When expanding a function (for example the determinant of the space-time metric g) as a functional of a perturbation from the flat metric ##h_{\mu \nu}##, i.e. ##g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} ## i would think that the thing to do is to recognize that ##g_{\mu \nu}## and thus also ##h_{\mu \nu}## is itself a function of x and thus ##g(g_{\mu \nu})## is a functional of ##g_{\mu \nu}## and therefore we can expand the functional Taylor series(adsbygoogle = window.adsbygoogle || []).push({});

$$ g(g_{\mu \nu}) = g(\eta_{\mu \nu}) + \int d^4 x \frac{\delta g}{\delta g_{\mu \nu}(x)}|_{h_{\mu \nu} =0} \delta g_{\mu \nu} + \ldots $$

However I've regularly encountered people seeming to treat ##g_{\mu \nu}## as a regular variable and thus expanding

$$ g(g_{\mu \nu})= g(\eta_{\mu \nu}) + \frac{\partial g}{\partial g_{\mu \nu}}|_{h_{\mu \nu} =0} \delta g_{\mu \nu} + \ldots $$

which also yields the correct answer. So I wondered is there something fundamentally wrong with the latter approach? I.e. what is the reason (if there is one) that one must thing of g as a functional and expand accordingly instead of thinking of it as a regular function?

(I realize that this is perhaps a question more suited to the mathematics forum, but I suspect those who knows a little about the mathematics of field theory can answer perhaps even better from a physicists point of view than those coming from a formal mathematics background.)

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# Functional or regular (partial) taylor series in Field theory

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