Short Calculus
This is a reprint of "A First Course in Calculus," which has gone through five editions since the early sixties. It covers all the topics traditionally taught in the first-year calculus sequence in a brief and elementary fashion. As sociological and educational conditions have evolved in various ways over the past four decades, it has been found worthwhile to make the original edition available again. The audience consists of those taking the first calculus course, in high school or college. The approach is the one which was successful decades ago, involving clarity, and adjusted to a time when the students'background was not as substantial as it might be. We are now back to those times, so it's time to start over again. There are no epsilon-deltas, but this does not imply that the book is not rigorous. Lang learned this attitude from Emil Artin, around 1950.
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- Serge Lang's "Short Calculus" is clear and concise, efficiently covering the basics in ~170 pages instead of hundreds of pages like some books. It then has a few more chapters going into a bit more depth but on topics that are also essential if you want to take things further
- Learning maths is all about having the proper prerequisites (and time and effort...). The concepts all build on simpler ones in a hierarchy leading down to our basic ideas of numbers and space, so if you're missing any of those simpler ones you simply won't be able to understand anything more advanced or exotic except in a very fragmentary or superficial way.
For differential geometry the prerequisites are linear algebra and multi-variable calculus, and the prerequisite for multi-variable calculus is single variable calculus, and the prerequisites for each of those is basic algebra, trigonometry, and elementary geometry.
You don't need to know everything about each of these to get things at the next level, but a thorough grounding in the basics is essential in my experience. There's a reason every STEM field begins with calculus and linear algebra - they're used everywhere in anything at a higher level. Once you get through those you will find things open up for you.
I don't know your level so it's difficult to make any concrete recommendations, but in general I find Lang's books to be clear and efficient sources. His Short Calculus covers all the basics of single variable calculus in less than 200 pages, instead of ~500 pages like many intro to calc books. Similarly his Calculus of Several Variables is ~300 pages instead of 500-700. Alternatively a mathematical methods for physicists book like the one by Riley, Hobson, and Bence might suit you. It's huge (~1300 pages), but you can pick out the chapters you want to learn from and it builds up from the basics to some quite sophisticated mathematics and has references if you want more depth on some topic, and great problems.
I find I don't really learn anything from watching videos. They can be complementary, but most of the learning with maths comes from doing problems after reading through an introduction to the concepts
- Another book on calculus for beginners I really liked [1].
[1] - https://www.amazon.com/Short-Calculus-Original-Undergraduate...
- >You should forget about all of that high school math and start directly with university math. It's actually easier in that it really starts from the beginning.
So, I think being able to do the problems in high school math is important; it's certainly expected for the college level books I've read. The books for highschoolers are completely useless, as far as I can tell, except as sources of exercises (and khan academy is way better for exercises than any book if you ask me) - My favorite books for high school math are Lang's "Basic Mathematics" and maybe "Short Calculus" which don't seem to be aimed at the high schooler, even though it covers math that the people I wish to associate myself with did in highschool. This might be what you mean about College math starting from the beginning? But my impression is that most college-level books presume you "speak the language" - presume that even if perhaps you can't prove all of high school math, you have memorized the notation and the formulas and know what formulae to apply to what situation.
I compare high school math, as conventionally taught, to the way a semi-literate reads; memorization without understanding. I Personally remain unconvinced that this has no value, though of course, it's clearly a lot less valuable than having real understanding through proofs and derivations. But value or no, most of the college-level math material seems to expect you to have spent some years memorizing and applying without understanding before you get to proofs and the other interesting stuff.
Personally, I've started working through the old memorization-and-drill because I can - though I am still functionally a layabout, I have worlds more willpower and motivation than when I was 16. Also, I've discovered spaced repetition flashcards, and between the two of those things, for the first time in my life, I can actually memorize things like the multiplication tables.
Right now, I'm looking at reading and comparing that with math; I started reading after I was expected to, and it was really quite difficult for me at first. But then I reached a certain level of proficiency, and after that? it became extremely easy, and really the foundation of all my further success. (and I am moderately successful, professionally. I've had a reasonable career as a UNIX SysAdmin/SRE/Programmer, though my competencies lean to the left of that list.)
My current working theory is that mathematics as a language is a similar sort of thing, and you certainly have to memorize the words and symbols... and memorizing the more important phrases will make you a better communicator, even before you gain true understanding of the depth of said phrase.
I guess the other thing is that when I first started learning to read... I would read books well above my level and actually understand very little of them; but I pretty quickly started filling in the holes. Perhaps I ought to utilize that strategy with math; I have plenty of good math books, but I tend to read until I find something I don't understand, then I stop and chew on it (or get help from a friend) - which is obviously way slower than just reading through even if I only understand 20%;
- This is an incredible book. I'm a visual thinker, and while this book lacks all the glossy pages of illustration found in a modern Calculus textbook, the writing style helps develop that visual intuition. In the same vein of concise Calculus books, Serge Lang's Short Calculus is also great (if you need a refresher, or if you're just starting out).
https://www.amazon.com/Short-Calculus-Original-Undergraduate...
- Lang's "Short Calculus" (a reprint of the 1st edition of his calculus textbook) is a welcome change from those 2234th edition doorstop textbooks that make calculus into a boring parade of examples, where the goal of the game seems to be becoming an ace at pattern-matching.
More recent, more idiosyncratic but quite nice is T.W. Körner's Calculus for the Ambitious. I could see a motivated high-schooler learn calculus from this.
