# Infinitesimal variation formula

I've read in some lecture notes that the (vertical) infinitesimal variation of a field w.r.t a local parameter epsilon can be expressed as $$\delta\phi^a = G^a\epsilon+G^{a,\mu}d_\mu\epsilon\tag{1}$$ for some (epsilon independent) functions $$G^a\hspace{0.2cm}and\hspace{0.2cm} G^{a,\mu} \tag{2}$$

I also know that the said variation can be expressed in terms of the Taylor expansion of the infinitesimal transformed field $$\delta\phi^a = {\phi^a}'(x)-\phi^a(x) = \epsilon\frac{\partial\phi^a}{\partial\epsilon} + \mathcal O(\epsilon^2)\tag{3}$$

My question is what is the formula for the $$G$$ functions? I am especially confused about the second one (how can a term involving the spacetime derivative of the parameter appear in the expansion)?

• Which lecture notes? Which page? Commented Jun 28, 2021 at 18:11
• I can't seem to find them, i think they cited DeWit or something like that. Sorry, it was some time ago. I don't remember the exact formula but i remember it said something about the derivative of the parameter Commented Jun 28, 2021 at 18:23
• I've managed to find these 2 posts that have the same formula physics.stackexchange.com/q/200340 , physics.stackexchange.com/q/66092 , don't know if that helps Commented Jun 30, 2021 at 15:07