Exercise · ZW-TIME-DERIVATIVE-011 · ★★★ Advanced
If \(v_t(x)=x\) for all \(t\) and \(\varphi_0=\operatorname{id}\), what is \(\varphi_1(x)\)?
Choose one option.
Show answer
- D. \(e\,x\)
Explanation
With \(v_t(x)=x\), the equation becomes \(\dot{\varphi}_t(x)=\varphi_t(x)\), whose solution with \(\varphi_0=\operatorname{id}\) is \(\varphi_t(x)=e^t x\). At \(t=1\) this gives \(e\,x\).
Review the concept: Time derivative
The time derivative \(\dot{\varphi}_t\) describes how a deformation changes as the time parameter \(t\) varies.
\[\dot{\varphi}_t=\frac{d\varphi_t}{dt}\]
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