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I'm currently reading these$^1$ lec. notes as an introduction to relativistic QFT. In chapter two (pp.57-61) he introduces the concept of field variations along with some formulas for the different kind of variations (I think at least that is what he is doing...). I'm having a really hard time with the math that he writes down there.

He discusses the example of a Poincare-transformation that sends $x$ to $x'$. He states that we can then write
$$x'^mu approx x^mu + delta x^mu = x^mu+delta omega^{munu}g_{nulambda}x^lambda+delta omega^mu,tag{2.12}$$
where $deltaomega^{munu}=-deltaomega^{numu}, |deltaomega^{munu}|ll 1$, $|deltaomega^mu|ll 1$ and $g_{munu}$ the metric tensor. I can understand the first approxmiation, why $deltaomega^{munu}$ has to be antisymmetric and much smaller than one, but I fail to see why $delta x^mu = delta omega^{munu}g_{nulambda}x^lambda+delta omega^mu$ should make sense as an approximation. He then goes on to say that
$$begin{align*}Delta u(x) &equiv u'(x+delta x)-u(x) equiv delta u(x+delta x) + du(x)\&=delta u(x)+delta x^{mu} partial_{mu} delta u(x)+cdots+mathrm{d} u(x) \&= delta u( x) +delta x^mupartial_mu u(x) + mathcal{O}(delta u delta x),end{align*}tag{2.13}$$
where $u:mathcal{M}tomathbb{K}=mathbb{R}$ or $mathbb{C}$ is a function on the Minkowski space and $u'$ the function after the Poincare-transformation. I honestly don't understand any part of the above calculations (mostly because he doesn't even mention what $delta$ and $d$ are supposed to be) and can even less imagine what $delta u(x)$ is supposed to be.

Can somebody maybe explain to me what exactly is going on here and why assumptions like $[delta,partial_mu]=0$ make sense? If you know of a better introduction to the topic feel free to suggest some books, paper, etc.

Math background: I'm not particularly familiar with the calculus of variations. I had some exposure to it in a classical machanics course, where we defined the variation of a functional ($delta S[f]=frac{d}{d epsilon}left.S[f+epsilon delta f]right|_{epsilon=0}$), but other than that I don't really now anything on the subject matter.

$^1$ Introduction to relativistic quantum field theory, R. Soldati, 2019.

1. The "three kinds of infinitesimal variations" are defined as follows.
   $$begin{align} Delta u(x) &~:=~ u^{prime}(x^{prime})-u(x) qquadtext{ total infinitesimal variation}, tag{2.2}cr delta u(x) &~:=~ u^{prime}(x)-u(x) qquad text{ local/vertical infinitesimal variation},tag{2.3}cr
   mathrm{d}u(x)&~:=~ u(x^{prime})- u(x) qquadtext{ differential/horizontal infinitesimal variation}.tag{2.4}
   end{align}$$
   Here the words horizontal and vertical spaces refer to spacetime and $u$-target space, respectively. While the terminology/names & notation vary from author to author, these three infinitesimal variations are often introduced in physics field theory textbooks.




2. The statement $[delta,partial_mu]=0$ means that vertical infinitesimal variations $delta$ commute with spacetime-derivatives $partial_{mu}equivfrac{partial}{partial x^{mu}}$. Be aware that total infinitesimal variations $Delta$ do not necessarily commute with spacetime-derivatives $partial_{mu}$.

$delta x^mu$ has one upper index, so you just write down all possible terms which give some object with an upper index. Also $delta x^mu$ should only depend on $x^mu$ and constants, because what else should it depend on? So you could come up with the idea to classify the contributions to $delta x^mu$ with respect to the order in $x$.

Zeroth order in $x$ is just a constant. Remember, it should carry an upper index, so it is a constant vector. $delta x^mu$ is infinitesimal, so each contribution should be infinitesimal. So you end up with an infinitesimal constant vector $$(delta x^mu)^{(0)}=delta omega^mu$$ to zeroth order in $x$.

First order in $x$ is something of the form "constant times x". Since $x$ is a vector, the most general form would be contracting it with a constant tensor. The tensor should be infinitesimal (since $x$ itself is not infinitesimal). To get back an object with one upper index, the index structure should be as follows

$$(delta x^mu)^{(1)}=deltaomega^mu_lambda x^lambda$$

We can lift one of the indices using the metric tensor: $deltaomega^mu_lambda=deltaomega^{munu}g_{nulambda}$ and will end up with

$$(delta x^mu)^{(1)}=deltaomega^{munu}g_{nulambda} x^lambda$$

In total, up to first order in $x$ we get exactly (2.12).