Exercise · ZW-DIFFEOMORPHIC-FLOW-010 · ★★☆ Intermediate
If the field \(v_t\) is identically zero for all \(t\), what happens to the flow \(\varphi_t\)?
Choose one option.
Show answer
- D. \(\varphi_t=\operatorname{id}\) for all \(t\)
Explanation
With \(v_t=0\) the equation gives \(\dot{\varphi}_t(x)=0\), so \(\varphi_t(x)\) stays at \(x\) from \(\varphi_0=\operatorname{id}\). A zero field does not deform space.
Review the concept: Diffeomorphic flow
A diffeomorphic flow \(\varphi_t\) is a family of deformations parametrized by time \(t\), where at each instant the transformation remains invertible and smooth.
\[\{\varphi_t\}_{t\in[0,1]},\quad \varphi_0=\operatorname{id}\]
More practice problems: diffeomorphic flow
- ZW-DIFFEOMORPHIC-FLOW-001 For a point x, the flow equation is read as φ_t(x)=v_t(φ_t(x)). What does it describe?
- ZW-DIFFEOMORPHIC-FLOW-002 What role does φ_0=id play in a flow?
- ZW-DIFFEOMORPHIC-FLOW-003 If two flows have the same final map but travel different paths, what can vary?
- ZW-DIFFEOMORPHIC-FLOW-004 Which hypothesis on v_t helps the flow equation have a smooth, well-determined trajectory?
- ZW-DIFFEOMORPHIC-FLOW-005 Which description is compatible with a diffeomorphic flow \φ_t\?
- ZW-DIFFEOMORPHIC-FLOW-006 What does it mean for φ_t to be a diffeomorphism?
- ZW-DIFFEOMORPHIC-FLOW-007 What is the field v_t in LDDMM?
- ZW-DIFFEOMORPHIC-FLOW-008 What does it mean for the flow to be parametrized by t∈[0,1]?