Exercise · ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-009 · ★☆☆ Basic
What does the inverse function theorem guarantee for a smooth map with invertible Jacobian?
Choose one option.
Show answer
- A. That the map is locally a diffeomorphism
Explanation
The inverse function theorem ensures that a smooth map with invertible differential is locally a diffeomorphism.
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
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-001 A map φ:M→ N is a candidate to be a diffeomorphism, but its inverse turns out to be only continuous…
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-002 During a registration, the Jacobian of φ vanishes in a region and folds appear. Which property is c…
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-003 If f and g are composable diffeomorphisms, why does g∘ f keep that structure?
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-004 Two deformations reach the same reference points, but one squashes a region until it forms a fold.…
- 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-007 What does it mean for the Jacobian of φ to be nonzero at every point?
- ZW-SMOOTH-DIFFEOMORPHIC-DEFORMATION-008 What is the velocity field in LDDMM?
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