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The Exercise and most of the notation is from the book "Introduction to smooth manifolds" by Lee. A smooth isotopy of M is a smooth map $H:M\times J\to M$ where $J\subset\mathbb{R}$ and $H_t:M\to M$ defined by $H_t(p)=H(p,t)$ is a diffeomorphism. The time-dependend flow $\Psi$ of a time-dependend vectorfield $V$ is defined by $$\frac{d}{ds}|_{s=t}\Psi(s,t_0,p)=V(t,\Psi(t,t_0,p))$$.

Now in the exercise 9-21 the claim is $$V(t,p):=\frac{\partial}{\partial t}H(p,t)$$ is a time-dependend vectorfield and its flow is $\Psi(t,t_0,p)=H_t\circ H_{t_0}^{-1}(p)$.

Now my question towards this. We need to show that $V(t,p)\in T_pM$ so that $V$ is a time-dependend vectorfield. But for some fixed $p$, the curve $s\mapsto H(p,s)$ evaluated at $t$ dose not need to be a curve through $p$. I use the notation $\frac{\partial }{\partial s}|_{s=t}f(s)=\frac{\partial }{\partial s}|_{s=t}(f(s))$ for $\frac{\partial}{\partial t}f(t)$. Let $f\in C^{\infty}(M)$, then $$V(t,p)f=\frac{\partial}{\partial s}|_{s=t}(H(p,s))f=\frac{\partial}{\partial s}|_{s=t}(f\circ H(p,s))=\lim_{s \to t}\frac{f(H(p,s))-f(H(p,t))}{s-t}$$ is the differential quotient for $f$ at $H(p,t)$ and $V(t,p)$ would be a tangentvector in $T_{H(t,p)}M$.

Should it be $$V(t,p)=\frac{\partial}{\partial s}|_{s=t} (H(H_t^{-1}(p),s))$$ or equivalent $$V(t,H(p,t))=\frac{\partial}{\partial s}|_{s=t} (H(p,s))$$ That would fit to $\Psi(t,t_0,p)=H_t\circ H_{t_0}^{-1}(p)$. With $$V(t,\Psi(t,t_0,p))=\frac{\partial}{\partial s}|_{s=t}\Psi(s,t_0,p)=\frac{\partial}{\partial s}|_{s=t}H(H_{t_0}^{-1}(p),s)=V(t,H_t(H_{t_0}^{-1}(p))$$

Or do I have a missunderstanding?

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