*To Question 1:* Just a conjugation. You need to observe that if $U:= \Phi(x,t)$ and $\displaystyle V:={\Phi(x + r y,t) - \Phi(x,t)\over r}$ then  $U+rV=\Phi(x + r y,t) $,  and $$\displaystyle\frac{f(U+r V,t) - f(U,t)}{r} = \frac{f(\Phi(x + r y,t),t) - f(\Phi(x,t),t)}{r} $$ 
so  
 
$$\tilde{f}_r(U,V,t) := \left(f(U,t), \frac{f(U+r V,t) - f(U,t)}{r}  \right)$$$$= \left(f(\Phi(x,t),t), \frac{f(\Phi(x + r y,t),t) - f(\Phi(x,t),t)}{r} \right)$$