Math Problem Statement
Show that the open interval (-2, 1) is diffeomorphic to ℝ.
Solution
To demonstrate that the open interval is diffeomorphic to , we need to find a smooth, bijective function with a smooth inverse .
Step-by-Step Solution:
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Diffeomorphism Definition:
- A diffeomorphism is a bijection between two manifolds that is smooth (infinitely differentiable), and its inverse is also smooth.
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Find a Suitable Function:
- We look for a smooth, bijective function that maps onto .
- A common choice for such a mapping is the following transformation: where the factor rescales and shifts the interval to , the domain of the tangent function.
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Verify Smoothness and Bijectivity:
- The tangent function is smooth on and maps this interval bijectively to .
- The rescaling and shifting by ensure that the domain of is exactly .
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Inverse Function:
- The inverse function of is given by:
- This inverse is also smooth because the arctangent function is smooth on .
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Conclusion:
- The function is a diffeomorphism between and , as it is bijective, smooth, and its inverse is smooth.
Would you like further clarifications or details on this? Here are some additional questions that expand on the topic:
- How can we verify smoothness of functions rigorously?
- Are there other functions that work as diffeomorphisms between open intervals and ?
- How does the choice of scaling and shifting affect the diffeomorphism?
- What are some applications of diffeomorphisms in real-world contexts?
- How does this relate to general manifold theory?
Tip: Always verify both the function and its inverse for smoothness when proving diffeomorphism!
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Math Problem Analysis
Mathematical Concepts
Diffeomorphism
Smooth Functions
Bijective Mappings
Formulas
f(x) = tan((π/3)(x + 2) - π/2)
f^{-1}(y) = (1/(π/3)) * (arctan(y) + π/2) - 2
Theorems
Diffeomorphism Definition
Suitable Grade Level
Undergraduate Mathematics