Hi I'm just wondering what steps Evans takes in his PDE book on page 50.

He defines

$\displaystyle u(x,t)=\int_0^t \int_{\mathbb{R}^n}\Phi(y,s)f(x-y,t-s)\,dy\,ds$

And then goes on to say:

$\displaystyle u_t(x,t)=\int_0^t\int_{\mathbb{R}^n}\Phi(y,s)f_t(x-y,t-s)\,dy\,ds + \int_{\mathbb{R}^n}\Phi(y,t)f(x-y,0)\,dy$.

I do not know how he got here:

I think I can get this far:

$\displaystyle \frac{u(x,t+h)-u(x,t)}{h} = \frac{1}{h}\left(\int_0^t \int_{\mathbb{R}^n}\Phi(y,s)[f(x-y,t+h-s)-f(x-y,t-s)]\,dy\,ds+\int_t^{t+h}\int_{\mathbb{R}^n}\Phi(y,s)f (x-y,t+h-s)\,dy\,ds\right)$

Taking the limit as $\displaystyle h\rightarrow 0$, I can kind of see how the left term on the RHS can go to $\displaystyle \Phi(y,s)f_t(x-y,t-s)$, but even this I'm not sure of, I don't really understand, I can't really see why it shouldn't it be $\displaystyle f_{t-s}(x-y,t-s)$ since isn't this the 'variables' that is having an infinitesimally small h added on to? I have no idea how the right term of the RHS of the above equation transforms as well. I think the thing that is most troubling me is that the variable t is in the integral and the integrand and what we are taking a limit of, h, is in the integral... and in the integrand once we take the 1/h back inside!

Can someone please help?! Thanks!