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This is my first time on Physics forums. I am a sophomore in high school who LOVES math. I have lots of free time this summer and would like to learn multivariable calculus and/or linear algebra (whichever is a prerequisite for the other, depending on the textbook I choose). I have two issues that each have a few questions:

I'm currently going through Spivak's Calculus at the moment - it isn't hard but it isn't easy either, which is a very happy medium for me (I feel appropriately challenged). I'm curious if there are good multivariable calculus textbooks that teach in a similar way (known for challenging material and are kind of chatty (not super terse)). Before you redirect me to a different thread, I have read them.

I've read that there are some books that are very good and teach a thing called vector calculus (which I've also read is the standard treatment of multivar). I've also read that there are other textbooks that teach a thing called differential forms which I've read are simpler, more elegant ways of dealing with the standard material, but are slightly unconventional.

Lastly, I WANT rigor but at the same time it would be great if I could get something more than just a theorem-book. I would like a book that discusses applications or at least physical intuition, if possible. Above all, I don't want some formula-book for people who don't care about proofs (nor do I want a book that lands in between theorem-book and engineer's guide to math). Is there some godly tome that combines both pure mathematics and interesting applications of theorems?

To end the rambling, here are my questions:

For someone who wants a mathematical challenge with some applications (if possible), what multivariable calculus text would you recommend? Should I study vector calc or differential forms, or both? If the third, which should I do first? Should I learn linear algebra beforehand, or does this depend on the text? If the text requires some knowledge of linear algebra, could you let me know?

I've attached a poll (although I'm not sure how it works or if it will), so that you can possibly nominate good textbooks and vote on them.

Thank you for your kindness and advice,

Brian