- #1
Hill
Gold Member
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- TL;DR Summary
- The example from "QFT and SM" by Schwartz
The example goes like this:
The group SO(2) is specified by angles ##\theta##. Let's parametrize a path by ##0 \leq t \lt 1## and consider the path ##\theta (t) = 2 \pi t##. Then it says, "There is no smooth function ##\theta (t,u)## for ##0 \leq u \leq 1##, such that ##\theta (t,0) = \theta (t)## and ##\theta (t,1) = 0##."
Why? What is wrong with ##\theta (t,u) = \theta (t) (1-u)##? What am I missing? I suspect it has something to do with the path being closed, but where does it appear in this example?
The group SO(2) is specified by angles ##\theta##. Let's parametrize a path by ##0 \leq t \lt 1## and consider the path ##\theta (t) = 2 \pi t##. Then it says, "There is no smooth function ##\theta (t,u)## for ##0 \leq u \leq 1##, such that ##\theta (t,0) = \theta (t)## and ##\theta (t,1) = 0##."
Why? What is wrong with ##\theta (t,u) = \theta (t) (1-u)##? What am I missing? I suspect it has something to do with the path being closed, but where does it appear in this example?