to grasp the basics. However, basically any classic text on algebra will serve the purpose.

Er--Hardy's "Course ... " can be rather daunting as a place to begin, although it's wonderful to read once you see what's going on. The huge Ami textbooks are probably better than Hardy's is for developing insight, which is particularly hard to acquire when you're learning independently, and there are some things in them that are better than Hardy's treatments. For example, the newer proofs of the mean-value theorem of the diff calc and of the Taylor expansion are much more intuitive than those in Hardy, and are just as rigorous; with a tweak or two, the same tricks also give you the error terms for polynomial interpolation.

Fortunately, the Ami textbooks go to a new edition almost every year (more money for the author), so second-hand copies of last year's version are usually available CHEAP! I would try some edition of the old classic by those two MIT guys (George B.) Thomas and (Ross L.) Finney.

ruveyn's recommendation to get into things via the MIT videos is very perceptive. Watching somebody do things can help you understand even if you don't want to interact with the person who's doing them.

jef

I couldn't disagree with you more. Once you have a good grounding in algebra (which is an absolute prerequisite, but for which Chrystal or any other classic algebra text will do just fine), Hardy's text is second only to the original English edittion of Courant's. (Unfortunately, the latter is more difficult to obtain nowadays). Yes, it may be difficult for beginners, but really not too difficult. And then again, Calculus just isn't easy and isn't meant to be also, however much some would like it to be. The intellectual reward which can be derived from solving even a few exercises in Hardy is much higher than that of solving a few standard problems from today's average textbooks. I very much believe in giving students a few hard exercises, rather than giving them loads of routine. This is because problems like Hardy's approximate actual research (be it mathematical or otherwise) not perfectly, but much more closely than, say, integrate this-or-that. And I do feel that student should be taught this from the beginning. Secondly, solving a difficult problem provides one with more motivation and self-confidence.


You may be right about Hardy's proof of the mean value theorem though. Also a few other points are a bit outdated perhaps (his presentation of complex numbers for example), but nowhere is it inadequate. No text is perfect however, and it's always useful to read more texts than just one.

A good place to begin is Mastering Technical Mathematics, Third Edition by Norman Crowhurst.

dddhgg (he of the G-major chord) and I surely disagree less than he may think. We both love Hardy's Course... and, probably, classical mathematics in general. I too am a fan of Courant's book. But we do differ about auto-pedagogy. Tollorin wanted recommendations for learning calculus on his own. His location is North-American; he's probably been away from math for a while. Courant and Hardy are Europeans of several generations ago: they may have been writing with the tacit assumption that the reader had already made the acquaintance of intuitive and formula-juggling calculus in Gymnasium or some such place.

I feel that the best suggestions are those that have the least chance of generating unnecessary frustration the first time that Tollorin works through the material. The century-and-a-quarter-old Cambridge Tripos problems (and the deltas and epsilons) can come later, if he wants to try them and has confidence based on doing easier problems from North-American textbooks and/or videos. If the Crowhurst recommendation works, that's fine too--but it shouldn't be viewed as the last word on calculus.

BTW, I looked at dddhgg's link to Chrystal's Algebra and got quite a shock: I have held that same (physical) copy in my hands as a student.

jef

Yes, you're right - I'm very fond of those classical math textbooks, like Hardy, Courant, Chrystal, Whittaker & Watson, etc. They're really unsurpassed in many respects, and I still maintain that some of these are suited for beginners. But I do see your point about different levels of exposure to the required prerequisites. As regards formula manipulation skills, education seems to have declined greatly since the early 20th Century. Also, purely geometric reasoning is gone almost completely. I was shocked to find that I had trouble understanding the elementary geometric proof on the first few pages of Whittaker's "Analytical Dynamics" (1917).

Even considering this, however, the standard American late 20th Century Calculus text looks like a bit of a waste of time - working through a telephone book size tome just to get to the point where you can begin reading Hardy's. If someone really has had zero exposure to mathematics, then reading a classic introductory text like "What is Mathematics" (Courant & Robbins) or "An Introduction to Mathematics" (Whitehead) looks like a better bet. My ideal maths self-education reading course looks like this:

1. "Mathematics for the Millions" (Hogben);
2. "What is Mathematics" (Courant & Robbins) and "An Introduction to Mathematics" (Whitehead);
3. Books I to VI of Euclid's Elements and "A Mathematician's Apology" (Hardy);
4. Algebra, vol I (Chrystal);
5. "A Course of Pure Mathematics" (Hardy) or "Differential and Integral Calculus", original edition (Courant);
6. "Concepts of Modern Mathematics" (Stewart) and "Linear Algebra" (many good books)
7. "Vector Calculus" (Marsden & Tromba), or similar; and "Groups and Symmetry" (Armstrong)
8. "Complex Analysis" (Lang)
9. "A Course in Modern Mathematical Physics" (Szekeres);
10. "An Introduction to the Theory of Numbers" (Hardy & Wright).

This reflects my personal preferences of course, and one could substitute other books at any point.

BTW, what a nice coincidence of you having held the copy of Chrystal's Algebra which is now on the internet. I own one myself. It was quite some work to obtain it.