
The classic book on infinitesimal calculus by Thompson, originally published in 1910; then updated with additional material in 1998 by Martin Gardner.
1910 book, but actually does the job well
I did chemical engineering long ago (how long? Computerized process control was just being introduced into the curriculum, and we learned to program in Fortran). I did several calculus courses, but it was always just a matter of memorizing techniques.
Much later, I came across Calculus Made Easy on Project Gutenberg, a textbook from 1914 that actually helped me to understand why calculus works, instead of just treating it like magic.
have been gifting the hardcopy of that book to all my friend's kids who are interested in these kinds of things.
highly recommended.
I have two resources in mind: Calculus Made Easy (available freely on Project Gutenberg) by Thompson, and simply Khan Academy.
Paul Lamar has detailed notes on Calculus, too [0].
[0]: https://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx
1. Sorry, I do not have a suggestion here. I'm Brazilian and I happened to find an amazing PDF of an obscure book (by a not well known professor). In Portuguese though.
2. "Calculus Made Easy" is my book of choice. Don't dismiss it because of the 'cheap' title, it's a very good book.
3. It's super important to get the intuitions first, and not just the math. So I mostly used two books, both heavily illustrated: "Deep Learning - A Visual Approach" and "Deep Learning Illustrated". Those are way better as an introduction to the field than the 'big names' (Goodfellow/Bengio, Bishop, etc).
Normally if I'm on youtube, it means I'm struggling with something. The statquest channel is amazing, it's the most 'baby steps' explanations I've ever seen. If I find a video on what I want to understand there, I don't waste my time trying other channels anymore.
What do you mean by "foundational"?
I love Calculus Made Easy, but in my opinion the reason it's so good is that it doesn't try to be foundational. It relies on quick and dirty tricks -- especially, treating dy/dx as a fraction when it can't be formally defined as such, but where doing so makes the subject much more intuitive and easy. It's not really a good foundation for further study, you need something that doesn't cut corners for that, but it's fantastic for what it is!
My favorite foundational book is Linear Algebra by Jim Hefferon:
https://hefferon.net/linearalgebra/
It is free, and I've seen the author on HN occasionally. It's foundational in the sense that it builds up the subject brick by brick, assuming the minimum realistic prerequisites needed, and it does an outstanding job of this.
But it's a slow read, essentially by design. If you want something quicker, you might try Interactive Linear Algebra by Margalit and Rabinoff:
https://textbooks.math.gatech.edu/ila/
Again free, with lots of widgets to play with.
Some other textbooks I really like:
Epp's Discrete Mathematics is fantastic. The newest edition is astronomically expensive, so go looking for used copies of previous editions.
Carter's Visual Group Theory is tremendous fun. Once again not foundational per se, but if you want to learn what group theory is all about without slogging through lots of formalities, that's a great choice. I'd say that roughly it's in the spirit of Calculus Made Easy.
Also, if you enjoy poker, check out Dan Harrington's books on the subject. In the course of explaining hold'em strategy, they will teach you lots about probability, counting, expected value, and game theory in an applied setting. They're fantastically written.