Exercise · ZW-VECTOR-FIELD-009 · ★☆☆ Basic
What does it mean for the field \(v_t\) to be smooth?
Choose one option.
Show answer
- D. Its components vary continuously and differentiably
Explanation
Smoothness means the field is continuous and differentiable, which allows regular flows to be generated. The other statements do not describe that property.
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?