I've some question about the theorem that Critical points preserved under diffeomorphism.
I have read Adriot Robot proof for that fact given in https://adroitrobot.wordpress.com/2010/10/07/critical-points-preserved-by-diffeomorphism/
I didn't get all notations there, but I'll denote the step where I cannot continue from:
Lets assume that $\phi$ is our diffeomorphism and f is given function which has local min at $\underline{0}$. Now assume that $ \phi(\underline{a}) =\underline{0} $.
( $ f \circ \phi (\underline{a}))'$ =$f'(\phi(\underline{a}) \phi(\underline{a}) $=0
By the formula of derevative of multiplication, we get that $ (f \circ \phi )''(\underline{a})$=$f'' \phi' (\underline{a}) $ since the second term including f gradient at the critical point.
Now , assuming that $f''$ is positive definite, how can we know that $ (f \circ \phi )''(\underline{a})$ is positive definite as well?