Is the Transition Map Smooth in the Intersecting Set?

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SUMMARY

The discussion focuses on determining the smoothness of the transition map in the intersecting set as depicted in the provided image link. Two methods are proposed for this verification: (1) demonstrating that the function Φ is a composition of two smooth functions, and (2) computing the composite function directly to prove its smoothness. The participant has partially explored the first method and is seeking advice on whether the second method may be more straightforward.

PREREQUISITES
  • Understanding of smooth functions in calculus
  • Familiarity with function composition
  • Basic knowledge of transition maps in topology
  • Ability to compute composite functions
NEXT STEPS
  • Research the properties of smooth functions in calculus
  • Explore function composition techniques in advanced mathematics
  • Study transition maps and their applications in topology
  • Learn methods for proving the smoothness of composite functions
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Mathematics students, particularly those studying calculus and topology, as well as researchers interested in smooth functions and transition maps.

sk1001
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Smooth transition map (easy!?)

Homework Statement


Check the transition map
http://img132.imageshack.us/img132/4341/18142532.png
is smooth in the set for which their images intersect

The Attempt at a Solution


I have thought of two ways to show this.

(1) Show that Φ is a composition of two smooth functions and is therefore smooth.
(2) compute the composite function and then prove that is smooth

Which way do you suggest?
I have attempted method (1) to some extent, but wondering if method (2) is easier.
 
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bump please
 

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