Exercise · ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-007 · ★☆☆ Basic
What does it mean for the Jacobian of \(\varphi\) to be nonzero at every point?
Choose one option.
Show answer
- C. That \(\varphi\) is locally invertible
Explanation
A nonzero Jacobian means the differential is invertible, which gives local invertibility by the inverse function theorem.
Review the concept: Smooth diffeomorphic deformation
A smooth diffeomorphic deformation \(\varphi\) is a bijective, smooth transformation whose inverse is also smooth.
\[\varphi^{-1}\text{ is smooth}\]
More practice problems: smooth diffeomorphic deformation
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- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-003 If f and g are composable diffeomorphisms, why does g∘ f keep that structure?
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- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-005 Which statement necessarily follows from φ being a smooth diffeomorphic deformation?
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-006 What is called a smooth, bijective transformation with a smooth inverse?
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-008 What is the velocity field in LDDMM?
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-009 What does the inverse function theorem guarantee for a smooth map with invertible Jacobian?
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