Advanced Calculus: A Differential Forms Approach

Since you mention analysis, I recommend Yet Another Introduction to Analysis and Metric Spaces by Victor Bryant. They should be at the right level and a lot of fun. Another analysis suggestion is Creative Mathematics by H.S. Wall. It is a book that walks a high school level student through creating the proofs themselves. The topic covered is a stripped down version of analysis, calculus, and later even differential geometry. It's really brilliant. Going slow and having fun when you're a teen would be much more productive than going fast and burning out. The Spivak book is a good suggestion and might be a little difficult depending on their actual background. The books by Gelfand mentioned by someone else (there's actually a series of them that cover algebra, functions, coordinates, trigonometry, etc.) would help provide the needed background. The book Conceptual Mathematics is claimed to be aimed at high school students. Maybe give it a whirl and see what happens. If they know calculus, then Advanced Calculus: A Differential Forms Approach by Harold Edwards is a gem. The first three chapters should be readable, as they give heuristic discussions of the topic.

1. La peste (Albert Camus) - entertaining if you're into French existentialism 2. Linked (Barabasi) - insightful and useful to understand that most phenomenon we see in real life follow a power law and what that means 3. Leonardo da Vinci (Walter Isaacson) - surprising how little of da Vinci's real personality is reflected in his modern image 4. Matrix computations – third edition (Gene Golub, Charles van Loan) - good read if you're into linear algebra 5. Giving effective feedback (Harvard business review) - useful for getting an understanding on how to handle human interaction 6. The old man and the sea (Ernest Hemingway) - enjoyed it a lot, a reflection of Hemingway's romantic spirit. Also, a quick read. 7. Screwjack (Hunter S. Thompson) - didn't resonate with me although I am a fan of existentialism 8. Sun and steel (Yukio Mishima) - raw and intimate. before reading this book I'd recommend reading up on Mishima's life 9. Stranger than fiction (Chuck Palahniuck) - somewhat entertaining, more so because it served as a glimpse into Palahniuck's creative process…