Math Problem Statement

Show that the open interval (-2, 1) is diffeomorphic to ℝ.

Solution

To demonstrate that the open interval (2,1)(-2, 1) is diffeomorphic to R\mathbb{R}, we need to find a smooth, bijective function f:(2,1)Rf: (-2, 1) \to \mathbb{R} with a smooth inverse f1:R(2,1)f^{-1}: \mathbb{R} \to (-2, 1).

Step-by-Step Solution:

  1. Diffeomorphism Definition:

    • A diffeomorphism is a bijection ff between two manifolds that is smooth (infinitely differentiable), and its inverse f1f^{-1} is also smooth.
  2. Find a Suitable Function:

    • We look for a smooth, bijective function f(x)f(x) that maps (2,1)(-2, 1) onto R\mathbb{R}.
    • A common choice for such a mapping is the following transformation: f(x)=tan(π3(x+2)π2),f(x) = \tan\left(\frac{\pi}{3}(x + 2) - \frac{\pi}{2}\right), where the factor π3(x+2)\frac{\pi}{3}(x + 2) rescales and shifts the interval (2,1)(-2, 1) to (π2,π2)(- \frac{\pi}{2}, \frac{\pi}{2}), the domain of the tangent function.
  3. Verify Smoothness and Bijectivity:

    • The tangent function is smooth on (π2,π2)(- \frac{\pi}{2}, \frac{\pi}{2}) and maps this interval bijectively to R\mathbb{R}.
    • The rescaling and shifting by π3(x+2)\frac{\pi}{3}(x + 2) ensure that the domain of ff is exactly (2,1)(-2, 1).
  4. Inverse Function:

    • The inverse function of f(x)f(x) is given by: f1(y)=1π/3(arctan(y)+π2)2.f^{-1}(y) = \frac{1}{\pi/3} \left(\arctan(y) + \frac{\pi}{2}\right) - 2.
    • This inverse is also smooth because the arctangent function is smooth on R\mathbb{R}.
  5. Conclusion:

    • The function f(x)=tan(π3(x+2)π2)f(x) = \tan\left(\frac{\pi}{3}(x + 2) - \frac{\pi}{2}\right) is a diffeomorphism between (2,1)(-2, 1) and R\mathbb{R}, 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:

  1. How can we verify smoothness of functions rigorously?
  2. Are there other functions that work as diffeomorphisms between open intervals and R\mathbb{R}?
  3. How does the choice of scaling and shifting affect the diffeomorphism?
  4. What are some applications of diffeomorphisms in real-world contexts?
  5. 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