Problem 8. Let fg,: I→ R2 be given by f(x) = (0,r), g(x) = (1,2). a. Specify a homotopy h: Ix I→ R2 from f to g. b. Specify a homotopy h: IxI→ R² from g to f.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Problem 8:** Let \( f, g : I \rightarrow \mathbb{R}^2 \) be given by 

\( f(x) = (0, x) \),  
\( g(x) = (1, x) \).

a. Specify a homotopy \( h : I \times I \rightarrow \mathbb{R}^2 \) from \( f \) to \( g \).  

b. Specify a homotopy \( i : I \times I \rightarrow \mathbb{R}^2 \) from \( g \) to \( f \).  

**Explanation:**
- The functions \( f \) and \( g \) represent paths from the initial point on the y-axis to a point along the x-axis.
- The task requires finding a continuous deformation (homotopy) from one path to the other and vice versa.
Transcribed Image Text:**Problem 8:** Let \( f, g : I \rightarrow \mathbb{R}^2 \) be given by \( f(x) = (0, x) \), \( g(x) = (1, x) \). a. Specify a homotopy \( h : I \times I \rightarrow \mathbb{R}^2 \) from \( f \) to \( g \). b. Specify a homotopy \( i : I \times I \rightarrow \mathbb{R}^2 \) from \( g \) to \( f \). **Explanation:** - The functions \( f \) and \( g \) represent paths from the initial point on the y-axis to a point along the x-axis. - The task requires finding a continuous deformation (homotopy) from one path to the other and vice versa.
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