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Let $v$ be the vector field $\sum_{i=1}^n x_i\dfrac{\partial}{\partial x_i}$, and let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be the projection map $(x_1,\ldots,x_n)\rightarrow x_1$. Show that $v$ and $w$ are $f$-related, where $w=x_1\dfrac{\partial}{\partial x_1}$.

By definition of $f$-related, we must check that for each point $p\in\mathbb{R}^n$, $f$ maps $v(p)$ to $Df(p)\cdot(v(p))$. Write $p=(p_1,\ldots,p_n)$. So $v(p)=(p_1,\ldots,p_n)$ also, and $f(v(p))=p_1$. On the other hand, $Df(p)$ is the $1\times n$ matrix $[1 0 \ldots 0]$. So $Df(p)\cdot v(p)=p_1$ also, and hence $v$ and $w$ are $f$-related.

Verify that $f$ maps integral curves of $v$ onto integral curves of $w$.

An integral curve of $v$ is a map $\gamma:(a,b)\rightarrow U$ such that for all $a<t<b$ and $p=\gamma(t)$, $$\left(p,\dfrac{d\gamma}{dt}(t)\right)=v(p).$$ Since $f$ is a mapping from $\mathbb{R}^n$ to $\mathbb{R}$, I am confused how it can map integral curves onto integral curves.

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Since $w$ lives in $\mathbb R$, so do its integral curves. They are of the form $\gamma(t)=ce^{t}$ for $c\in\mathbb R$. The integral curves of $v$ live in $\mathbb R^n$, they are of the form $\gamma(t)=\zeta e^t$ for $\zeta\in\mathbb R^n$. Projection sends the latter onto $t\mapsto \zeta_1 e^t$.

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