I am finding some of the notation confusing in Chapter 3 on Classical Field Theory in Schwartz' QFT book a bit confusing.
First off, on page 34 he defines a translation of a field to first order as $$\phi(x) \rightarrow \phi(x+\xi) = \phi(x) +\xi^{\nu}\partial_v \phi(x)+...\tag{3.29}$$ So good so far, but then he writes $$\frac{\delta\phi}{\delta \xi^{\nu}} = \partial_v\phi.\tag{3.30}$$ To me that looks like an ordinary partial derivative, so I don't get why he writes it as $\frac{\delta\phi}{\delta \xi^{\nu}}$ instead of $\frac{\partial\phi}{\partial \xi^{\nu}}$.
My second question is that he later writes
$$\frac{\delta\mathscr{L}}{\delta \xi^{\nu}} = \partial_{\nu}\mathscr{L}.\tag{3.31}$$ Since this is a total derivative, $$ \delta S = \int d^4x\delta\mathscr{L}=\xi^{\nu}\int d^4x\partial_{\nu}\mathscr{L}=0.$$
I wouldn't interpret $\partial_{\nu}\mathscr{L}$ as a total derivative. Wouldn't that typically be the total divergence $d_{_\nu}\mathscr{L}$?
He seems to use the term loosely.
