Calculus, Volume 1

Supposing you have at least high school mathematics under the belt: 1. Book of Proof by Hammack [1] 2. Tom Apostol Calculus Vol 1 (stop at Linear Algebra) 3. Linear Algebra by Insel, Spence & Friedberg 4. Understanding Analysis by Stephen Abbott 5. Tom Apostol Calculus Vol 2 These are books with excellent exposition, and solution manuals are available when needed. You are absolutely required to do all the exercises. To learn mathematics is to do mathematics. Two other excellent books are Spivak - Calculus, the problems in this book are very good, but much more difficult than Apostol. For more advanced analysis there's Walter Rudin's Principles of Mathematical Analysis (baby Rudin), which on its own is too difficult as a first exposure to analysis, but there's a cool professor that recorded an entire semester of baby Rudin lecture videos [2]. This is as close as it gets without being enrolled in a university. [1] https://www.people.vcu.edu/~rhammack/BookOfProof/ [2] https://www.youtube.com/watch?v=ab41LEw9oiI&list=PLun8-Z_lTk... (https://www.youtube.com/watch?v=ab41LEw9oiI&list=PLun8-Z_lTkC6qJF1sVh3_Hx7aL6FPd0IN) (the first video is a bit blurry, it gets better)

Apostol Calculus Vol 1 The introduction gives a history lesson, a great description of the method of exhaustion with descriptive drawings, and some very straightforward proofs on finding the area under the curve for parabolic segments via some basic infinite sums (no limits), all in 10 pages in a conversational style. And it just gets better from there. My dad gave me his book he used in the 70s and I read it in high school in parallel to the textbook the school issued. Everyone thought I was some genius because I thought calculus was so obvious and I could explain it so well but I was just parroting the text and proofs from this book, basically verbatum. I told the other kids to use it as well but no one did.