Studying for an exam, and trying to get my head around the concepts of push forwards.
The question I'm attempting to answer is:
"Give an example of a continuously differentiable diffeomorphism F and a continuously differentiable vector field X, such that the push forward of X is continuous but not differentiable." The answer given in the back of the book is:
On the real numbers, take
$F(x) = $ $x^2$ if $x \ge 0$; $−x^2$ if $x \gt 0$,
and $X(x) = 1$.
(Apologies for the awful formatting - I'll edit again in future when I work out how to use MathJaX to get piecewise functions!)
I'd really appreciate if someone could explain why this example satisfies the question.