Integrals over chained functions

[SchroedingersLion]

Good evening!

Going through a bunch of calculations in Ashcroft's and Mermin's Solid State Physics, I have come across either an error on their part or a missunderstanding on my part.

Suppose we have a concatenated function, say the fermi function $f(\epsilon)$ that goes from R to R. We know that $\epsilon$ is a function of the 3 dimensional wave vector k . If there are applied electromagnetic fields, the wave vector depends on time t', i.e. k(t').

Now suppose we want to calculate an integral $$\int_{-\infty}^t p(t')\frac {df}{dt'} \, dt' $$ with some scalar function $p(t')$.
Chain rule tells us that $\frac {df}{dt'} = \frac {df}{d\epsilon} \frac {d\epsilon}{d\mathbf k}\frac {d\mathbf k}{dt'}$, so we get $$ \int_{-\infty}^t p(t')\frac {df}{d\epsilon} \frac {d\epsilon}{d\mathbf k}\frac {d\mathbf k}{dt'} \, dt' $$

So, assuming that $\frac {df}{d\epsilon}$ would still depend on $\epsilon$ and $\frac {d\epsilon}{d\mathbf k}$ would still depend on $\mathbf k$, am I right in assuming that both of these terms would also be considered as being dependent on t'?
Because $\epsilon$ depends on $\mathbf k$ and the latter on t'. So, under the integral, I would express $\epsilon$ and $\mathbf k$ in terms of t', before I integrate.

Yet, the book keeps acting as if the time dependence only comes from $p(t)$ and $\frac {d\mathbf k}{dt'}$.SL

 

[Abdullah Almosalami]

Wow I'm surprised this question hasn't gotten any answers at all.

Anyways, I'm not too knowledgeable but your reasoning makes total sense to me, and I want to ask if you can elaborate on what you mean by the book insisting that the "time dependence only comes from $p(t)$ and $\frac {d\vec k} {dt'}$"? Do you mean the integrand's time dependence?