Let v_t(x)=2x on ℝ with φ_0=id. What is φ_1∘φ_1/2^-1(3)?

Exercise · ZW-DIFFEOMORPHIC-FLOW-011 · ★★★ Advanced

Let \(v_t(x)=2x\) on \(\mathbb{R}\) with \(\varphi_0=\operatorname{id}\). What is \(\varphi_1\circ\varphi_{1/2}^{-1}(3)\)?

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C. \(3e\)

Explanation

With \(v_t(x)=2x\) the flow is \(\varphi_t(x)=e^{2t}x\), so \(\varphi_{1/2}^{-1}(3)=3/e\) and \(\varphi_1(3/e)=3e\). Applying \(\varphi_1\) without first undoing \(\varphi_{1/2}\) gives a different value.

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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.

\[\{\varphi_t\}_{t\in[0,1]},\quad \varphi_0=\operatorname{id}\]

Let \(v_t(x)=2x\) on \(\mathbb{R}\) with \(\varphi_0=\operatorname{id}\). What is \(\varphi_1\circ\varphi_{1/2}^{-1}(3)\)?

Explanation

Review the concept: Diffeomorphic flow

More practice problems: diffeomorphic flow