Discussion Overview
The discussion revolves around proving whether a specific mapping from R² to R², defined as (x/(x²+y²), y/(x²+y²)), is a diffeomorphism. Participants explore the continuity and differentiability of the mapping, particularly in relation to the origin, and consider potential restrictions to make it a diffeomorphism.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant asserts that the mapping is not defined at the origin, suggesting it cannot be a diffeomorphism on R².
- Another participant proposes that the mapping is smooth away from the origin and hints at the existence of a smooth inverse.
- A participant suggests finding the inverse by expressing u and v in terms of x and y, indicating a method to derive the inverse mapping.
- One participant emphasizes understanding the geometric interpretation of the mapping, relating it to polar coordinates and the transformation of points based on their distance from the origin.
- Another participant notes that the function behaves as its own inverse when excluding the origin.
- A later reply discusses the potential of using polar coordinates to analyze the mapping further, suggesting a method to derive equations for r' and u'.
Areas of Agreement / Disagreement
Participants generally agree that the mapping is not a diffeomorphism on R² due to its undefined nature at the origin. However, there is no consensus on the specific values to restrict it to in order to establish a diffeomorphism, nor on the exact method to prove it.
Contextual Notes
Some participants express uncertainty about the continuity and differentiability of the mapping, particularly in relation to the origin, and the steps required to find the inverse are not fully resolved.