How to show a mapping is a diffeomorphism

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To show a mapping is a diffeomorphism, it's essential to define terms clearly and demonstrate your own attempts at solving the problem. For complex questions, particularly those involving topology and differential geometry, the "calculus and beyond" section is appropriate unless the material is graduate-level. It's important to avoid editing posts after receiving responses, as this can confuse the discussion. Acknowledging minor edits with explanations is acceptable. Clear communication and structured problem-solving approaches are key to effective assistance in mathematical forums.
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Hint: What is |f(x,y)|?
 
Uh, that was a strange way to edit your post. Does it mean that you have solved the problem now?

A few tips for the next time: When you need help with textbook-style questions, you should usually put them in the "calculus and beyond" section of the homework forum. The exception is if the topic is too advanced, e.g. if it's a difficult problem from a course for graduate students of mathematics. The basics of topology and differential geometry is not too advanced in my opinion. Many of the homework helpers will be able to assist you with that. You are also expected to define your terms, and to show us your own attempt to solve the problem, up to the point where you got stuck. If you have no idea where to begin, then it's OK to say that, and we'll give you a hint about where to begin. Also, don't change your post after you have received replies to it. Minor changes are OK, if you acknowledge that by adding an explanatory text, like "I fixed a typo in this equation after X's reply below".
 

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