Exercise · ZW-VECTOR-FIELD-010 · ★★☆ Intermediate
If \(v_t\) is smooth and bounded, what can be guaranteed about \(\varphi_t\)?
Choose one option.
Show answer
- C. That \(\varphi_t\) is a diffeomorphism for each \(t\)
Explanation
A smooth bounded field generates a flow of diffeomorphisms. It need not be an isometry, which would require preserving all distances.
Review the concept: Velocity field
A velocity field \(v_t\) is a vector field that changes with time: at each instant \(t\) it assigns to every point the velocity at which it moves. In LDDMM it guides the smooth motion of the deformation.
\[v_t(x)\]
More practice problems: velocity field
- ZW-VECTOR-FIELD-001 In φ_t=v_t∘φ_t, what object does v_t assign to each position?
- ZW-VECTOR-FIELD-002 Why is it not equivalent to replace v_t(φ_t(x)) by v_t(x) throughout the flow?
- ZW-VECTOR-FIELD-003 If v_t depends on t, what does it mean?
- ZW-VECTOR-FIELD-004 A smooth vector field can generate a flow, but that does not mean the flow preserves all distances.…
- ZW-VECTOR-FIELD-005 Select the only correct statement about the field v_t.
- ZW-VECTOR-FIELD-006 What does a vector field v_t assign to each point x of the space?
- ZW-VECTOR-FIELD-007 In the equation φ_t=v_t∘φ_t, what does φ_t represent?
- ZW-VECTOR-FIELD-008 What is the velocity at time t of a point following the flow φ_t from x?