I once heard someone (perhaps from someone famous -- anyone have a citation?) say that there ought to be a series of books called "__ for Mathematicians," each one of which would explain a different topic or discipline using the tools of mathematics. (The idea is that knowledge of higher mathematics helps to clarify the exposition or simplify issues which would otherwise be inaccessible to beginners.)
Even though this book series doesn't actually exist, there are certainly some books which fall into this category. (I've listed two below.) What are the best books of this type?
- Economics with Calculus by Lovell (an introductory economics textbook which assumes calculus knowledge)
- Mathematical Methods of Classic Mechanics by Arnold (adopts an axiomatic, mathematical approach to classical mechanics)
As you can see, the above examples vary pretty widely in what type of mathematical sophistication they expect from the reader; for me the key fact is that they both significantly alter the normal presentation of material to make it more suitable for readers comfortable with mathematical reasoning.