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Both of them are applied mathematics books. What are the main differences between them? Which is more mathematical i.e. mathematically advanced, mathematically rigorous?

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Here is a (legitimate, pre-publication) pdf of the Stone and Goldbart book, which shows that it describes the mathematical toolbox that a physicist would need. It is explicitly targeted at graduate students in physics, I don't think it would be suitable for a math course. Notice also that this book describes the math that appears in physics problems, but it contains hardly any physics applications.

The Princeton companion is an encyclopedic work giving a comprehensive and concise overview of a large variety of topics in physics with an advanced mathematical aspect. It does not seem really suited as the main text for a course in applied math, but it would be a very useful reference for someone who has taken such a course. It is explicitly targeted at mathematicians, although I imagine physicists would find it a valuable reference as well. (I would.)

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