How to show that for every vector field $v$ there exists a constant $M_{ > 0}$ such that $||v(x)|| \leq M$ $\space \forall x \in \gamma([a,b])$?

Question

Given is that the length $L(\gamma)$ of a $C^1$-curve $\gamma : [a, b] \longrightarrow \mathbb{R^n}$ is defined as:

$L(\gamma) = \int^b_a||\gamma'(t)||dt$

Now I need to show that for every continuous vector field $v$ defined on $\gamma([a, b])$ there exists a constant $M > 0$ such that $||v(x)|| \leq M$ $\space \forall x \in \gamma([a,b])$.

I've been stuck on this question for a while and I'm not sure how to show this correctly. Thank you very much in advance for your time and help!

score 1 · Accepted Answer · answered Oct 27 '21 at 15:19

In this case vector field is a continous map $v:\gamma ([a,b] ) \to \mathbb{R^m} $ and since $\gamma ([a,b] )$ is a compact space therefore $v$ is a bounded hence $$||v(x)||\leq M \hspace{0.5cm}\forall x\in \gamma ([a,b] )$$

How to show that for every vector field $v$ there exists a constant $M_{ > 0}$ such that $||v(x)|| \leq M$ $\space \forall x \in \gamma([a,b])$?

1 Answers1