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I am trying to understand this post about left invariant vector field. I don't understand in the example with the river flow. What is exactly $F$ and $f$ in this example?

Now if the two games coincide then you have that the vector field is invariant i.e. $X_{F(p)}(f)=X_p(f∘F)$

You can think about this with an hypersimple physical example: Consider a river and a thermometer. The river is flowing so a particle of water is moved along the flow. If you're interested on the changing of temperature along the flow of the river, in any given time you can either measure the temperature few istants after the time given measuring the temperature in the new place where the particles will be or you can measure now the temperature of the particles and assign this temperature to the place where you guess they will be given the velocity they have.

roi_saumon
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  • The English is clumsy. But I would say $F$ is the velocity of the river (as a function of position and time) and $f$ is the temperature (as a function of position). – Ted Shifrin Jun 13 '20 at 20:56
  • Isn't the vector field $X$ the velocity? – roi_saumon Jun 14 '20 at 18:33
  • Yes, I was not careful. $F$ is the flow. I'm used to putting a time parameter in there. And, of course, $X$ is the velocity field. My apologies. – Ted Shifrin Jun 14 '20 at 19:04

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